├── Makefile ├── .gitmodules ├── LICENSE ├── docs ├── poly.pdf.html └── index.html ├── README ├── .gitignore ├── unicode.sty └── refs.bib /Makefile: -------------------------------------------------------------------------------- 1 | poly.pdf: FORCE 2 | latexmk -pdf -halt-on-error -file-line-error poly.tex 3 | 4 | FORCE: 5 | -------------------------------------------------------------------------------- /.gitmodules: -------------------------------------------------------------------------------- 1 | [submodule "isogenies_bib"] 2 | path = isogenies_bib 3 | url = git@github.com:defeo/isogenies.bib.git 4 | [submodule "cryptobib"] 5 | path = cryptobib 6 | url = https://github.com/cryptobib/export 7 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | This work is licensed under the Creative Commons Attribution-NonCommercial 2 | 4.0 International License. To view a copy of this license, visit 3 | . 4 | -------------------------------------------------------------------------------- /docs/poly.pdf.html: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | Mathematics of Isogeny Based Cryptography 6 | 7 | 8 | 9 |

10 | Download here: https://arxiv.org/pdf/1711.04062 11 |

12 | 13 | 14 | -------------------------------------------------------------------------------- /README: -------------------------------------------------------------------------------- 1 | # Isogeny Graphs in Cryptography 2 | 3 | These lectures notes were written for the summer school on *Graph 4 | Theory Meets Cryptography* in Würzburg, Germany, in July 2019. Their 5 | goal is to survey the mathematical background of isogeny-based 6 | cryptography, a blossoming new field with applications to post-quantum 7 | cryptography and blockchains. They are by no means a reference text 8 | on the theory of elliptic curves, nor on cryptography; readers are 9 | encouraged to complement these notes with the references given in the 10 | bibliography. 11 | -------------------------------------------------------------------------------- /docs/index.html: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | EMA 2017 – Isogeny-based cryptography 6 | 11 | 12 | 13 |

Isogeny Based Cryptography

14 |

15 | Luca De Feo
16 | Université de Versailles & Inria Saclay 17 |

18 | 19 |

20 | École mathématique 21 | africaine
22 | May 10 – 23, 2017, Thiès, Senegal 23 |

24 | 25 |

Télecharger le polycopié

26 | 27 | 28 | 32 | 33 | 34 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | 14 | ## Intermediate documents: 15 | *.dvi 16 | *-converted-to.* 17 | # these rules might exclude image files for figures etc. 18 | # *.ps 19 | # *.eps 20 | # *.pdf 21 | 22 | ## Generated if empty string is given at "Please type another file name for output:" 23 | poly.pdf 24 | 25 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 26 | *.bbl 27 | *.bcf 28 | *.blg 29 | *-blx.aux 30 | *-blx.bib 31 | *.run.xml 32 | 33 | ## Build tool auxiliary files: 34 | *.fdb_latexmk 35 | *.synctex 36 | *.synctex(busy) 37 | *.synctex.gz 38 | *.synctex.gz(busy) 39 | *.pdfsync 40 | 41 | ## Auxiliary and intermediate files from other packages: 42 | # algorithms 43 | *.alg 44 | *.loa 45 | 46 | # achemso 47 | acs-*.bib 48 | 49 | # amsthm 50 | *.thm 51 | 52 | # beamer 53 | *.nav 54 | *.pre 55 | *.snm 56 | *.vrb 57 | 58 | # changes 59 | *.soc 60 | 61 | # cprotect 62 | *.cpt 63 | 64 | # elsarticle (documentclass of Elsevier journals) 65 | *.spl 66 | 67 | # endnotes 68 | *.ent 69 | 70 | # fixme 71 | *.lox 72 | 73 | # feynmf/feynmp 74 | *.mf 75 | *.mp 76 | *.t[1-9] 77 | *.t[1-9][0-9] 78 | *.tfm 79 | 80 | #(r)(e)ledmac/(r)(e)ledpar 81 | *.end 82 | *.?end 83 | *.[1-9] 84 | *.[1-9][0-9] 85 | *.[1-9][0-9][0-9] 86 | *.[1-9]R 87 | *.[1-9][0-9]R 88 | *.[1-9][0-9][0-9]R 89 | *.eledsec[1-9] 90 | *.eledsec[1-9]R 91 | *.eledsec[1-9][0-9] 92 | *.eledsec[1-9][0-9]R 93 | *.eledsec[1-9][0-9][0-9] 94 | *.eledsec[1-9][0-9][0-9]R 95 | 96 | # glossaries 97 | *.acn 98 | *.acr 99 | *.glg 100 | *.glo 101 | *.gls 102 | *.glsdefs 103 | 104 | # gnuplottex 105 | *-gnuplottex-* 106 | 107 | # gregoriotex 108 | *.gaux 109 | *.gtex 110 | 111 | # hyperref 112 | *.brf 113 | 114 | # knitr 115 | *-concordance.tex 116 | # TODO Comment the next line if you want to keep your tikz graphics files 117 | *.tikz 118 | *-tikzDictionary 119 | 120 | # listings 121 | *.lol 122 | 123 | # makeidx 124 | *.idx 125 | *.ilg 126 | *.ind 127 | *.ist 128 | 129 | # minitoc 130 | *.maf 131 | *.mlf 132 | *.mlt 133 | *.mtc[0-9]* 134 | *.slf[0-9]* 135 | *.slt[0-9]* 136 | *.stc[0-9]* 137 | 138 | # minted 139 | _minted* 140 | *.pyg 141 | 142 | # morewrites 143 | *.mw 144 | 145 | # nomencl 146 | *.nlo 147 | 148 | # pax 149 | *.pax 150 | 151 | # pdfpcnotes 152 | *.pdfpc 153 | 154 | # sagetex 155 | *.sagetex.sage 156 | *.sagetex.py 157 | *.sagetex.scmd 158 | 159 | # scrwfile 160 | *.wrt 161 | 162 | # sympy 163 | *.sout 164 | *.sympy 165 | sympy-plots-for-*.tex/ 166 | 167 | # pdfcomment 168 | *.upa 169 | *.upb 170 | 171 | # pythontex 172 | *.pytxcode 173 | pythontex-files-*/ 174 | 175 | # thmtools 176 | *.loe 177 | 178 | # TikZ & PGF 179 | *.dpth 180 | *.md5 181 | *.auxlock 182 | 183 | # todonotes 184 | *.tdo 185 | 186 | # easy-todo 187 | *.lod 188 | 189 | # xindy 190 | *.xdy 191 | 192 | # xypic precompiled matrices 193 | *.xyc 194 | 195 | # endfloat 196 | *.ttt 197 | *.fff 198 | 199 | # Latexian 200 | TSWLatexianTemp* 201 | 202 | ## Editors: 203 | # WinEdt 204 | *.bak 205 | *.sav 206 | *~ 207 | _region_* 208 | 209 | # Texpad 210 | .texpadtmp 211 | 212 | # Kile 213 | *.backup 214 | 215 | # KBibTeX 216 | *~[0-9]* 217 | 218 | # auto folder when using emacs and auctex 219 | /auto/* 220 | 221 | # expex forward references with \gathertags 222 | *-tags.tex 223 | -------------------------------------------------------------------------------- /unicode.sty: -------------------------------------------------------------------------------- 1 | 2 | % Preamble for Unicode characters. 3 | \RequirePackage{amsmath} 4 | \ifx\directlua\undefined 5 | \usepackage[T1]{fontenc} 6 | \usepackage[utf8]{inputenc} 7 | \else 8 | \usepackage{fontspec} 9 | % unicode-math bugue ici 10 | \def\DeclareUnicodeCharacter#1#2{% 11 | \def\tmp{#2}\uccode`\~="#1 \catcode"#1 \active 12 | \uppercase{\global\let~\tmp}% 13 | \uccode`\~=0} 14 | \fi 15 | \newcommand{\@ifdisplay}[2]{\mathchoice{#1}{#2}{#2}{#2}}% 16 | % Extensible arrows%<<< 17 | \def\mapstofill@{\arrowfill@{\mapstochar\relbar}\relbar\rightarrow} 18 | \newcommand{\xmapsto}[2][]{\ext@arrow 0399\mapstofill@{#1}{#2}} 19 | \def\mapsfromfill@{\arrowfill@\leftarrow\relbar{\relbar\mapsfromchar}} 20 | \newcommand{\xmapsfrom}[2][]{\ext@arrow 0399\mapsfromfill@{#1}{#2}} 21 | \def\longhookrightarrow{\DOTSB\lhook\protect\relbar\protect\joinrel\rightarrow} 22 | \def\hookrightarrowfill@{\arrowfill@{\lhook\mkern 3mu}\relbar\rightarrow} 23 | \newcommand{\xhookrightarrow}[2][]{\ext@arrow 0399\hookrightarrowfill@{#1}{#2}} 24 | %>>> 25 | % Automatic arrows%<<< 26 | \def\@matharrow#1#2#3{% 27 | \let\@matharr@short@#1 \let\@matharr@long@#2 \let\@matharr@x@#3 28 | \let\@matharr@up@\relax \let\@matharr@down@\relax 29 | \@matharr@step} 30 | \def\@matharr@step{\futurelet\@matharr@what \@matharr@} 31 | \def\@matharr@{\let\next\@matharr@do 32 | \ifx ^\@matharr@what\let\next\@matharr@up \fi 33 | \ifx _\@matharr@what\let\next\@matharr@down \fi 34 | \next} 35 | \def\@matharr@up^#1{\def\@matharr@up@{#1}\@matharr@step} 36 | \def\@matharr@down_#1{\def\@matharr@down@{#1}\@matharr@step} 37 | \def\@matharr@do{% 38 | \let\@matharr@do@\@matharr@x 39 | \ifx\@matharr@up@\relax \ifx\@matharr@down@\relax 40 | % Make an unlabeled arrow. Make it long if in display mode, short 41 | % otherwise. 42 | \def\@matharr@do@{\@ifdisplay{\@matharr@long@}{\@matharr@short@}}% 43 | \fi \fi 44 | \@matharr@do@} 45 | % Make a labeled extensible arrow. 46 | \def\@matharr@x{% 47 | \@ifdisplay 48 | {\@matharr@x@[\mkern 8mu\@matharr@down@\mkern 8mu]% 49 | {\mkern 8mu\@matharr@up@\mkern 8mu}}% 50 | {\@matharr@x@[\mkern 12mu\@matharr@down@\mkern 12mu]% 51 | {\mkern 12mu\@matharr@up@\mkern 12mu}}% 52 | } 53 | 54 | \def\autorightarrow{\@matharrow\rightarrow\longrightarrow\xrightarrow}% 55 | \def\autoleftarrow{\@matharrow\leftarrow\longleftarrow\xleftarrow}% 56 | \def\automapsto{\@matharrow\mapsto\longmapsto\xmapsto}% 57 | \def\automapsfrom{\@matharrow\mapsfrom\longmapsfrom\xmapsfrom}% 58 | \def\autohookrightarrow{\@matharrow\hookrightarrow\longhookrightarrow 59 | \xhookrightarrow} 60 | %>>> 61 | % Capital Greek letters%<<< 62 | \def\Alpha{A} 63 | \def\Beta{B} 64 | \def\Epsilon{E} 65 | \def\Zeta{Z} 66 | \def\Eta{H} 67 | \def\Iota{I} 68 | \def\Kappa{K} 69 | \def\Mu{M} 70 | \def\Nu{N} 71 | \def\Omicron{O} 72 | \def\Rho{P} 73 | \def\Tau{T} 74 | \def\Chi{X} 75 | \def\omicron{o} 76 | %%>>> 77 | % This is a generated file, do not edit ! 78 | % 00A0   NO-BREAK SPACE 79 | \DeclareUnicodeCharacter{00A0}{~} 80 | % 00A1 ¡ INVERTED EXCLAMATION MARK 81 | \DeclareUnicodeCharacter{00A1}{\textexclamdown} 82 | % 00A2 ¢ CENT SIGN 83 | \DeclareUnicodeCharacter{00A2}{\textcent} 84 | % 00A3 £ POUND SIGN 85 | \DeclareUnicodeCharacter{00A3}{\pounds} 86 | % 00A4 ¤ CURRENCY SIGN 87 | \DeclareUnicodeCharacter{00A4}{\textcurrency} 88 | % 00A5 ¥ YEN SIGN 89 | \DeclareUnicodeCharacter{00A5}{\textyen} 90 | % 00A6 ¦ BROKEN BAR 91 | \DeclareUnicodeCharacter{00A6}{\textbrokenbar} 92 | % 00A7 § SECTION SIGN 93 | \DeclareUnicodeCharacter{00A7}{{\mathhexbox 278}} 94 | % 00A8 ¨ DIAERESIS 95 | \DeclareUnicodeCharacter{00A8}{\"{ }} 96 | % 00A9 © COPYRIGHT SIGN 97 | \DeclareUnicodeCharacter{00A9}{\copyright} 98 | % 00AA ª FEMININE ORDINAL INDICATOR 99 | \DeclareUnicodeCharacter{00AA}{\textordfeminine} 100 | % 00AB « LEFT-POINTING DOUBLE ANGLE QUOTATION MARK 101 | \DeclareUnicodeCharacter{00AB}{\guillemotleft} 102 | % 00AC ¬ NOT SIGN 103 | \DeclareUnicodeCharacter{00AC}{\neg} 104 | % 00AE ® REGISTERED SIGN 105 | \DeclareUnicodeCharacter{00AE}{\textregistered} 106 | % 00AF ¯ MACRON 107 | \DeclareUnicodeCharacter{00AF}{\textasciimacron­} 108 | % 00B0 ° DEGREE SIGN 109 | \DeclareUnicodeCharacter{00B0}{\textsuperscript{o}} 110 | % 00B1 ± PLUS-MINUS SIGN 111 | \DeclareUnicodeCharacter{00B1}{\pm} 112 | % 00B2 ² SUPERSCRIPT TWO 113 | \DeclareUnicodeCharacter{00B2}{\textsuperscript{2}} 114 | % 00B3 ³ SUPERSCRIPT THREE 115 | \DeclareUnicodeCharacter{00B3}{\textsuperscript{3}} 116 | % 00B5 µ MICRO SIGN 117 | \DeclareUnicodeCharacter{00B5}{\textmu} 118 | % 00B6 ¶ PILCROW SIGN 119 | \DeclareUnicodeCharacter{00B6}{{\mathhexbox 27B}} 120 | % 00B7 · MIDDLE DOT 121 | \DeclareUnicodeCharacter{00B7}{\cdot} 122 | % 00B9 ¹ SUPERSCRIPT ONE 123 | \DeclareUnicodeCharacter{00B9}{\textsuperscript{1}} 124 | % 00BA º MASCULINE ORDINAL INDICATOR 125 | \DeclareUnicodeCharacter{00BA}{\textordmasculine} 126 | % 00BB » RIGHT-POINTING DOUBLE ANGLE QUOTATION MARK 127 | \DeclareUnicodeCharacter{00BB}{\guillemotright} 128 | % 00BC ¼ VULGAR FRACTION ONE QUARTER 129 | \DeclareUnicodeCharacter{00BC}{\ensuremath{\sfrac{1}{4}}} 130 | % 00BD ½ VULGAR FRACTION ONE HALF 131 | \DeclareUnicodeCharacter{00BD}{\ensuremath{\sfrac{1}{2}}} 132 | % 00BE ¾ VULGAR FRACTION THREE QUARTERS 133 | \DeclareUnicodeCharacter{00BE}{\ensuremath{\sfrac{3}{4}}} 134 | % 00BF ¿ INVERTED QUESTION MARK 135 | \DeclareUnicodeCharacter{00BF}{\textquestiondown} 136 | % 00D7 × MULTIPLICATION SIGN 137 | \DeclareUnicodeCharacter{00D7}{\times} 138 | % 00F7 ÷ DIVISION SIGN 139 | \DeclareUnicodeCharacter{00F7}{\div} 140 | % 0131 ı LATIN SMALL LETTER DOTLESS I 141 | \DeclareUnicodeCharacter{0131}{\imath} 142 | % 0237 ȷ LATIN SMALL LETTER DOTLESS J 143 | \DeclareUnicodeCharacter{0237}{\jmath} 144 | % 0391 Α GREEK CAPITAL LETTER ALPHA 145 | \DeclareUnicodeCharacter{0391}{\Alpha} 146 | % 0392 Β GREEK CAPITAL LETTER BETA 147 | \DeclareUnicodeCharacter{0392}{\Beta} 148 | % 0393 Γ GREEK CAPITAL LETTER GAMMA 149 | \DeclareUnicodeCharacter{0393}{\Gamma} 150 | % 0394 Δ GREEK CAPITAL LETTER DELTA 151 | \DeclareUnicodeCharacter{0394}{\Delta} 152 | % 0395 Ε GREEK CAPITAL LETTER EPSILON 153 | \DeclareUnicodeCharacter{0395}{\Epsilon} 154 | % 0396 Ζ GREEK CAPITAL LETTER ZETA 155 | \DeclareUnicodeCharacter{0396}{\Zeta} 156 | % 0397 Η GREEK CAPITAL LETTER ETA 157 | \DeclareUnicodeCharacter{0397}{\Eta} 158 | % 0398 Θ GREEK CAPITAL LETTER THETA 159 | \DeclareUnicodeCharacter{0398}{\Theta} 160 | % 0399 Ι GREEK CAPITAL LETTER IOTA 161 | \DeclareUnicodeCharacter{0399}{\Iota} 162 | % 039A Κ GREEK CAPITAL LETTER KAPPA 163 | \DeclareUnicodeCharacter{039A}{\Kappa} 164 | % 039B Λ GREEK CAPITAL LETTER LAMDA 165 | \DeclareUnicodeCharacter{039B}{\Lambda} 166 | % 039C Μ GREEK CAPITAL LETTER MU 167 | \DeclareUnicodeCharacter{039C}{\Mu} 168 | % 039D Ν GREEK CAPITAL LETTER NU 169 | \DeclareUnicodeCharacter{039D}{\Nu} 170 | % 039E Ξ GREEK CAPITAL LETTER XI 171 | \DeclareUnicodeCharacter{039E}{\Xi} 172 | % 039F Ο GREEK CAPITAL LETTER OMICRON 173 | \DeclareUnicodeCharacter{039F}{\Omicron} 174 | % 03A0 Π GREEK CAPITAL LETTER PI 175 | \DeclareUnicodeCharacter{03A0}{\Pi} 176 | % 03A1 Ρ GREEK CAPITAL LETTER RHO 177 | \DeclareUnicodeCharacter{03A1}{\Rho} 178 | % 03A3 Σ GREEK CAPITAL LETTER SIGMA 179 | \DeclareUnicodeCharacter{03A3}{\Sigma} 180 | % 03A4 Τ GREEK CAPITAL LETTER TAU 181 | \DeclareUnicodeCharacter{03A4}{\Tau} 182 | % 03A5 Υ GREEK CAPITAL LETTER UPSILON 183 | \DeclareUnicodeCharacter{03A5}{\Upsilon} 184 | % 03A6 Φ GREEK CAPITAL LETTER PHI 185 | \DeclareUnicodeCharacter{03A6}{\Phi} 186 | % 03A7 Χ GREEK CAPITAL LETTER CHI 187 | \DeclareUnicodeCharacter{03A7}{\Chi} 188 | % 03A8 Ψ GREEK CAPITAL LETTER PSI 189 | \DeclareUnicodeCharacter{03A8}{\Psi} 190 | % 03A9 Ω GREEK CAPITAL LETTER OMEGA 191 | \DeclareUnicodeCharacter{03A9}{\Omega} 192 | % 03B1 α GREEK SMALL LETTER ALPHA 193 | \DeclareUnicodeCharacter{03B1}{\alpha} 194 | % 03B2 β GREEK SMALL LETTER BETA 195 | \DeclareUnicodeCharacter{03B2}{\beta} 196 | % 03B3 γ GREEK SMALL LETTER GAMMA 197 | \DeclareUnicodeCharacter{03B3}{\gamma} 198 | % 03B4 δ GREEK SMALL LETTER DELTA 199 | \DeclareUnicodeCharacter{03B4}{\delta} 200 | % 03B5 ε GREEK SMALL LETTER EPSILON 201 | \DeclareUnicodeCharacter{03B5}{\varepsilon} 202 | % 03B6 ζ GREEK SMALL LETTER ZETA 203 | \DeclareUnicodeCharacter{03B6}{\zeta} 204 | % 03B7 η GREEK SMALL LETTER ETA 205 | \DeclareUnicodeCharacter{03B7}{\eta} 206 | % 03B8 θ GREEK SMALL LETTER THETA 207 | \DeclareUnicodeCharacter{03B8}{\theta} 208 | % 03B9 ι GREEK SMALL LETTER IOTA 209 | \DeclareUnicodeCharacter{03B9}{\iota} 210 | % 03BA κ GREEK SMALL LETTER KAPPA 211 | \DeclareUnicodeCharacter{03BA}{\kappa} 212 | % 03BB λ GREEK SMALL LETTER LAMDA 213 | \DeclareUnicodeCharacter{03BB}{\lambda} 214 | % 03BC μ GREEK SMALL LETTER MU 215 | \DeclareUnicodeCharacter{03BC}{\mu} 216 | % 03BD ν GREEK SMALL LETTER NU 217 | \DeclareUnicodeCharacter{03BD}{\nu} 218 | % 03BE ξ GREEK SMALL LETTER XI 219 | \DeclareUnicodeCharacter{03BE}{\xi} 220 | % 03BF ο GREEK SMALL LETTER OMICRON 221 | \DeclareUnicodeCharacter{03BF}{\omicron} 222 | % 03C0 π GREEK SMALL LETTER PI 223 | \DeclareUnicodeCharacter{03C0}{\pi} 224 | % 03C1 ρ GREEK SMALL LETTER RHO 225 | \DeclareUnicodeCharacter{03C1}{\rho} 226 | % 03C2 ς GREEK SMALL LETTER FINAL SIGMA 227 | \DeclareUnicodeCharacter{03C2}{\varsigma} 228 | % 03C3 σ GREEK SMALL LETTER SIGMA 229 | \DeclareUnicodeCharacter{03C3}{\sigma} 230 | % 03C4 τ GREEK SMALL LETTER TAU 231 | \DeclareUnicodeCharacter{03C4}{\tau} 232 | % 03C5 υ GREEK SMALL LETTER UPSILON 233 | \DeclareUnicodeCharacter{03C5}{\upsilon} 234 | % 03C6 φ GREEK SMALL LETTER PHI 235 | \DeclareUnicodeCharacter{03C6}{\varphi} 236 | % 03C7 χ GREEK SMALL LETTER CHI 237 | \DeclareUnicodeCharacter{03C7}{\chi} 238 | % 03C8 ψ GREEK SMALL LETTER PSI 239 | \DeclareUnicodeCharacter{03C8}{\psi} 240 | % 03C9 ω GREEK SMALL LETTER OMEGA 241 | \DeclareUnicodeCharacter{03C9}{\omega} 242 | % 03D0 ϐ GREEK BETA SYMBOL 243 | \DeclareUnicodeCharacter{03D0}{\ensuremath\varbeta} 244 | % 03D1 ϑ GREEK THETA SYMBOL 245 | \DeclareUnicodeCharacter{03D1}{\vartheta} 246 | % 03D5 ϕ GREEK PHI SYMBOL 247 | \DeclareUnicodeCharacter{03D5}{\phi} 248 | % 03D6 ϖ GREEK PI SYMBOL 249 | \DeclareUnicodeCharacter{03D6}{\ensuremath\varpi} 250 | % 03D8 Ϙ GREEK LETTER ARCHAIC KOPPA 251 | \DeclareUnicodeCharacter{03D8}{\ensuremath{\Qoppa}} 252 | % 03D9 ϙ GREEK SMALL LETTER ARCHAIC KOPPA 253 | \DeclareUnicodeCharacter{03D9}{\ensuremath{\qoppa}} 254 | % 03DA Ϛ GREEK LETTER STIGMA 255 | \DeclareUnicodeCharacter{03DA}{\ensuremath{\Stigma}} 256 | % 03DB ϛ GREEK SMALL LETTER STIGMA 257 | \DeclareUnicodeCharacter{03DB}{\ensuremath{\stigma}} 258 | % 03DC Ϝ GREEK LETTER DIGAMMA 259 | \DeclareUnicodeCharacter{03DC}{\ensuremath{\Digamma}} 260 | % 03DD ϝ GREEK SMALL LETTER DIGAMMA 261 | \DeclareUnicodeCharacter{03DD}{\ensuremath{\digamma}} 262 | % 03DE Ϟ GREEK LETTER KOPPA 263 | \DeclareUnicodeCharacter{03DE}{\ensuremath{\Koppa}} 264 | % 03DF ϟ GREEK SMALL LETTER KOPPA 265 | \DeclareUnicodeCharacter{03DF}{\ensuremath{\koppa}} 266 | % 03E0 Ϡ GREEK LETTER SAMPI 267 | \DeclareUnicodeCharacter{03E0}{\ensuremath{\Sampi}} 268 | % 03E1 ϡ GREEK SMALL LETTER SAMPI 269 | \DeclareUnicodeCharacter{03E1}{\ensuremath{\sampi}} 270 | % 03F0 ϰ GREEK KAPPA SYMBOL 271 | \DeclareUnicodeCharacter{03F0}{\ensuremath{\varkappa}} 272 | % 03F1 ϱ GREEK RHO SYMBOL 273 | \DeclareUnicodeCharacter{03F1}{\ensuremath{\varrho}} 274 | % 03F5 ϵ GREEK LUNATE EPSILON SYMBOL 275 | \DeclareUnicodeCharacter{03F5}{\epsilon} 276 | % 03F6 ϶ GREEK REVERSED LUNATE EPSILON SYMBOL 277 | \DeclareUnicodeCharacter{03F6}{\ensuremath{\backepsilon}} 278 | % 1D62 ᵢ LATIN SUBSCRIPT SMALL LETTER I 279 | \DeclareUnicodeCharacter{1D62}{\ensuremath{_i}} 280 | % 1D63 ᵣ LATIN SUBSCRIPT SMALL LETTER R 281 | \DeclareUnicodeCharacter{1D63}{\ensuremath{_r}} 282 | % 1D64 ᵤ LATIN SUBSCRIPT SMALL LETTER U 283 | \DeclareUnicodeCharacter{1D64}{\ensuremath{_u}} 284 | % 1D65 ᵥ LATIN SUBSCRIPT SMALL LETTER V 285 | \DeclareUnicodeCharacter{1D65}{\ensuremath{_v}} 286 | % 2013 – EN DASH 287 | \DeclareUnicodeCharacter{2013}{\text{--}} 288 | % 2014 — EM DASH 289 | \DeclareUnicodeCharacter{2014}{\text{---}} 290 | % 2016 ‖ DOUBLE VERTICAL LINE 291 | \DeclareUnicodeCharacter{2016}{\textbardbl} 292 | % 2018 ‘ LEFT SINGLE QUOTATION MARK 293 | \DeclareUnicodeCharacter{2018}{\textquoteleft} 294 | % 2019 ’ RIGHT SINGLE QUOTATION MARK 295 | \DeclareUnicodeCharacter{2019}{\textquoteright} 296 | % 201A ‚ SINGLE LOW-9 QUOTATION MARK 297 | \DeclareUnicodeCharacter{201A}{\quotesinglbase} 298 | % 201C “ LEFT DOUBLE QUOTATION MARK 299 | \DeclareUnicodeCharacter{201C}{\textquotedblleft} 300 | % 201D ” RIGHT DOUBLE QUOTATION MARK 301 | \DeclareUnicodeCharacter{201D}{\textquotedblright} 302 | % 201E „ DOUBLE LOW-9 QUOTATION MARK 303 | \DeclareUnicodeCharacter{201E}{\quotedblbase} 304 | % 2020 † DAGGER 305 | \DeclareUnicodeCharacter{2020}{\dag} 306 | % 2021 ‡ DOUBLE DAGGER 307 | \DeclareUnicodeCharacter{2021}{\ddag} 308 | % 2022 • BULLET 309 | \DeclareUnicodeCharacter{2022}{\bullet} 310 | % 2026 … HORIZONTAL ELLIPSIS 311 | \DeclareUnicodeCharacter{2026}{\dots} 312 | % 2030 ‰ PER MILLE SIGN 313 | \DeclareUnicodeCharacter{2030}{\textperthousand} 314 | % 2031 ‱ PER TEN THOUSAND SIGN 315 | \DeclareUnicodeCharacter{2031}{\textpertenthousand} 316 | % 2032 ′ PRIME 317 | \DeclareUnicodeCharacter{2032}{\prime} 318 | % 2033 ″ DOUBLE PRIME 319 | \DeclareUnicodeCharacter{2033}{\second} 320 | % 2034 ‴ TRIPLE PRIME 321 | \DeclareUnicodeCharacter{2034}{\third} 322 | % 2035 ‵ REVERSED PRIME 323 | \DeclareUnicodeCharacter{2035}{\backprime} 324 | % 2038 ‸ CARET 325 | \DeclareUnicodeCharacter{2038}{\ifmmode\widehat{}\else\textasciicircum\fi} 326 | % 2039 ‹ SINGLE LEFT-POINTING ANGLE QUOTATION MARK 327 | \DeclareUnicodeCharacter{2039}{\guilsinglleft} 328 | % 203A › SINGLE RIGHT-POINTING ANGLE QUOTATION MARK 329 | \DeclareUnicodeCharacter{203A}{\guilsinglright} 330 | % 203B ※ REFERENCE MARK 331 | \DeclareUnicodeCharacter{203B}{\textreferencemark} 332 | % 203C ‼ DOUBLE EXCLAMATION MARK 333 | \DeclareUnicodeCharacter{203C}{{!\kern -.5ex!}} 334 | % 203D ‽ INTERROBANG 335 | \DeclareUnicodeCharacter{203D}{\textinterrobang} 336 | % 203E ‾ OVERLINE 337 | \DeclareUnicodeCharacter{203E}{\overline} 338 | % 2042 ⁂ ASTERISM 339 | \DeclareUnicodeCharacter{2042}{\asterism} 340 | % 2045 ⁅ LEFT SQUARE BRACKET WITH QUILL 341 | \DeclareUnicodeCharacter{2045}{\textlquill} 342 | % 2046 ⁆ RIGHT SQUARE BRACKET WITH QUILL 343 | \DeclareUnicodeCharacter{2046}{\textrquill} 344 | % 2047 ⁇ DOUBLE QUESTION MARK 345 | \DeclareUnicodeCharacter{2047}{{?\kern -.5ex?}} 346 | % 2048 ⁈ QUESTION EXCLAMATION MARK 347 | \DeclareUnicodeCharacter{2048}{{?\kern -.5ex!}} 348 | % 2049 ⁉ EXCLAMATION QUESTION MARK 349 | \DeclareUnicodeCharacter{2049}{{!\kern -.5ex?}} 350 | % 2052 ⁒ COMMERCIAL MINUS SIGN 351 | \DeclareUnicodeCharacter{2052}{\textdiscount} 352 | % 2062 ⁢ INVISIBLE TIMES 353 | \DeclareUnicodeCharacter{2062}{{}} 354 | % 2070 ⁰ SUPERSCRIPT ZERO 355 | \DeclareUnicodeCharacter{2070}{^0} 356 | % 2074 ⁴ SUPERSCRIPT FOUR 357 | \DeclareUnicodeCharacter{2074}{^4} 358 | % 2075 ⁵ SUPERSCRIPT FIVE 359 | \DeclareUnicodeCharacter{2075}{^5} 360 | % 2076 ⁶ SUPERSCRIPT SIX 361 | \DeclareUnicodeCharacter{2076}{^6} 362 | % 2077 ⁷ SUPERSCRIPT SEVEN 363 | \DeclareUnicodeCharacter{2077}{^7} 364 | % 2078 ⁸ SUPERSCRIPT EIGHT 365 | \DeclareUnicodeCharacter{2078}{^8} 366 | % 2079 ⁹ SUPERSCRIPT NINE 367 | \DeclareUnicodeCharacter{2079}{^9} 368 | % 207F ⁿ SUPERSCRIPT LATIN SMALL LETTER N 369 | \DeclareUnicodeCharacter{207F}{^n} 370 | % 2080 ₀ SUBSCRIPT ZERO 371 | \DeclareUnicodeCharacter{2080}{_0} 372 | % 2081 ₁ SUBSCRIPT ONE 373 | \DeclareUnicodeCharacter{2081}{_1} 374 | % 2082 ₂ SUBSCRIPT TWO 375 | \DeclareUnicodeCharacter{2082}{_2} 376 | % 2083 ₃ SUBSCRIPT THREE 377 | \DeclareUnicodeCharacter{2083}{_3} 378 | % 2084 ₄ SUBSCRIPT FOUR 379 | \DeclareUnicodeCharacter{2084}{_4} 380 | % 2085 ₅ SUBSCRIPT FIVE 381 | \DeclareUnicodeCharacter{2085}{_5} 382 | % 2086 ₆ SUBSCRIPT SIX 383 | \DeclareUnicodeCharacter{2086}{_6} 384 | % 2087 ₇ SUBSCRIPT SEVEN 385 | \DeclareUnicodeCharacter{2087}{_7} 386 | % 2088 ₈ SUBSCRIPT EIGHT 387 | \DeclareUnicodeCharacter{2088}{_8} 388 | % 2089 ₉ SUBSCRIPT NINE 389 | \DeclareUnicodeCharacter{2089}{_9} 390 | % 208A ₊ SUBSCRIPT PLUS SIGN 391 | \DeclareUnicodeCharacter{208A}{_+} 392 | % 208B ₋ SUBSCRIPT MINUS 393 | \DeclareUnicodeCharacter{208B}{_-} 394 | % 208C ₌ SUBSCRIPT EQUALS SIGN 395 | \DeclareUnicodeCharacter{208C}{_=} 396 | % 208D ₍ SUBSCRIPT LEFT PARENTHESIS 397 | \DeclareUnicodeCharacter{208D}{_(} 398 | % 208E ₎ SUBSCRIPT RIGHT PARENTHESIS 399 | \DeclareUnicodeCharacter{208E}{_)} 400 | % 2090 ₐ LATIN SUBSCRIPT SMALL LETTER A 401 | \DeclareUnicodeCharacter{2090}{_a} 402 | % 2091 ₑ LATIN SUBSCRIPT SMALL LETTER E 403 | \DeclareUnicodeCharacter{2091}{_e} 404 | % 2092 ₒ LATIN SUBSCRIPT SMALL LETTER O 405 | \DeclareUnicodeCharacter{2092}{_o} 406 | % 2093 ₓ LATIN SUBSCRIPT SMALL LETTER X 407 | \DeclareUnicodeCharacter{2093}{_x} 408 | % 2095 ₕ LATIN SUBSCRIPT SMALL LETTER H 409 | \DeclareUnicodeCharacter{2095}{_h} 410 | % 2096 ₖ LATIN SUBSCRIPT SMALL LETTER K 411 | \DeclareUnicodeCharacter{2096}{_k} 412 | % 2097 ₗ LATIN SUBSCRIPT SMALL LETTER L 413 | \DeclareUnicodeCharacter{2097}{_l} 414 | % 2098 ₘ LATIN SUBSCRIPT SMALL LETTER M 415 | \DeclareUnicodeCharacter{2098}{_m} 416 | % 2099 ₙ LATIN SUBSCRIPT SMALL LETTER N 417 | \DeclareUnicodeCharacter{2099}{_n} 418 | % 209A ₚ LATIN SUBSCRIPT SMALL LETTER P 419 | \DeclareUnicodeCharacter{209A}{_p} 420 | % 209B ₛ LATIN SUBSCRIPT SMALL LETTER S 421 | \DeclareUnicodeCharacter{209B}{_s} 422 | % 209C ₜ LATIN SUBSCRIPT SMALL LETTER T 423 | \DeclareUnicodeCharacter{209C}{_t} 424 | % 2102 ℂ DOUBLE-STRUCK CAPITAL C 425 | \DeclareUnicodeCharacter{2102}{\ensuremath{\mathbb{C}}} 426 | % 2107 ℇ EULER CONSTANT 427 | \DeclareUnicodeCharacter{2107}{\ensuremath{\Euler}} 428 | % 210A ℊ SCRIPT SMALL G 429 | \DeclareUnicodeCharacter{210A}{\ensuremath{\mathcal g}} 430 | % 210B ℋ SCRIPT CAPITAL H 431 | \DeclareUnicodeCharacter{210B}{\ensuremath{\mathrsfs H}} 432 | % 210C ℌ BLACK-LETTER CAPITAL H 433 | \DeclareUnicodeCharacter{210C}{\ensuremath{\mathfrak H}} 434 | % 210D ℍ DOUBLE-STRUCK CAPITAL H 435 | \DeclareUnicodeCharacter{210D}{\ensuremath{\mathbb{H}}} 436 | % 210F ℏ PLANCK CONSTANT OVER TWO PI 437 | \DeclareUnicodeCharacter{210F}{\hbar} 438 | % 2110 ℐ SCRIPT CAPITAL I 439 | \DeclareUnicodeCharacter{2110}{\ensuremath{\mathrsfs I}} 440 | % 2111 ℑ BLACK-LETTER CAPITAL I 441 | \DeclareUnicodeCharacter{2111}{\Im} 442 | % 2112 ℒ SCRIPT CAPITAL L 443 | \DeclareUnicodeCharacter{2112}{\ensuremath{\mathrsfs L}} 444 | % 2113 ℓ SCRIPT SMALL L 445 | \DeclareUnicodeCharacter{2113}{\ensuremath{\ell}} 446 | % 2115 ℕ DOUBLE-STRUCK CAPITAL N 447 | \DeclareUnicodeCharacter{2115}{\ensuremath{\mathbb{N}}} 448 | % 2118 ℘ SCRIPT CAPITAL P 449 | \DeclareUnicodeCharacter{2118}{\ensuremath{\wp}} 450 | % 2119 ℙ DOUBLE-STRUCK CAPITAL P 451 | \DeclareUnicodeCharacter{2119}{\ensuremath{\mathbb{P}}} 452 | % 211A ℚ DOUBLE-STRUCK CAPITAL Q 453 | \DeclareUnicodeCharacter{211A}{\ensuremath{\mathbb{Q}}} 454 | % 211D ℝ DOUBLE-STRUCK CAPITAL R 455 | \DeclareUnicodeCharacter{211D}{\ensuremath{\mathbb{R}}} 456 | % 2122 ™ TRADE MARK SIGN 457 | \DeclareUnicodeCharacter{2122}{\texttrademark} 458 | % 2124 ℤ DOUBLE-STRUCK CAPITAL Z 459 | \DeclareUnicodeCharacter{2124}{\ensuremath{\mathbb{Z}}} 460 | % 2126 Ω OHM SIGN 461 | \DeclareUnicodeCharacter{2126}{\Omega} 462 | % 2127 ℧ INVERTED OHM SIGN 463 | \DeclareUnicodeCharacter{2127}{\mho} 464 | % 2128 ℨ BLACK-LETTER CAPITAL Z 465 | \DeclareUnicodeCharacter{2128}{\ensuremath{\mathfrak Z}} 466 | % 212A K KELVIN SIGN 467 | \DeclareUnicodeCharacter{212A}{\ensuremath{\mathrm K}} 468 | % 212B Å ANGSTROM SIGN 469 | \DeclareUnicodeCharacter{212B}{\ensuremath{\mathring{\mathrm A}}} 470 | % 212C ℬ SCRIPT CAPITAL B 471 | \DeclareUnicodeCharacter{212C}{\ensuremath{\mathrsfs B}} 472 | % 212D ℭ BLACK-LETTER CAPITAL C 473 | \DeclareUnicodeCharacter{212D}{\ensuremath{\mathfrak C}} 474 | % 212E ℮ ESTIMATED SYMBOL 475 | \DeclareUnicodeCharacter{212E}{\textestimated} 476 | % 212F ℯ SCRIPT SMALL E 477 | \DeclareUnicodeCharacter{212F}{\ensuremath{\mathrsfs e}} 478 | % 2130 ℰ SCRIPT CAPITAL E 479 | \DeclareUnicodeCharacter{2130}{\ensuremath{\mathrsfs E}} 480 | % 2131 ℱ SCRIPT CAPITAL F 481 | \DeclareUnicodeCharacter{2131}{\ensuremath{\mathrsfs F}} 482 | % 2132 Ⅎ TURNED CAPITAL F 483 | \DeclareUnicodeCharacter{2132}{\Finv} 484 | % 2135 ℵ ALEF SYMBOL 485 | \DeclareUnicodeCharacter{2135}{\aleph} 486 | % 2136 ℶ BET SYMBOL 487 | \DeclareUnicodeCharacter{2136}{\beth} 488 | % 2137 ℷ GIMEL SYMBOL 489 | \DeclareUnicodeCharacter{2137}{\gimel} 490 | % 2138 ℸ DALET SYMBOL 491 | \DeclareUnicodeCharacter{2138}{\daleth} 492 | % 213C ℼ DOUBLE-STRUCK SMALL PI 493 | \DeclareUnicodeCharacter{213C}{\mathbb{\pi}} 494 | % 213D ℽ DOUBLE-STRUCK SMALL GAMMA 495 | \DeclareUnicodeCharacter{213D}{\mathbb{\gamma}} 496 | % 213E ℾ DOUBLE-STRUCK CAPITAL GAMMA 497 | \DeclareUnicodeCharacter{213E}{\mathbb{\Pi}} 498 | % 213F ℿ DOUBLE-STRUCK CAPITAL PI 499 | \DeclareUnicodeCharacter{213F}{\mathbb{\Gamma}} 500 | % 2140 ⅀ DOUBLE-STRUCK N-ARY SUMMATION 501 | \DeclareUnicodeCharacter{2140}{\mathbb{\Sigma}} 502 | % 2141 ⅁ TURNED SANS-SERIF CAPITAL G 503 | \DeclareUnicodeCharacter{2141}{\Game} 504 | % 2144 ⅄ TURNED SANS-SERIF CAPITAL Y 505 | \DeclareUnicodeCharacter{2144}{\Yup} 506 | % 2146 ⅆ DOUBLE-STRUCK ITALIC SMALL D 507 | \DeclareUnicodeCharacter{2146}{\mathrm{d}} 508 | % 2148 ⅈ DOUBLE-STRUCK ITALIC SMALL I 509 | \DeclareUnicodeCharacter{2148}{\imath} 510 | % 2149 ⅉ DOUBLE-STRUCK ITALIC SMALL J 511 | \DeclareUnicodeCharacter{2149}{\jmath} 512 | % 214B ⅋ TURNED AMPERSAND 513 | \DeclareUnicodeCharacter{214B}{\invamp} 514 | % 2190 ← LEFTWARDS ARROW 515 | \DeclareUnicodeCharacter{2190}{\autoleftarrow} 516 | % 2191 ↑ UPWARDS ARROW 517 | \DeclareUnicodeCharacter{2191}{\uparrow} 518 | % 2192 → RIGHTWARDS ARROW 519 | \DeclareUnicodeCharacter{2192}{\autorightarrow} 520 | % 2193 ↓ DOWNWARDS ARROW 521 | \DeclareUnicodeCharacter{2193}{\downarrow} 522 | % 2194 ↔ LEFT RIGHT ARROW 523 | \DeclareUnicodeCharacter{2194}{\leftrightarrow} 524 | % 2195 ↕ UP DOWN ARROW 525 | \DeclareUnicodeCharacter{2195}{\updownarrow} 526 | % 2196 ↖ NORTH WEST ARROW 527 | \DeclareUnicodeCharacter{2196}{\nwarrow} 528 | % 2197 ↗ NORTH EAST ARROW 529 | \DeclareUnicodeCharacter{2197}{\nearrow} 530 | % 2198 ↘ SOUTH EAST ARROW 531 | \DeclareUnicodeCharacter{2198}{\searrow} 532 | % 2199 ↙ SOUTH WEST ARROW 533 | \DeclareUnicodeCharacter{2199}{\swarrow} 534 | % 219A ↚ LEFTWARDS ARROW WITH STROKE 535 | \DeclareUnicodeCharacter{219A}{\nleftarrow} 536 | % 219B ↛ RIGHTWARDS ARROW WITH STROKE 537 | \DeclareUnicodeCharacter{219B}{\nrightarrow} 538 | % 219E ↞ LEFTWARDS TWO HEADED ARROW 539 | \DeclareUnicodeCharacter{219E}{\twoheadleftarrow} 540 | % 21A0 ↠ RIGHTWARDS TWO HEADED ARROW 541 | \DeclareUnicodeCharacter{21A0}{\twoheadrightarrow} 542 | % 21A2 ↢ LEFTWARDS ARROW WITH TAIL 543 | \DeclareUnicodeCharacter{21A2}{\leftarrowtail} 544 | % 21A3 ↣ RIGHTWARDS ARROW WITH TAIL 545 | \DeclareUnicodeCharacter{21A3}{\rightarrowtail} 546 | % 21A4 ↤ LEFTWARDS ARROW FROM BAR 547 | \DeclareUnicodeCharacter{21A4}{\automapsfrom} 548 | % 21A6 ↦ RIGHTWARDS ARROW FROM BAR 549 | \DeclareUnicodeCharacter{21A6}{\automapsto} 550 | % 21A9 ↩ LEFTWARDS ARROW WITH HOOK 551 | \DeclareUnicodeCharacter{21A9}{\hookleftarrow} 552 | % 21AA ↪ RIGHTWARDS ARROW WITH HOOK 553 | \DeclareUnicodeCharacter{21AA}{\autohookrightarrow} 554 | % 21AB ↫ LEFTWARDS ARROW WITH LOOP 555 | \DeclareUnicodeCharacter{21AB}{\looparrowleft} 556 | % 21AC ↬ RIGHTWARDS ARROW WITH LOOP 557 | \DeclareUnicodeCharacter{21AC}{\looparrowright} 558 | % 21AD ↭ LEFT RIGHT WAVE ARROW 559 | \DeclareUnicodeCharacter{21AD}{\leftrightsquigarrow} 560 | % 21AE ↮ LEFT RIGHT ARROW WITH STROKE 561 | \DeclareUnicodeCharacter{21AE}{\nleftrightarrow} 562 | % 21AF ↯ DOWNWARDS ZIGZAG ARROW 563 | \DeclareUnicodeCharacter{21AF}{\lightning} 564 | % 21B0 ↰ UPWARDS ARROW WITH TIP LEFTWARDS 565 | \DeclareUnicodeCharacter{21B0}{\Lsh} 566 | % 21B1 ↱ UPWARDS ARROW WITH TIP RIGHTWARDS 567 | \DeclareUnicodeCharacter{21B1}{\Rsh} 568 | % 21B6 ↶ ANTICLOCKWISE TOP SEMICIRCLE ARROW 569 | \DeclareUnicodeCharacter{21B6}{\curvearrowleft} 570 | % 21B7 ↷ CLOCKWISE TOP SEMICIRCLE ARROW 571 | \DeclareUnicodeCharacter{21B7}{\curvearrowright} 572 | % 21BA ↺ ANTICLOCKWISE OPEN CIRCLE ARROW 573 | \DeclareUnicodeCharacter{21BA}{\circlearrowleft} 574 | % 21BB ↻ CLOCKWISE OPEN CIRCLE ARROW 575 | \DeclareUnicodeCharacter{21BB}{\circlearrowright} 576 | % 21BC ↼ LEFTWARDS HARPOON WITH BARB UPWARDS 577 | \DeclareUnicodeCharacter{21BC}{\leftharpoonup} 578 | % 21BD ↽ LEFTWARDS HARPOON WITH BARB DOWNWARDS 579 | \DeclareUnicodeCharacter{21BD}{\leftharpoondown} 580 | % 21BE ↾ UPWARDS HARPOON WITH BARB RIGHTWARDS 581 | \DeclareUnicodeCharacter{21BE}{\upharpoonright} 582 | % 21BF ↿ UPWARDS HARPOON WITH BARB LEFTWARDS 583 | \DeclareUnicodeCharacter{21BF}{\upharpoonleft} 584 | % 21C0 ⇀ RIGHTWARDS HARPOON WITH BARB UPWARDS 585 | \DeclareUnicodeCharacter{21C0}{\rightharpoonup} 586 | % 21C1 ⇁ RIGHTWARDS HARPOON WITH BARB DOWNWARDS 587 | \DeclareUnicodeCharacter{21C1}{\rightharpoondown} 588 | % 21C2 ⇂ DOWNWARDS HARPOON WITH BARB RIGHTWARDS 589 | \DeclareUnicodeCharacter{21C2}{\downharpoonright} 590 | % 21C3 ⇃ DOWNWARDS HARPOON WITH BARB LEFTWARDS 591 | \DeclareUnicodeCharacter{21C3}{\downharpoonleft} 592 | % 21C4 ⇄ RIGHTWARDS ARROW OVER LEFTWARDS ARROW 593 | \DeclareUnicodeCharacter{21C4}{\rightleftarrows} 594 | % 21C5 ⇅ UPWARDS ARROW LEFTWARDS OF DOWNWARDS ARROW 595 | \DeclareUnicodeCharacter{21C5}{\updownarrows} 596 | % 21C6 ⇆ LEFTWARDS ARROW OVER RIGHTWARDS ARROW 597 | \DeclareUnicodeCharacter{21C6}{\leftrightarrows} 598 | % 21C7 ⇇ LEFTWARDS PAIRED ARROWS 599 | \DeclareUnicodeCharacter{21C7}{\leftleftarrows} 600 | % 21C8 ⇈ UPWARDS PAIRED ARROWS 601 | \DeclareUnicodeCharacter{21C8}{\upuparrows} 602 | % 21C9 ⇉ RIGHTWARDS PAIRED ARROWS 603 | \DeclareUnicodeCharacter{21C9}{\rightrightarrows} 604 | % 21CA ⇊ DOWNWARDS PAIRED ARROWS 605 | \DeclareUnicodeCharacter{21CA}{\downdownarrows} 606 | % 21CB ⇋ LEFTWARDS HARPOON OVER RIGHTWARDS HARPOON 607 | \DeclareUnicodeCharacter{21CB}{\leftrightharpoons} 608 | % 21CC ⇌ RIGHTWARDS HARPOON OVER LEFTWARDS HARPOON 609 | \DeclareUnicodeCharacter{21CC}{\rightleftharpoons} 610 | % 21CD ⇍ LEFTWARDS DOUBLE ARROW WITH STROKE 611 | \DeclareUnicodeCharacter{21CD}{\nLeftarrow} 612 | % 21CE ⇎ LEFT RIGHT DOUBLE ARROW WITH STROKE 613 | \DeclareUnicodeCharacter{21CE}{\nLeftrightarrow} 614 | % 21CF ⇏ RIGHTWARDS DOUBLE ARROW WITH STROKE 615 | \DeclareUnicodeCharacter{21CF}{\nRightarrow} 616 | % 21D0 ⇐ LEFTWARDS DOUBLE ARROW 617 | \DeclareUnicodeCharacter{21D0}{\Leftarrow} 618 | % 21D1 ⇑ UPWARDS DOUBLE ARROW 619 | \DeclareUnicodeCharacter{21D1}{\Uparrow} 620 | % 21D2 ⇒ RIGHTWARDS DOUBLE ARROW 621 | \DeclareUnicodeCharacter{21D2}{\Rightarrow} 622 | % 21D3 ⇓ DOWNWARDS DOUBLE ARROW 623 | \DeclareUnicodeCharacter{21D3}{\Downarrow} 624 | % 21D4 ⇔ LEFT RIGHT DOUBLE ARROW 625 | \DeclareUnicodeCharacter{21D4}{\Leftrightarrow} 626 | % 21D5 ⇕ UP DOWN DOUBLE ARROW 627 | \DeclareUnicodeCharacter{21D5}{\Updownarrow} 628 | % 21D6 ⇖ NORTH WEST DOUBLE ARROW 629 | \DeclareUnicodeCharacter{21D6}{\Nwarrow} 630 | % 21D7 ⇗ NORTH EAST DOUBLE ARROW 631 | \DeclareUnicodeCharacter{21D7}{\Nearrow} 632 | % 21D8 ⇘ SOUTH EAST DOUBLE ARROW 633 | \DeclareUnicodeCharacter{21D8}{\Searrow} 634 | % 21D9 ⇙ SOUTH WEST DOUBLE ARROW 635 | \DeclareUnicodeCharacter{21D9}{\Swarrow} 636 | % 21DA ⇚ LEFTWARDS TRIPLE ARROW 637 | \DeclareUnicodeCharacter{21DA}{\Lleftarrow} 638 | % 21DB ⇛ RIGHTWARDS TRIPLE ARROW 639 | \DeclareUnicodeCharacter{21DB}{\Rrightarrow} 640 | % 21DC ⇜ LEFTWARDS SQUIGGLE ARROW 641 | \DeclareUnicodeCharacter{21DC}{\leftsquigarrow} 642 | % 21DD ⇝ RIGHTWARDS SQUIGGLE ARROW 643 | \DeclareUnicodeCharacter{21DD}{\rightsquigarrow} 644 | % 21E0 ⇠ LEFTWARDS DASHED ARROW 645 | \DeclareUnicodeCharacter{21E0}{\dashleftarrow} 646 | % 21E2 ⇢ RIGHTWARDS DASHED ARROW 647 | \DeclareUnicodeCharacter{21E2}{\dashrightarrow} 648 | % 21E4 ⇤ LEFTWARDS ARROW TO BAR 649 | \DeclareUnicodeCharacter{21E4}{\LeftArrowBar} 650 | % 21E5 ⇥ RIGHTWARDS ARROW TO BAR 651 | \DeclareUnicodeCharacter{21E5}{\RightArrowBar} 652 | % 21F0 ⇰ RIGHTWARDS WHITE ARROW FROM WALL 653 | \DeclareUnicodeCharacter{21F0}{\Mapsto} 654 | % 21FD ⇽ LEFTWARDS OPEN-HEADED ARROW 655 | \DeclareUnicodeCharacter{21FD}{\leftarrowtriangle} 656 | % 21FE ⇾ RIGHTWARDS OPEN-HEADED ARROW 657 | \DeclareUnicodeCharacter{21FE}{\rightarrowtriangle} 658 | % 21FF ⇿ LEFT RIGHT OPEN-HEADED ARROW 659 | \DeclareUnicodeCharacter{21FF}{\leftrightarrowtriangle} 660 | % 2200 ∀ FOR ALL 661 | \DeclareUnicodeCharacter{2200}{\forall} 662 | % 2201 ∁ COMPLEMENT 663 | \DeclareUnicodeCharacter{2201}{\complement} 664 | % 2202 ∂ PARTIAL DIFFERENTIAL 665 | \DeclareUnicodeCharacter{2202}{\partial} 666 | % 2203 ∃ THERE EXISTS 667 | \DeclareUnicodeCharacter{2203}{\exists} 668 | % 2204 ∄ THERE DOES NOT EXIST 669 | \DeclareUnicodeCharacter{2204}{\nexists} 670 | % 2205 ∅ EMPTY SET 671 | \DeclareUnicodeCharacter{2205}{\varnothing} 672 | % 2207 ∇ NABLA 673 | \DeclareUnicodeCharacter{2207}{\nabla} 674 | % 2208 ∈ ELEMENT OF 675 | \DeclareUnicodeCharacter{2208}{\in} 676 | % 2209 ∉ NOT AN ELEMENT OF 677 | \DeclareUnicodeCharacter{2209}{\notin} 678 | % 220B ∋ CONTAINS AS MEMBER 679 | \DeclareUnicodeCharacter{220B}{\ni} 680 | % 220C ∌ DOES NOT CONTAIN AS MEMBER 681 | \DeclareUnicodeCharacter{220C}{\notni} 682 | % 220D ∍ SMALL CONTAINS AS MEMBER 683 | \DeclareUnicodeCharacter{220D}{\sqbullet} 684 | % 220F ∏ N-ARY PRODUCT 685 | \DeclareUnicodeCharacter{220F}{\prod} 686 | % 2210 ∐ N-ARY COPRODUCT 687 | \DeclareUnicodeCharacter{2210}{\coprod} 688 | % 2211 ∑ N-ARY SUMMATION 689 | \DeclareUnicodeCharacter{2211}{\sum} 690 | % 2213 ∓ MINUS-OR-PLUS SIGN 691 | \DeclareUnicodeCharacter{2213}{\mp} 692 | % 2214 ∔ DOT PLUS 693 | \DeclareUnicodeCharacter{2214}{\dotplus} 694 | % 2216 ∖ SET MINUS 695 | \DeclareUnicodeCharacter{2216}{\smallsetminus} 696 | % 2217 ∗ ASTERISK OPERATOR 697 | \DeclareUnicodeCharacter{2217}{\ast} 698 | % 2218 ∘ RING OPERATOR 699 | \DeclareUnicodeCharacter{2218}{\circ} 700 | % 2219 ∙ BULLET OPERATOR 701 | \DeclareUnicodeCharacter{2219}{\bullet} 702 | % 221A √ SQUARE ROOT 703 | \DeclareUnicodeCharacter{221A}{\sqrt} 704 | % 221B ∛ CUBE ROOT 705 | \DeclareUnicodeCharacter{221B}{\sqrt[3]} 706 | % 221C ∜ FOURTH ROOT 707 | \DeclareUnicodeCharacter{221C}{\sqrt[4]} 708 | % 221D ∝ PROPORTIONAL TO 709 | \DeclareUnicodeCharacter{221D}{\propto} 710 | % 221E ∞ INFINITY 711 | \DeclareUnicodeCharacter{221E}{\infty} 712 | % 2220 ∠ ANGLE 713 | \DeclareUnicodeCharacter{2220}{\angle} 714 | % 2221 ∡ MEASURED ANGLE 715 | \DeclareUnicodeCharacter{2221}{\measuredangle} 716 | % 2222 ∢ SPHERICAL ANGLE 717 | \DeclareUnicodeCharacter{2222}{\sphericalangle} 718 | % 2223 ∣ DIVIDES 719 | \DeclareUnicodeCharacter{2223}{\mid} 720 | % 2224 ∤ DOES NOT DIVIDE 721 | \DeclareUnicodeCharacter{2224}{\nmid} 722 | % 2225 ∥ PARALLEL TO 723 | \DeclareUnicodeCharacter{2225}{\parallel} 724 | % 2226 ∦ NOT PARALLEL TO 725 | \DeclareUnicodeCharacter{2226}{\nparallel} 726 | % 2227 ∧ LOGICAL AND 727 | \DeclareUnicodeCharacter{2227}{\wedge} 728 | % 2228 ∨ LOGICAL OR 729 | \DeclareUnicodeCharacter{2228}{\vee} 730 | % 2229 ∩ INTERSECTION 731 | \DeclareUnicodeCharacter{2229}{\cap} 732 | % 222A ∪ UNION 733 | \DeclareUnicodeCharacter{222A}{\cup} 734 | % 222B ∫ INTEGRAL 735 | \DeclareUnicodeCharacter{222B}{\int} 736 | % 222C ∬ DOUBLE INTEGRAL 737 | \DeclareUnicodeCharacter{222C}{\iint} 738 | % 222D ∭ TRIPLE INTEGRAL 739 | \DeclareUnicodeCharacter{222D}{\iiint} 740 | % 222E ∮ CONTOUR INTEGRAL 741 | \DeclareUnicodeCharacter{222E}{\oint} 742 | % 222F ∯ SURFACE INTEGRAL 743 | \DeclareUnicodeCharacter{222F}{\oiint} 744 | % 2230 ∰ VOLUME INTEGRAL 745 | \DeclareUnicodeCharacter{2230}{\oiiint} 746 | % 2232 ∲ CLOCKWISE CONTOUR INTEGRAL 747 | \DeclareUnicodeCharacter{2232}{\ointclockwise} 748 | % 2233 ∳ ANTICLOCKWISE CONTOUR INTEGRAL 749 | \DeclareUnicodeCharacter{2233}{\oiintctrclockwise} 750 | % 2234 ∴ THEREFORE 751 | \DeclareUnicodeCharacter{2234}{\therefore} 752 | % 2235 ∵ BECAUSE 753 | \DeclareUnicodeCharacter{2235}{\because} 754 | % 2236 ∶ RATIO 755 | \DeclareUnicodeCharacter{2236}{:} 756 | % 2237 ∷ PROPORTION 757 | \DeclareUnicodeCharacter{2237}{\Proportion} 758 | % 2238 ∸ DOT MINUS 759 | \DeclareUnicodeCharacter{2238}{\dotminus} 760 | % 2239 ∹ EXCESS 761 | \DeclareUnicodeCharacter{2239}{\eqcolon} 762 | % 223C ∼ TILDE OPERATOR 763 | \DeclareUnicodeCharacter{223C}{\sim} 764 | % 223D ∽ REVERSED TILDE 765 | \DeclareUnicodeCharacter{223D}{\backsim} 766 | % 223F ∿ SINE WAVE 767 | \DeclareUnicodeCharacter{223F}{\AC} 768 | % 2240 ≀ WREATH PRODUCT 769 | \DeclareUnicodeCharacter{2240}{\wr} 770 | % 2241 ≁ NOT TILDE 771 | \DeclareUnicodeCharacter{2241}{\nsim} 772 | % 2243 ≃ ASYMPTOTICALLY EQUAL TO 773 | \DeclareUnicodeCharacter{2243}{\simeq} 774 | % 2244 ≄ NOT ASYMPTOTICALLY EQUAL TO 775 | \DeclareUnicodeCharacter{2244}{\nsimeq} 776 | % 2245 ≅ APPROXIMATELY EQUAL TO 777 | \DeclareUnicodeCharacter{2245}{\cong} 778 | % 2247 ≇ NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO 779 | \DeclareUnicodeCharacter{2247}{\ncong} 780 | % 2248 ≈ ALMOST EQUAL TO 781 | \DeclareUnicodeCharacter{2248}{\approx} 782 | % 2249 ≉ NOT ALMOST EQUAL TO 783 | \DeclareUnicodeCharacter{2249}{\napprox} 784 | % 224A ≊ ALMOST EQUAL OR EQUAL TO 785 | \DeclareUnicodeCharacter{224A}{\approxeq} 786 | % 224D ≍ EQUIVALENT TO 787 | \DeclareUnicodeCharacter{224D}{\asymp} 788 | % 224E ≎ GEOMETRICALLY EQUIVALENT TO 789 | \DeclareUnicodeCharacter{224E}{\Bumpeq} 790 | % 224F ≏ DIFFERENCE BETWEEN 791 | \DeclareUnicodeCharacter{224F}{\bumpeq} 792 | % 2250 ≐ APPROACHES THE LIMIT 793 | \DeclareUnicodeCharacter{2250}{\doteq} 794 | % 2251 ≑ GEOMETRICALLY EQUAL TO 795 | \DeclareUnicodeCharacter{2251}{\doteqdot} 796 | % 2252 ≒ APPROXIMATELY EQUAL TO OR THE IMAGE OF 797 | \DeclareUnicodeCharacter{2252}{\fallingdotseq} 798 | % 2253 ≓ IMAGE OF OR APPROXIMATELY EQUAL TO 799 | \DeclareUnicodeCharacter{2253}{\risingdotseq} 800 | % 2254 ≔ COLON EQUALS 801 | \DeclareUnicodeCharacter{2254}{\coloneqq} 802 | % 2255 ≕ EQUALS COLON 803 | \DeclareUnicodeCharacter{2255}{\eqqcolon} 804 | % 2256 ≖ RING IN EQUAL TO 805 | \DeclareUnicodeCharacter{2256}{\eqcirc} 806 | % 2257 ≗ RING EQUAL TO 807 | \DeclareUnicodeCharacter{2257}{\circeq} 808 | % 2258 ≘ CORRESPONDS TO 809 | \DeclareUnicodeCharacter{2258}{\stackrel{\frown}{=}} 810 | % 2259 ≙ ESTIMATES 811 | \DeclareUnicodeCharacter{2259}{\stackrel{\wedge}{=}} 812 | % 225A ≚ EQUIANGULAR TO 813 | \DeclareUnicodeCharacter{225A}{\stackrel{\vee}{=}} 814 | % 225B ≛ STAR EQUALS 815 | \DeclareUnicodeCharacter{225B}{\stackrel{\star}{=}} 816 | % 225C ≜ DELTA EQUAL TO 817 | \DeclareUnicodeCharacter{225C}{\triangleq} 818 | % 225D ≝ EQUAL TO BY DEFINITION 819 | \DeclareUnicodeCharacter{225D}{\stackrel{\text{\tiny def}}{=}} 820 | % 225F ≟ QUESTIONED EQUAL TO 821 | \DeclareUnicodeCharacter{225F}{\stackrel{\text{\tiny ?}}{=}} 822 | % 2260 ≠ NOT EQUAL TO 823 | \DeclareUnicodeCharacter{2260}{\neq} 824 | % 2261 ≡ IDENTICAL TO 825 | \DeclareUnicodeCharacter{2261}{\equiv} 826 | % 2262 ≢ NOT IDENTICAL TO 827 | \DeclareUnicodeCharacter{2262}{\not\equiv} 828 | % 2263 ≣ STRICTLY EQUIVALENT TO 829 | \DeclareUnicodeCharacter{2263}{\stackrel{=}{=}} 830 | % 2264 ≤ LESS-THAN OR EQUAL TO 831 | \DeclareUnicodeCharacter{2264}{\leq} 832 | % 2265 ≥ GREATER-THAN OR EQUAL TO 833 | \DeclareUnicodeCharacter{2265}{\geq} 834 | % 2266 ≦ LESS-THAN OVER EQUAL TO 835 | \DeclareUnicodeCharacter{2266}{\leqq} 836 | % 2267 ≧ GREATER-THAN OVER EQUAL TO 837 | \DeclareUnicodeCharacter{2267}{\geqq} 838 | % 2268 ≨ LESS-THAN BUT NOT EQUAL TO 839 | \DeclareUnicodeCharacter{2268}{\lneqq} 840 | % 2269 ≩ GREATER-THAN BUT NOT EQUAL TO 841 | \DeclareUnicodeCharacter{2269}{\gneqq} 842 | % 226A ≪ MUCH LESS-THAN 843 | \DeclareUnicodeCharacter{226A}{\ll} 844 | % 226B ≫ MUCH GREATER-THAN 845 | \DeclareUnicodeCharacter{226B}{\gg} 846 | % 226C ≬ BETWEEN 847 | \DeclareUnicodeCharacter{226C}{\between} 848 | % 226D ≭ NOT EQUIVALENT TO 849 | \DeclareUnicodeCharacter{226D}{\nasymp} 850 | % 226E ≮ NOT LESS-THAN 851 | \DeclareUnicodeCharacter{226E}{\nless} 852 | % 226F ≯ NOT GREATER-THAN 853 | \DeclareUnicodeCharacter{226F}{\ngtr} 854 | % 2270 ≰ NEITHER LESS-THAN NOR EQUAL TO 855 | \DeclareUnicodeCharacter{2270}{\nleq} 856 | % 2271 ≱ NEITHER GREATER-THAN NOR EQUAL TO 857 | \DeclareUnicodeCharacter{2271}{\ngeq} 858 | % 2272 ≲ LESS-THAN OR EQUIVALENT TO 859 | \DeclareUnicodeCharacter{2272}{\lesssim} 860 | % 2273 ≳ GREATER-THAN OR EQUIVALENT TO 861 | \DeclareUnicodeCharacter{2273}{\gtrsim} 862 | % 2274 ≴ NEITHER LESS-THAN NOR EQUIVALENT TO 863 | \DeclareUnicodeCharacter{2274}{\nlesssim} 864 | % 2275 ≵ NEITHER GREATER-THAN NOR EQUIVALENT TO 865 | \DeclareUnicodeCharacter{2275}{\ngtrsim} 866 | % 2276 ≶ LESS-THAN OR GREATER-THAN 867 | \DeclareUnicodeCharacter{2276}{\lessgtr} 868 | % 2277 ≷ GREATER-THAN OR LESS-THAN 869 | \DeclareUnicodeCharacter{2277}{\gtrless} 870 | % 2278 ≸ NEITHER LESS-THAN NOR GREATER-THAN 871 | \DeclareUnicodeCharacter{2278}{\nlessgtr} 872 | % 2279 ≹ NEITHER GREATER-THAN NOR LESS-THAN 873 | \DeclareUnicodeCharacter{2279}{\ngtrless} 874 | % 227A ≺ PRECEDES 875 | \DeclareUnicodeCharacter{227A}{\prec} 876 | % 227B ≻ SUCCEEDS 877 | \DeclareUnicodeCharacter{227B}{\succ} 878 | % 227C ≼ PRECEDES OR EQUAL TO 879 | \DeclareUnicodeCharacter{227C}{\preccurlyeq} 880 | % 227D ≽ SUCCEEDS OR EQUAL TO 881 | \DeclareUnicodeCharacter{227D}{\succccurlyeq} 882 | % 227E ≾ PRECEDES OR EQUIVALENT TO 883 | \DeclareUnicodeCharacter{227E}{\precsim} 884 | % 227F ≿ SUCCEEDS OR EQUIVALENT TO 885 | \DeclareUnicodeCharacter{227F}{\succsim} 886 | % 2280 ⊀ DOES NOT PRECEDE 887 | \DeclareUnicodeCharacter{2280}{\nprec} 888 | % 2281 ⊁ DOES NOT SUCCEED 889 | \DeclareUnicodeCharacter{2281}{\nsucc} 890 | % 2282 ⊂ SUBSET OF 891 | \DeclareUnicodeCharacter{2282}{\subset} 892 | % 2283 ⊃ SUPERSET OF 893 | \DeclareUnicodeCharacter{2283}{\supset} 894 | % 2284 ⊄ NOT A SUBSET OF 895 | \DeclareUnicodeCharacter{2284}{\nsubset} 896 | % 2285 ⊅ NOT A SUPERSET OF 897 | \DeclareUnicodeCharacter{2285}{\nsupset} 898 | % 2286 ⊆ SUBSET OF OR EQUAL TO 899 | \DeclareUnicodeCharacter{2286}{\subseteq} 900 | % 2287 ⊇ SUPERSET OF OR EQUAL TO 901 | \DeclareUnicodeCharacter{2287}{\supseteq} 902 | % 2288 ⊈ NEITHER A SUBSET OF NOR EQUAL TO 903 | \DeclareUnicodeCharacter{2288}{\nsubseteq} 904 | % 2289 ⊉ NEITHER A SUPERSET OF NOR EQUAL TO 905 | \DeclareUnicodeCharacter{2289}{\nsupseteq} 906 | % 228A ⊊ SUBSET OF WITH NOT EQUAL TO 907 | \DeclareUnicodeCharacter{228A}{\subsetneq} 908 | % 228B ⊋ SUPERSET OF WITH NOT EQUAL TO 909 | \DeclareUnicodeCharacter{228B}{\supsetneq} 910 | % 228E ⊎ MULTISET UNION 911 | \DeclareUnicodeCharacter{228E}{\uplus} 912 | % 228F ⊏ SQUARE IMAGE OF 913 | \DeclareUnicodeCharacter{228F}{\sqsubset} 914 | % 2290 ⊐ SQUARE ORIGINAL OF 915 | \DeclareUnicodeCharacter{2290}{\sqsupset} 916 | % 2291 ⊑ SQUARE IMAGE OF OR EQUAL TO 917 | \DeclareUnicodeCharacter{2291}{\sqsubseteq} 918 | % 2292 ⊒ SQUARE ORIGINAL OF OR EQUAL TO 919 | \DeclareUnicodeCharacter{2292}{\sqsupseteq} 920 | % 2293 ⊓ SQUARE CAP 921 | \DeclareUnicodeCharacter{2293}{\sqcap} 922 | % 2294 ⊔ SQUARE CUP 923 | \DeclareUnicodeCharacter{2294}{\sqcup} 924 | % 2295 ⊕ CIRCLED PLUS 925 | \DeclareUnicodeCharacter{2295}{\oplus} 926 | % 2296 ⊖ CIRCLED MINUS 927 | \DeclareUnicodeCharacter{2296}{\ominus} 928 | % 2297 ⊗ CIRCLED TIMES 929 | \DeclareUnicodeCharacter{2297}{\otimes} 930 | % 2298 ⊘ CIRCLED DIVISION SLASH 931 | \DeclareUnicodeCharacter{2298}{\oslash} 932 | % 2299 ⊙ CIRCLED DOT OPERATOR 933 | \DeclareUnicodeCharacter{2299}{\odot} 934 | % 229A ⊚ CIRCLED RING OPERATOR 935 | \DeclareUnicodeCharacter{229A}{\circledcirc} 936 | % 229B ⊛ CIRCLED ASTERISK OPERATOR 937 | \DeclareUnicodeCharacter{229B}{\circledast} 938 | % 229D ⊝ CIRCLED DASH 939 | \DeclareUnicodeCharacter{229D}{\circleddash} 940 | % 229E ⊞ SQUARED PLUS 941 | \DeclareUnicodeCharacter{229E}{\boxplus} 942 | % 229F ⊟ SQUARED MINUS 943 | \DeclareUnicodeCharacter{229F}{\boxminus} 944 | % 22A0 ⊠ SQUARED TIMES 945 | \DeclareUnicodeCharacter{22A0}{\boxtimes} 946 | % 22A1 ⊡ SQUARED DOT OPERATOR 947 | \DeclareUnicodeCharacter{22A1}{\boxdot} 948 | % 22A2 ⊢ RIGHT TACK 949 | \DeclareUnicodeCharacter{22A2}{\vdash} 950 | % 22A3 ⊣ LEFT TACK 951 | \DeclareUnicodeCharacter{22A3}{\dashv} 952 | % 22A4 ⊤ DOWN TACK 953 | \DeclareUnicodeCharacter{22A4}{\top} 954 | % 22A5 ⊥ UP TACK 955 | \DeclareUnicodeCharacter{22A5}{\bot} 956 | % 22A6 ⊦ ASSERTION 957 | \DeclareUnicodeCharacter{22A6}{\vdash} 958 | % 22A7 ⊧ MODELS 959 | \DeclareUnicodeCharacter{22A7}{\models} 960 | % 22A9 ⊩ FORCES 961 | \DeclareUnicodeCharacter{22A9}{\Vdash} 962 | % 22AA ⊪ TRIPLE VERTICAL BAR RIGHT TURNSTILE 963 | \DeclareUnicodeCharacter{22AA}{\Vvdash} 964 | % 22AB ⊫ DOUBLE VERTICAL BAR DOUBLE RIGHT TURNSTILE 965 | \DeclareUnicodeCharacter{22AB}{\VDash} 966 | % 22AC ⊬ DOES NOT PROVE 967 | \DeclareUnicodeCharacter{22AC}{\nvdash} 968 | % 22AD ⊭ NOT TRUE 969 | \DeclareUnicodeCharacter{22AD}{\nvDash} 970 | % 22AE ⊮ DOES NOT FORCE 971 | \DeclareUnicodeCharacter{22AE}{\nVdash} 972 | % 22AF ⊯ NEGATED DOUBLE VERTICAL BAR DOUBLE RIGHT TURNSTILE 973 | \DeclareUnicodeCharacter{22AF}{\nVDash} 974 | % 22B2 ⊲ NORMAL SUBGROUP OF 975 | \DeclareUnicodeCharacter{22B2}{\triangleleft} 976 | % 22B3 ⊳ CONTAINS AS NORMAL SUBGROUP 977 | \DeclareUnicodeCharacter{22B3}{\triangleright} 978 | % 22B4 ⊴ NORMAL SUBGROUP OF OR EQUAL TO 979 | \DeclareUnicodeCharacter{22B4}{\unlhd} 980 | % 22B8 ⊸ MULTIMAP 981 | \DeclareUnicodeCharacter{22B8}{\multimap} 982 | % 22BA ⊺ INTERCALATE 983 | \DeclareUnicodeCharacter{22BA}{\intercal} 984 | % 22BB ⊻ XOR 985 | \DeclareUnicodeCharacter{22BB}{\veebar} 986 | % 22BC ⊼ NAND 987 | \DeclareUnicodeCharacter{22BC}{\barwedge} 988 | % 22B5 ⊵ CONTAINS AS NORMAL SUBGROUP OR EQUAL TO 989 | \DeclareUnicodeCharacter{22B5}{\unrhd} 990 | % 22C0 ⋀ N-ARY LOGICAL AND 991 | \DeclareUnicodeCharacter{22C0}{\bigwedge} 992 | % 22C1 ⋁ N-ARY LOGICAL OR 993 | \DeclareUnicodeCharacter{22C1}{\bigvee} 994 | % 22C2 ⋂ N-ARY INTERSECTION 995 | \DeclareUnicodeCharacter{22C2}{\bigcap} 996 | % 22C3 ⋃ N-ARY UNION 997 | \DeclareUnicodeCharacter{22C3}{\bigcup} 998 | % 22C4 ⋄ DIAMOND OPERATOR 999 | \DeclareUnicodeCharacter{22C4}{\diamond} 1000 | % 22C5 ⋅ DOT OPERATOR 1001 | \DeclareUnicodeCharacter{22C5}{\cdot} 1002 | % 22C6 ⋆ STAR OPERATOR 1003 | \DeclareUnicodeCharacter{22C6}{\star} 1004 | % 22C7 ⋇ DIVISION TIMES 1005 | \DeclareUnicodeCharacter{22C7}{\divideontimes} 1006 | % 22C8 ⋈ BOWTIE 1007 | \DeclareUnicodeCharacter{22C8}{\bowtie} 1008 | % 22C9 ⋉ LEFT NORMAL FACTOR SEMIDIRECT PRODUCT 1009 | \DeclareUnicodeCharacter{22C9}{\ltimes} 1010 | % 22CA ⋊ RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT 1011 | \DeclareUnicodeCharacter{22CA}{\rtimes} 1012 | % 22CB ⋋ LEFT SEMIDIRECT PRODUCT 1013 | \DeclareUnicodeCharacter{22CB}{\leftthreetimes} 1014 | % 22CC ⋌ RIGHT SEMIDIRECT PRODUCT 1015 | \DeclareUnicodeCharacter{22CC}{\rightthreetimes} 1016 | % 22CD ⋍ REVERSED TILDE EQUALS 1017 | \DeclareUnicodeCharacter{22CD}{\backsimeq} 1018 | % 22CE ⋎ CURLY LOGICAL OR 1019 | \DeclareUnicodeCharacter{22CE}{\curlyvee} 1020 | % 22CF ⋏ CURLY LOGICAL AND 1021 | \DeclareUnicodeCharacter{22CF}{\curlywedge} 1022 | % 22D0 ⋐ DOUBLE SUBSET 1023 | \DeclareUnicodeCharacter{22D0}{\Subset} 1024 | % 22D1 ⋑ DOUBLE SUPERSET 1025 | \DeclareUnicodeCharacter{22D1}{\Supset} 1026 | % 22D2 ⋒ DOUBLE INTERSECTION 1027 | \DeclareUnicodeCharacter{22D2}{\Cap} 1028 | % 22D3 ⋓ DOUBLE UNION 1029 | \DeclareUnicodeCharacter{22D3}{\Cup} 1030 | % 22D4 ⋔ PITCHFORK 1031 | \DeclareUnicodeCharacter{22D4}{\pitchfork} 1032 | % 22D6 ⋖ LESS-THAN WITH DOT 1033 | \DeclareUnicodeCharacter{22D6}{\lessdot} 1034 | % 22D7 ⋗ GREATER-THAN WITH DOT 1035 | \DeclareUnicodeCharacter{22D7}{\gtrdot} 1036 | % 22D8 ⋘ VERY MUCH LESS-THAN 1037 | \DeclareUnicodeCharacter{22D8}{\lll} 1038 | % 22D9 ⋙ VERY MUCH GREATER-THAN 1039 | \DeclareUnicodeCharacter{22D9}{\ggg} 1040 | % 22DA ⋚ LESS-THAN EQUAL TO OR GREATER-THAN 1041 | \DeclareUnicodeCharacter{22DA}{\lesseqgtr} 1042 | % 22DB ⋛ GREATER-THAN EQUAL TO OR LESS-THAN 1043 | \DeclareUnicodeCharacter{22DB}{\gtreqless} 1044 | % 22DE ⋞ EQUAL TO OR PRECEDES 1045 | \DeclareUnicodeCharacter{22DE}{\curlyeqprec} 1046 | % 22DF ⋟ EQUAL TO OR SUCCEEDS 1047 | \DeclareUnicodeCharacter{22DF}{\curlyeqsucc} 1048 | % 22E0 ⋠ DOES NOT PRECEDE OR EQUAL 1049 | \DeclareUnicodeCharacter{22E0}{\npreceq} 1050 | % 22E1 ⋡ DOES NOT SUCCEED OR EQUAL 1051 | \DeclareUnicodeCharacter{22E1}{\nsucceq} 1052 | % 22E2 ⋢ NOT SQUARE IMAGE OF OR EQUAL TO 1053 | \DeclareUnicodeCharacter{22E2}{\nsqsubseteq} 1054 | % 22E3 ⋣ NOT SQUARE ORIGINAL OF OR EQUAL TO 1055 | \DeclareUnicodeCharacter{22E3}{\nsqsupseteq} 1056 | % 22E4 ⋤ SQUARE IMAGE OF OR NOT EQUAL TO 1057 | \DeclareUnicodeCharacter{22E4}{\sqsubsetneq} 1058 | % 22E5 ⋥ SQUARE ORIGINAL OF OR NOT EQUAL TO 1059 | \DeclareUnicodeCharacter{22E5}{\sqsupsetneq} 1060 | % 22E6 ⋦ LESS-THAN BUT NOT EQUIVALENT TO 1061 | \DeclareUnicodeCharacter{22E6}{\lnsim} 1062 | % 22E7 ⋧ GREATER-THAN BUT NOT EQUIVALENT TO 1063 | \DeclareUnicodeCharacter{22E7}{\gnsim} 1064 | % 22E8 ⋨ PRECEDES BUT NOT EQUIVALENT TO 1065 | \DeclareUnicodeCharacter{22E8}{\precnsim} 1066 | % 22E9 ⋩ SUCCEEDS BUT NOT EQUIVALENT TO 1067 | \DeclareUnicodeCharacter{22E9}{\succnsim} 1068 | % 22EA ⋪ NOT NORMAL SUBGROUP OF 1069 | \DeclareUnicodeCharacter{22EA}{\ntriangleleft} 1070 | % 22EB ⋫ DOES NOT CONTAIN AS NORMAL SUBGROUP 1071 | \DeclareUnicodeCharacter{22EB}{\ntriangleright} 1072 | % 22EC ⋬ NOT NORMAL SUBGROUP OF OR EQUAL TO 1073 | \DeclareUnicodeCharacter{22EC}{\ntrianglelefteq} 1074 | % 22ED ⋭ DOES NOT CONTAIN AS NORMAL SUBGROUP OR EQUAL 1075 | \DeclareUnicodeCharacter{22ED}{\ntrianglerighteq} 1076 | % 22EE ⋮ VERTICAL ELLIPSIS 1077 | \DeclareUnicodeCharacter{22EE}{\vdots} 1078 | % 22EF ⋯ MIDLINE HORIZONTAL ELLIPSIS 1079 | \DeclareUnicodeCharacter{22EF}{\cdots} 1080 | % 22F0 ⋰ UP RIGHT DIAGONAL ELLIPSIS 1081 | \DeclareUnicodeCharacter{22F0}{\iddots} 1082 | % 22F1 ⋱ DOWN RIGHT DIAGONAL ELLIPSIS 1083 | \DeclareUnicodeCharacter{22F1}{\ddots} 1084 | % 2300 ⌀ DIAMETER SIGN 1085 | \DeclareUnicodeCharacter{2300}{\diameter} 1086 | % 2308 ⌈ LEFT CEILING 1087 | \DeclareUnicodeCharacter{2308}{\lceil} 1088 | % 2309 ⌉ RIGHT CEILING 1089 | \DeclareUnicodeCharacter{2309}{\rceil} 1090 | % 230A ⌊ LEFT FLOOR 1091 | \DeclareUnicodeCharacter{230A}{\lfloor} 1092 | % 230B ⌋ RIGHT FLOOR 1093 | \DeclareUnicodeCharacter{230B}{\rfloor} 1094 | % 2322 ⌢ FROWN 1095 | \DeclareUnicodeCharacter{2322}{\frown} 1096 | % 2323 ⌣ SMILE 1097 | \DeclareUnicodeCharacter{2323}{\smile} 1098 | % 2329 〈 LEFT ANGLE BRAKET 1099 | \DeclareUnicodeCharacter{2329}{\langle} 1100 | % 232a 〉 RIGHT ANGLE BRAKET 1101 | \DeclareUnicodeCharacter{232A}{\rangle} 1102 | % 25A1 □ WHITE SQUARE 1103 | \DeclareUnicodeCharacter{25A1}{\square} 1104 | % 25B3 △ WHITE UP-POINTING TRIANGLE 1105 | \DeclareUnicodeCharacter{25B3}{\triangle} 1106 | % 2615 ☕ HOT BEVERAGE 1107 | \DeclareUnicodeCharacter{2615}{NONE " hot beverage} 1108 | % 2621 ☡ CAUTION SIGN 1109 | \DeclareUnicodeCharacter{2621}{\dbend} 1110 | % 2627 ☧ CHI RHO 1111 | \DeclareUnicodeCharacter{2627}{NONE " labarum} 1112 | % 2639 ☹ WHITE FROWNING FACE 1113 | \DeclareUnicodeCharacter{2639}{\frownie " wasysym} 1114 | % 263A ☺ WHITE SMILING FACE 1115 | \DeclareUnicodeCharacter{263A}{\smiley " wasysym} 1116 | % 2660 ♠ BLACK SPADE SUIT 1117 | \DeclareUnicodeCharacter{2660}{\spadesuit} 1118 | % 2661 ♡ WHITE HEART SUIT 1119 | \DeclareUnicodeCharacter{2661}{\heartsuit} 1120 | % 2662 ♢ WHITE DIAMOND SUIT 1121 | \DeclareUnicodeCharacter{2662}{\diamondsuit} 1122 | % 2663 ♣ BLACK CLUB SUIT 1123 | \DeclareUnicodeCharacter{2663}{\clubsuit} 1124 | % 266D ♭ MUSIC FLAT SIGN 1125 | \DeclareUnicodeCharacter{266D}{\flat} 1126 | % 266E ♮ MUSIC NATURAL SIGN 1127 | \DeclareUnicodeCharacter{266E}{\natural} 1128 | % 266F ♯ MUSIC SHARP SIGN 1129 | \DeclareUnicodeCharacter{266F}{\sharp} 1130 | % 26A0 ⚠ WARNING SIGN 1131 | \DeclareUnicodeCharacter{26A0}{\ensuremath{\lower .25ex\hbox{\Large $\triangle$\hskip -1.25ex}!\;\,}} 1132 | % 27C2 ⟂ PERPENDICULAR 1133 | \DeclareUnicodeCharacter{27C2}{\perp} 1134 | % 2A00 ⨀ N-ARY CIRCLED DOT OPERATOR 1135 | \DeclareUnicodeCharacter{2A00}{\bigodot} 1136 | % 2A01 ⨁ N-ARY CIRCLED PLUS OPERATOR 1137 | \DeclareUnicodeCharacter{2A01}{\bigoplus} 1138 | % 2A02 ⨂ N-ARY CIRCLED TIMES OPERATOR 1139 | \DeclareUnicodeCharacter{2A02}{\bigotimes} 1140 | % 2A05 ⨅ N-ARY SQUARE INTERSECTION OPERATOR 1141 | \DeclareUnicodeCharacter{2A05}{\bigsqcap} 1142 | % 2A06 ⨆ N-ARY SQUARE UNION OPERATOR 1143 | \DeclareUnicodeCharacter{2A06}{\bigsqcup} 1144 | % 2A1D ⨝ JOIN 1145 | \DeclareUnicodeCharacter{2A1D}{\join} 1146 | % 2A3F ⨿ AMALGAMATION OR COPRODUCT 1147 | \DeclareUnicodeCharacter{2A3F}{\amalg} 1148 | % 2A7D ⩽ LESS-THAN OR SLANTED EQUAL TO 1149 | \DeclareUnicodeCharacter{2A7D}{\leqslant} 1150 | % 2A7E ⩾ GREATER-THAN OR SLANTED EQUAL TO 1151 | \DeclareUnicodeCharacter{2A7E}{\geqslant} 1152 | % 2E18 ⸘ INVERTED INTERROBANG 1153 | \DeclareUnicodeCharacter{2E18}{\textinterrobangdown} 1154 | -------------------------------------------------------------------------------- /refs.bib: -------------------------------------------------------------------------------- 1 | % Encoding: UTF-8 2 | 3 | @book{Lam, 4 | AUTHOR = {Lam, T. Y.}, 5 | TITLE = {Introduction to quadratic forms over fields}, 6 | SERIES = {Graduate Studies in Mathematics}, 7 | VOLUME = {67}, 8 | PUBLISHER = {American Mathematical Society, Providence, RI}, 9 | YEAR = {2005}, 10 | PAGES = {xxii+550}, 11 | ISBN = {0-8218-1095-2}, 12 | MRCLASS = {11Exx}, 13 | MRNUMBER = {2104929}, 14 | MRREVIEWER = {K. Szymiczek}, 15 | } 16 | 17 | @Book{silverman:advanced, 18 | title = {Advanced Topics in the Arithmetic of Elliptic Curves}, 19 | publisher = {Springer}, 20 | year = {1994}, 21 | author = {Silverman, Joseph H.}, 22 | volume = {151}, 23 | series = {Graduate Texts in Mathematics}, 24 | month = jan, 25 | isbn = {0387943285}, 26 | abstract = {{This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Gr\"{o}ssencharacters and L-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and N\'{e}ron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q-models for elliptic curves over C and R, followed by Tate's theory of q-models for elliptic curves with non-integral j-invariant over p-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.}}, 27 | citeulike-article-id = {789887}, 28 | citeulike-linkout-0 = {http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&path=ASIN/0387943285}, 29 | citeulike-linkout-1 = {http://www.amazon.de/exec/obidos/redirect?tag=citeulike01-21&path=ASIN/0387943285}, 30 | citeulike-linkout-2 = {http://www.amazon.fr/exec/obidos/redirect?tag=citeulike06-21&path=ASIN/0387943285}, 31 | citeulike-linkout-3 = {http://www.amazon.co.uk/exec/obidos/ASIN/0387943285/citeulike00-21}, 32 | citeulike-linkout-4 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387943285}, 33 | citeulike-linkout-5 = {http://www.worldcat.org/isbn/0387943285}, 34 | citeulike-linkout-6 = {http://books.google.com/books?vid=ISBN0387943285}, 35 | citeulike-linkout-7 = {http://www.amazon.com/gp/search?keywords=0387943285&index=books&linkCode=qs}, 36 | citeulike-linkout-8 = {http://www.librarything.com/isbn/0387943285}, 37 | day = {01}, 38 | groups = {Isogenies}, 39 | howpublished = {Paperback}, 40 | keywords = {elliptic\_curve}, 41 | posted-at = {2010-06-20 19:57:19}, 42 | url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0387943285}, 43 | } 44 | 45 | @Book{silverman:elliptic, 46 | title = {The arithmetic of elliptic curves}, 47 | publisher = {Springer-Verlag}, 48 | year = {1992}, 49 | author = {Silverman, Joseph H.}, 50 | volume = {106}, 51 | series = {Graduate Texts in Mathematics}, 52 | address = {New York}, 53 | citeulike-article-id = {10862495}, 54 | comment = {Corrected reprint of the 1986 original}, 55 | groups = {Isogenies}, 56 | mrnumber = {MR1329092 (95m:11054)}, 57 | posted-at = {2012-07-06 21:19:47}, 58 | } 59 | 60 | @Book{langANT, 61 | title = {Algebraic number theory}, 62 | publisher = {Springer-Verlag}, 63 | year = {1994}, 64 | author = {Lang, Serge}, 65 | volume = {110}, 66 | series = {Graduate Texts in Mathematics}, 67 | isbn = {978-0-387-94225-4}, 68 | doi = {10.1007/978-1-4612-0853-2}, 69 | location = {New York}, 70 | pagetotal = {XIII, 357}, 71 | } 72 | 73 | @Book{lang1987elliptic, 74 | title = {Elliptic Functions}, 75 | publisher = {Springer}, 76 | year = {1987}, 77 | author = {Lang, Serge}, 78 | volume = {112}, 79 | series = {Graduate texts in mathematics}, 80 | isbn = {9780387965086}, 81 | groups = {Isogenies}, 82 | lccn = {87004514}, 83 | } 84 | 85 | @Book{neukirch2013algebraic, 86 | title = {Algebraic number theory}, 87 | publisher = {Springer Verlag}, 88 | year = {1999}, 89 | author = {Neukirch, J{\"u}rgen}, 90 | volume = {322}, 91 | isbn = {978-3-642-08473-7}, 92 | doi = {10.1007/978-3-662-03983-0}, 93 | location = {Berlin Heidelberg}, 94 | } 95 | 96 | @InProceedings{10.1007/3-540-44448-3_18, 97 | author = {Hamdy, Safuat and M{\"o}ller, Bodo}, 98 | title = {Security of Cryptosystems Based on Class Groups of Imaginary Quadratic Orders}, 99 | booktitle = {Advances in Cryptology --- ASIACRYPT 2000}, 100 | year = {2000}, 101 | editor = {Okamoto, Tatsuaki}, 102 | pages = {234--247}, 103 | address = {Berlin, Heidelberg}, 104 | publisher = {Springer Berlin Heidelberg}, 105 | abstract = {In this work we investigate the dificulty of the 106 | discrete logarithm problem in class groups of 107 | imaginary quadratic orders.In particular, we discuss 108 | several strategies to compute discrete logarithms in 109 | those class groups.Based on heuristic reasoning, we 110 | give advice for selecting the cryptographic 111 | parameter, i.e. the discriminant, such that 112 | cryptosystems based on class groups of imaginary 113 | quadratic orders would offer a similar security as 114 | commonly used cryptosystems.}, 115 | isbn = {978-3-540-44448-0}, 116 | } 117 | 118 | @InProceedings{10.1007/3-540-44598-6_10, 119 | author = {Ko, Ki Hyoung and Lee, Sang Jin and Cheon, Jung Hee and Han, Jae Woo and Kang, Ju-sung and Park, Choonsik}, 120 | title = {New Public-Key Cryptosystem Using Braid Groups}, 121 | booktitle = {Advances in Cryptology --- CRYPTO 2000}, 122 | year = {2000}, 123 | editor = {Bellare, Mihir}, 124 | pages = {166--183}, 125 | address = {Berlin, Heidelberg}, 126 | publisher = {Springer Berlin Heidelberg}, 127 | abstract = {The braid groups are infinite non-commutative groups 128 | naturally arising from geometric braids. The aim of 129 | this article is twofold. One is to show that the 130 | braid groups can serve as a good source to enrich 131 | cryptography. The feature that makes the braid groups 132 | useful to cryptography includes the followings: (i) 133 | The word problem is solved via a fast algorithm which 134 | computes the canonical form which can be efficiently 135 | manipulated by computers. (ii) The group operations 136 | can be performed efficiently. (iii) The braid groups 137 | have many mathematically hard problems that can be 138 | utilized to design cryptographic primitives. The 139 | other is to propose and implement a new key agreement 140 | scheme and public key cryptosystem based on these 141 | primitives in the braid groups. The efficiency of our 142 | systems is demonstrated by their speed and 143 | information rate. The security of our systems is 144 | based on topological, combinatorial and 145 | group-theoretical problems that are intractible 146 | according to our current mathematical knowledge. The 147 | foundation of our systems is quite different from 148 | widely used cryptosystems based on number theory, but 149 | there are some similarities in design.}, 150 | isbn = {978-3-540-44598-2}, 151 | } 152 | 153 | @InProceedings{10.1007/3-540-44598-6_8, 154 | author = {Biehl, Ingrid and Meyer, Bernd and M{\"u}ller, Volker}, 155 | title = {Differential Fault Attacks on Elliptic Curve Cryptosystems}, 156 | booktitle = {Advances in Cryptology --- CRYPTO 2000}, 157 | year = {2000}, 158 | editor = {Bellare, Mihir}, 159 | pages = {131--146}, 160 | address = {Berlin, Heidelberg}, 161 | publisher = {Springer Berlin Heidelberg}, 162 | isbn = {978-3-540-44598-2}, 163 | } 164 | 165 | @InProceedings{10.1007/3-540-45353-9_12, 166 | author = {Abdalla, Michel and Bellare, Mihir and Rogaway, Phillip}, 167 | title = {The Oracle {D}iffie--{H}ellman Assumptions and an Analysis of {DHIES}}, 168 | booktitle = {Topics in Cryptology --- CT-RSA 2001}, 169 | year = {2001}, 170 | editor = {Naccache, David}, 171 | pages = {143--158}, 172 | address = {Berlin, Heidelberg}, 173 | publisher = {Springer Berlin Heidelberg}, 174 | abstract = {This paper provides security analysis for the 175 | public-key encryption scheme DHIES (formerly named 176 | DHES and DHAES), which was proposed in [7] and is now 177 | in several draft standards. DHIES is a Diffie-Hellman 178 | based scheme that combines a symmetric encryption 179 | method, a message authentication code, and a hash 180 | function, in addition to number-theoretic operations, 181 | in a way which is intended to provide security 182 | against chosen-ciphertext attacks. In this paper we 183 | find natural assumptions under which DHIES achieves 184 | security under chosen-ciphertext attack. The 185 | assumptions we make about the Diffie-Hellman problem 186 | are interesting variants of the customary ones, and 187 | we investigate relationships among them, and provide 188 | security lower bounds. Our proofs are in the standard 189 | model; no random-oracle assumption is required.}, 190 | isbn = {978-3-540-45353-6}, 191 | } 192 | 193 | @InProceedings{10.1007/3-540-48405-1_34, 194 | author = {Fujisaki, Eiichiro and Okamoto, Tatsuaki}, 195 | title = {Secure Integration of Asymmetric and Symmetric Encryption Schemes}, 196 | booktitle = {Advances in Cryptology --- CRYPTO' 99}, 197 | year = {1999}, 198 | editor = {Wiener, Michael}, 199 | pages = {537--554}, 200 | address = {Berlin, Heidelberg}, 201 | publisher = {Springer Berlin Heidelberg}, 202 | abstract = {This paper shows a generic and simple conversion from 203 | weak asymmetric and symmetric encryption schemes into 204 | an asymmetric encryption scheme which is secure in a 205 | very strong sense --- indistinguishability against 206 | adaptive chosen-ciphertext attacks in the random 207 | oracle model. In particular, this conversion can be 208 | applied efficiently to an asymmetric encryption 209 | scheme that provides a large enough coin space and, 210 | for every message, many enough variants of the 211 | encryption, like the ElGamal encryption scheme.}, 212 | isbn = {978-3-540-48405-9}, 213 | } 214 | 215 | @InProceedings{10.1007/978-3-319-70500-2_12, 216 | author = {Hofheinz, Dennis and H{\"o}velmanns, Kathrin and Kiltz, Eike}, 217 | title = {A Modular Analysis of the {Fujisaki-Okamoto} Transformation}, 218 | booktitle = {Theory of Cryptography}, 219 | year = {2017}, 220 | editor = {Kalai, Yael and Reyzin, Leonid}, 221 | pages = {341--371}, 222 | publisher = {Springer International Publishing}, 223 | isbn = {978-3-319-70500-2}, 224 | } 225 | 226 | @InProceedings{10.1007/978-3-642-14081-5_15, 227 | author = {Biasse, Jean-Fran{\c{c}}ois and Jacobson, Michael J. and Silvester, Alan K.}, 228 | title = {Security Estimates for Quadratic Field Based Cryptosystems}, 229 | booktitle = {Information Security and Privacy}, 230 | year = {2010}, 231 | editor = {Steinfeld, Ron and Hawkes, Philip}, 232 | pages = {233--247}, 233 | address = {Berlin, Heidelberg}, 234 | publisher = {Springer Berlin Heidelberg}, 235 | abstract = {We describe implementations for solving the discrete 236 | logarithm problem in the class group of an imaginary 237 | quadratic field and in the infrastructure of a real 238 | quadratic field. The algorithms used incorporate 239 | improvements over previously-used algorithms, and 240 | extensive numerical results are presented 241 | demonstrating their efficiency. This data is used as 242 | the basis for extrapolations, used to provide 243 | recommendations for parameter sizes providing 244 | approximately the same level of security as block 245 | ciphers with 80, 112, 128, 192, and 256-bit symmetric 246 | keys.}, 247 | isbn = {978-3-642-14081-5}, 248 | } 249 | 250 | @InProceedings{10.1007/978-3-642-60539-0_27, 251 | author = {Gao, Shuhong and Panario, Daniel}, 252 | title = {Tests and Constructions of Irreducible Polynomials over Finite Fields}, 253 | booktitle = {Foundations of Computational Mathematics}, 254 | year = {1997}, 255 | editor = {Cucker, Felipe and Shub, Michael}, 256 | pages = {346--361}, 257 | address = {Berlin, Heidelberg}, 258 | publisher = {Springer Berlin Heidelberg}, 259 | abstract = {In this paper we focus on tests and constructions of 260 | irreducible polynomials over finite fields. We 261 | revisit Rabin's (1980) algorithm providing a variant 262 | of it that improves Rabin's cost estimate by a log n 263 | factor. We give a precise analysis of the probability 264 | that a random polynomial of degree n contains no 265 | irreducible factors of degree less than O(log n). 266 | This probability is naturally related to Ben-Or's 267 | (1981) algorithm for testing irreducibility of 268 | polynomials over finite fields. We also compute the 269 | probability of a polynomial being irreducible when it 270 | has no irreducible factors of low degree. This 271 | probability is useful in the analysis of various 272 | algorithms for factoring polynomials over finite 273 | fields. We present an experimental comparison of 274 | these irreducibility methods when testing random 275 | polynomials.}, 276 | isbn = {978-3-642-60539-0}, 277 | } 278 | 279 | @InProceedings{10.1007/BFb0052240, 280 | author = {Lim, Chae Hoon and Lee, Pil Joong}, 281 | title = {A key recovery attack on discrete log-based schemes using a prime order subgroup}, 282 | booktitle = {Advances in Cryptology --- CRYPTO '97}, 283 | year = {1997}, 284 | editor = {Kaliski, Burton S.}, 285 | pages = {249--263}, 286 | address = {Berlin, Heidelberg}, 287 | publisher = {Springer Berlin Heidelberg}, 288 | abstract = {Consider the well-known oracle attack: somehow one 289 | gets a certain computation result as a function of a 290 | secret key from the secret key owner and tries to 291 | extract some information on the secret key. This 292 | attacking scenario is well understood in the 293 | cryptographic community. However, there are many 294 | protocols based on the discrete logarithm problem 295 | that turn out to leak many of the secret key bits 296 | from this oracle attack, unless suitable checkings 297 | are carried out. In this paper we present a key 298 | recovery attack on various discrete log-based schemes 299 | working in a prime order subgroup. Our attack may 300 | reveal part of, or the whole secret key in most 301 | Diffie-Hellman-type key exchange protocols and some 302 | applications of ElGamal encryption and signature 303 | schemes.}, 304 | isbn = {978-3-540-69528-8}, 305 | } 306 | 307 | @InProceedings{10.1007/BFb0099440, 308 | author = {Cohen, Henri and Lenstra, Hendrik W.}, 309 | title = {Heuristics on class groups of number fields}, 310 | booktitle = {Number Theory Noordwijkerhout 1983}, 311 | year = {1984}, 312 | editor = {Jager, Hendrik}, 313 | pages = {33--62}, 314 | address = {Berlin, Heidelberg}, 315 | publisher = {Springer Berlin Heidelberg}, 316 | isbn = {978-3-540-38906-4}, 317 | } 318 | 319 | @Article{10.2307/24522768, 320 | author = {David Harvey}, 321 | title = {Counting points on hyperelliptic curves in average polynomial time}, 322 | journal = {Annals of Mathematics}, 323 | year = {2014}, 324 | volume = {179}, 325 | number = {2}, 326 | pages = {783--803}, 327 | issn = {0003486X}, 328 | abstract = {Let g ≥ 1, and let Q ∈ Z[x] be a monic, 329 | squarefree polynomial of degree 2g + 1. For an odd 330 | prime p not dividing the discriminant of Q, let Zp(T) 331 | denote the zeta function of the hyperelliptic curve 332 | of genus g over the finite field Fp obtained by 333 | reducing the coefficients of the equation y2 = Q(x) 334 | modulo p. We present an explicit deterministic 335 | algorithm that given as input Q and a positive 336 | integer N, computes Zp(T) simultaneously for all such 337 | primes p < N, whose average complexity per prime is 338 | polynomial in g, log N, and the number of bits 339 | required to represent Q.}, 340 | publisher = {Annals of Mathematics}, 341 | url = {http://www.jstor.org/stable/24522768}, 342 | } 343 | 344 | @InProceedings{Adleman-Lenstra, 345 | author = {Adleman, Leonard M. and Lenstra, Hendrik W.}, 346 | title = {Finding Irreducible Polynomials over Finite Fields}, 347 | booktitle = {Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing}, 348 | year = {1986}, 349 | series = {STOC '86}, 350 | pages = {350--355}, 351 | address = {New York, NY, USA}, 352 | publisher = {ACM}, 353 | doi = {10.1145/12130.12166}, 354 | isbn = {0-89791-193-8}, 355 | } 356 | 357 | @Article{Allombert02, 358 | author = {Bill Allombert}, 359 | title = {Explicit Computation of Isomorphisms between Finite Fields}, 360 | journal = {Finite Fields and Their Applications}, 361 | year = {2002}, 362 | volume = {8}, 363 | number = {3}, 364 | pages = {332--342}, 365 | doi = {10.1006/ffta.2001.0344}, 366 | } 367 | 368 | @Electronic{Allombert02-rev, 369 | author = {Bill Allombert}, 370 | year = {2002}, 371 | title = {Explicit Computation of Isomorphisms between Finite Fields}, 372 | note = {Revised version}, 373 | url = {https://www.math.u-bordeaux.fr/~ballombe/fpisom.ps}, 374 | number = {3}, 375 | pages = {332--342}, 376 | volume = {8}, 377 | } 378 | 379 | @InProceedings{antipa+brown+gallant+lambert+struik+vanstone06, 380 | author = {Antipa, Adrian and Brown, Daniel and Gallant, Robert and Lambert, Rob and Struik, Ren\'{e} and Vanstone, Scott}, 381 | title = {Accelerated Verification of {ECDSA} Signatures}, 382 | booktitle = {Selected Areas in Cryptography 2005}, 383 | year = {2006}, 384 | volume = {3897}, 385 | series = {Lecture Notes in Computer Science}, 386 | pages = {307--318}, 387 | address = {Berlin, Heidelberg}, 388 | publisher = {Springer Berlin / Heidelberg}, 389 | abstract = {{Verification of ECDSA signatures is considerably 390 | slower than generation of ECDSA signatures. This 391 | paper describes a method that can be used to 392 | accelerate verification of ECDSA signatures by more 393 | than 40\% with virtually no added implementation 394 | complexity. The method can also be used to accelerate 395 | verification for other ElGamal-like signature 396 | algorithms, including DSA.}}, 397 | chapter = {21}, 398 | doi = {10.1007/11693383_21}, 399 | isbn = {978-3-540-33108-7}, 400 | } 401 | 402 | @Electronic{atkin91, 403 | author = {Atkin, Arthur O. L.}, 404 | year = {1991}, 405 | title = {The number of points on an elliptic curve modulo a prime}, 406 | howpublished = {Manuscript, Chicago IL}, 407 | url = {http://www.lix.polytechnique.fr/Labo/Francois.Morain/AtkinEmails/19910614.txt}, 408 | } 409 | 410 | @Electronic{atkin92, 411 | author = {Atkin, Arthur O. L.}, 412 | year = {1992}, 413 | title = {The number of points on an elliptic curve modulo a prime (II)}, 414 | howpublished = {\url{http://www.lix.polytechnique.fr/Labo/Francois.Morain/AtkinEmails/19910614.txt}}, 415 | url = {http://www.lix.polytechnique.fr/Labo/Francois.Morain/AtkinEmails/19920319.txt}, 416 | } 417 | 418 | @Article{AtkinMorain93, 419 | author = {Atkin, Arthur O. L. and Morain, François}, 420 | title = {Elliptic curves and primality proving}, 421 | journal = {Mathematics of Computation}, 422 | year = {1993}, 423 | volume = {61}, 424 | number = {203}, 425 | pages = {29--68}, 426 | issn = {0025-5718}, 427 | doi = {10.2307/2152935}, 428 | } 429 | 430 | @Article{Aubry:1999:TTS:2947511.2947551, 431 | author = {Aubry, Philippe and Lazard, Daniel and Moreno Maza, Marc}, 432 | title = {On the Theories of Triangular Sets}, 433 | journal = {Journal of Symbolic Computation}, 434 | year = {1999}, 435 | volume = {28}, 436 | number = {1}, 437 | pages = {105--124}, 438 | month = jul, 439 | issn = {0747-7171}, 440 | address = {Duluth, MN, USA}, 441 | doi = {10.1006/jsco.1999.0269}, 442 | publisher = {Academic Press, Inc.}, 443 | } 444 | 445 | @PhdThesis{belding08-thesis, 446 | author = {Belding, Juliana V.}, 447 | title = {Number Theoretic Algorithms for Elliptic Curves}, 448 | school = {University of Maryland}, 449 | year = {2008}, 450 | abstract = {We present new algorithms related to both theoretical 451 | and practical questions in the area of elliptic 452 | curves and class field theory. The dissertation has 453 | two main parts, as described below. Let O be an 454 | imaginary quadratic order of discriminant D < 0, and 455 | let K = √ Q( D). The class polynomial HD of O is 456 | the polynomial whose roots are precisely the 457 | j-invariants of elliptic curves with complex 458 | multiplication by O. Computing this polynomial is 459 | useful in constructing elliptic curves suitable for 460 | cryptography, as well as in the context of explicit 461 | class field theory. In the first part of the 462 | dissertation, we present an algorithm to compute HD 463 | p-adically where p is a prime inert in K and not 464 | dividing D. ̃ This involves computing the canonical 465 | lift E of a pair (E, f ) where E is a supersingular 466 | elliptic curve and f is an embedding of O into the 467 | endomorphism ring of E. We also present an algorithm 468 | to compute HD modulo p for p inert which is used in 469 | the Chinese remainder theorem algorithm to compute HD 470 | . For an elliptic curve E over any field K, the Weil 471 | pairing en is a bilinear map on the points of order n 472 | of E. The Weil pairing is a useful tool in both the 473 | theory of elliptic curves and the application of 474 | elliptic curves to cryptography. However, for K of 475 | characteristic p, the classical Weil pairing on the 476 | points of order p is trivial. In the second part of 477 | the dissertation, we consider E over the dual numbers 478 | K[ ] and define a non-degenerate ” Weil pairing on 479 | p-torsion.” We show that this pairing satisfies 480 | many of the same properties of the classical pairing. 481 | Moreover, we show that it directly relates to recent 482 | attacks on the discrete logarithm problem on the 483 | p-torsion subgroup of an elliptic curve over the 484 | finite field Fq . We also present a new attack on the 485 | discrete logarithm problem on anomalous curves using 486 | a lift of E over Fp [ ].}, 487 | } 488 | 489 | @InProceedings{Ben-Or1981, 490 | author = {Ben-Or, Michael}, 491 | title = {Probabilistic algorithms in finite fields}, 492 | booktitle = {22nd Annual Symposium on Foundations of Computer Science ({SFCS} 1981)}, 493 | year = {1981}, 494 | pages = {394--398}, 495 | month = oct, 496 | doi = {10.1109/SFCS.1981.37}, 497 | issn = {0272-5428}, 498 | } 499 | 500 | @Article{berlekamp1970factoring, 501 | author = {Berlekamp, Elwyn R.}, 502 | title = {Factoring polynomials over large finite fields}, 503 | journal = {Mathematics of computation}, 504 | year = {1970}, 505 | volume = {24}, 506 | number = {111}, 507 | pages = {713--735}, 508 | doi = {10.1090/S0025-5718-1970-0276200-X}, 509 | } 510 | 511 | @Article{Berlekamp82, 512 | author = {Berlekamp, Elwyn R.}, 513 | title = {Bit-serial {R}eed--{S}olomon encoders}, 514 | journal = {IEEE Transactions on Information Theory}, 515 | year = {1982}, 516 | volume = {28}, 517 | number = {6}, 518 | pages = {869--874}, 519 | } 520 | 521 | @InProceedings{BGPS05, 522 | author = {Bostan, Alin and Gonz{\'a}lez-Vega, Laureano and Perdry, Hervé and Schost, {\'E}ric}, 523 | title = {From {N}ewton sums to coefficients: complexity issues in characteristic $p$}, 524 | booktitle = {MEGA'05}, 525 | year = {2005}, 526 | } 527 | 528 | @Article{bisson+sutherland11, 529 | author = {Bisson, Gaetan and Sutherland, Andrew V.}, 530 | title = {Computing the endomorphism ring of an ordinary elliptic curve over a finite field}, 531 | journal = {Journal of Number Theory}, 532 | year = {2011}, 533 | volume = {131}, 534 | number = {5}, 535 | pages = {815--831}, 536 | month = may, 537 | issn = {0022314X}, 538 | abstract = {We present two algorithms to compute the endomorphism 539 | ring of an ordinary elliptic curve E defined over a 540 | finite field . Under suitable heuristic assumptions, 541 | both have subexponential complexity. We bound the 542 | complexity of the first algorithm in terms of , while 543 | our bound for the second algorithm depends primarily 544 | on {log|DE}|, where {DE} is the discriminant of the 545 | order isomorphic to {End(E}). As a byproduct, our 546 | method yields a short certificate that may be used to 547 | verify that the endomorphism ring is as claimed.}, 548 | doi = {10.1016/j.jnt.2009.11.003}, 549 | } 550 | 551 | @Book{blake+seroussi+smart, 552 | title = {Elliptic curves in cryptography}, 553 | publisher = {Cambridge University Press}, 554 | year = {1999}, 555 | author = {Blake, Ian F. and Seroussi, Gadiel and Smart, Nigel P.}, 556 | address = {New York, NY, USA}, 557 | isbn = {0-521-65374-6}, 558 | } 559 | 560 | @Article{BoFlSaSc06, 561 | author = {Alin Bostan and Philippe Flajolet and Bruno Salvy and Éric Schost}, 562 | title = {Fast computation of special resultants}, 563 | journal = {Journal of Symbolic Computation}, 564 | year = {2006}, 565 | volume = {41}, 566 | number = {1}, 567 | pages = {1--29}, 568 | issn = {0747-7171}, 569 | abstract = {We propose fast algorithms for computing composed 570 | products and composed sums, as well as diamond 571 | products of univariate polynomials. These operations 572 | correspond to special multivariate resultants, that 573 | we compute using power sums of roots of polynomials, 574 | by means of their generating series.}, 575 | doi = {10.1016/j.jsc.2005.07.001}, 576 | } 577 | 578 | @Article{bosma+cannon+steel97, 579 | author = {Bosma, Wieb and Cannon, John and Steel, Allan}, 580 | title = {Lattices of compatibly embedded finite fields}, 581 | journal = {Journal of Symbolic Computation}, 582 | year = {1997}, 583 | volume = {24}, 584 | number = {3-4}, 585 | pages = {351--369}, 586 | issn = {0747-7171}, 587 | address = {Duluth, MN, USA}, 588 | doi = {10.1006/jsco.1997.0138}, 589 | publisher = {Academic Press, Inc.}, 590 | } 591 | 592 | @InProceedings{bostan+lecerf+schost:tellegen, 593 | author = {Bostan, Alin and Lecerf, Gr{\'e}goire and Schost, \'{E}ric}, 594 | title = {{T}ellegen's principle into practice}, 595 | booktitle = {ISSAC'03}, 596 | year = {2003}, 597 | pages = {37--44}, 598 | publisher = {ACM}, 599 | abstract = {The transposition principle, also called Tellegen's 600 | principle, is a set of transformation rules for 601 | linear programs. Yet, though well known, it is not 602 | used systematically, and few practical 603 | implementations rely on it. In this article, we 604 | propose explicit transposed versions of polynomial 605 | multiplication and division but also new faster 606 | algorithms for multipoint evaluation, interpolation 607 | and their transposes. We report on their 608 | implementation in Shoup's {NTL} C++ library.}, 609 | doi = {10.1145/860854.860870}, 610 | isbn = {1-58113-641-2}, 611 | } 612 | 613 | @Article{bostan+morain+salvy+schost08, 614 | author = {Bostan, Alin and Morain, Fran\c{c}ois and Salvy, Bruno and Schost, {\'{E}}ric}, 615 | title = {Fast algorithms for computing isogenies between elliptic curves}, 616 | journal = {Mathematics of Computation}, 617 | year = {2008}, 618 | volume = {77}, 619 | number = {263}, 620 | pages = {1755--1778}, 621 | month = sep, 622 | issn = {0025-5718}, 623 | abstract = {We survey algorithms for computing isogenies between 624 | elliptic curvesdefined over a field of characteristic 625 | either 0 or a large prime. Weintroduce a new 626 | algorithm that computes an isogeny of degree ell ( 627 | elldifferent from the characteristic) in time 628 | quasi-linear with respect to ell E This is based in 629 | particular on fast algorithms for power 630 | seriesexpansion of the Weierstrass wp -function and 631 | related functions.}, 632 | doi = {10.1090/S0025-5718-08-02066-8}, 633 | } 634 | 635 | @Article{bostan+salvy+schost03, 636 | author = {Bostan, Alin and Salvy, Bruno and Schost, \'{E}ric}, 637 | title = {Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality}, 638 | journal = {Applicable Algebra in Engineering, Communication and Computing}, 639 | year = {2003}, 640 | volume = {14}, 641 | number = {4}, 642 | pages = {239--272}, 643 | month = nov, 644 | issn = {0938-1279}, 645 | abstract = {Many questions concerning a zero-dimensional 646 | polynomial system can be reduced to linear algebra 647 | operations in the quotient algebra A= k[ X 1,…, X 648 | n]/I, where I is the ideal generated by the input 649 | system. Assuming that the multiplicative structure of 650 | the algebra A is (partly) known, we address the 651 | question of speeding up the linear algebra phase for 652 | the computation of minimal polynomials and rational 653 | parametrizations in A. We present new formul{\ae} for 654 | the rational parametrizations, extending those of 655 | Rouillier, and algorithms extending ideas introduced 656 | by Shoup in the univariate case. Our approach is 657 | based on the A-module structure of the dual space 658 | \$\widehat{A}\$ . An important feature of our 659 | algorithms is that we do not require \$\widehat{A}\$ 660 | to be free and of rank 1. The complexity of our 661 | algorithms for computing the minimal polynomial and 662 | the rational parametrizations are O(2 nD 5/2) and O( 663 | n2 nD 5/2) respectively, where D is the dimension of 664 | A. For fixed n, this is better than algorithms based 665 | on linear algebra except when the complexity of the 666 | available matrix product has exponent less than 5/2.}, 667 | doi = {10.1007/s00200-003-0133-5}, 668 | } 669 | 670 | @Book{Bostan10, 671 | title = {Algorithmes rapides pour les polyn\^omes, s\'eries formelles et matrices}, 672 | year = {2010}, 673 | author = {Alin Bostan}, 674 | volume = {1}, 675 | number = {2}, 676 | series = {Les cours du CIRM}, 677 | url = {https://hal.inria.fr/hal-00780433/}, 678 | } 679 | 680 | @Book{BourbakiAlgCom9, 681 | title = {\'{E}l\'ements de math\'ematique}, 682 | publisher = {Springer}, 683 | year = {2007}, 684 | author = {Bourbaki, Nicolas}, 685 | note = {Alg{\`e}bre. Chapitre 9}, 686 | } 687 | 688 | @Article{BrCa87, 689 | author = {Joel V. Brawley and Leonard Carlitz}, 690 | title = {Irreducibles and the composed product for polynomials over a finite field}, 691 | journal = {Discrete Mathematics}, 692 | year = {1987}, 693 | volume = {65}, 694 | number = {2}, 695 | pages = {115--139}, 696 | issn = {0012-365X}, 697 | abstract = {Let GF(q) denote the finite field of q elements and 698 | let GF[q,x] denote the integral domain of polynomials 699 | in an indeterminate x over GF(q). Further, let Γ = 700 | Γ(q) denote the algebraic closure of GF(q) so that 701 | every polynomial in GF[q,x] on which there is defined 702 | a binary considers certain sets of monic polynomials 703 | from GF[q,x] on which there is defined a binary 704 | operation called the composed product. Here, if f and 705 | g are monics in GF[q,x] with deg f = m and deg g=n, 706 | then the composed product, denoted by f♦g and 707 | defined in terms of the roots of f and g, is also in 708 | GF[q,x] and has degree mn. In the present paper, the 709 | two most important composed products, denoted by the 710 | special symbols Ō and ∗, are those induced by the 711 | field multiplication and the field addition on Γ and 712 | defined by: f∘g = Π Παβ (x − αβ), f∗g = 713 | Π Παβ (x − (α+β)), where the products 714 | indicated by П are the usual products in Γ[x] and 715 | are taken over all the roots α of f and β of g, 716 | (including multiplicities). These two composed 717 | products are called composed multiplication and 718 | composed addition, respectively. After introducing 719 | and developing some theory concerning a more general 720 | notion of composed product, this paper moves to the 721 | special composed products above and asks whether the 722 | irreducibles over GF(q) can be factored uniquely into 723 | indecomposables with respect to each of these 724 | products. Here, the term “irreducible” is used in 725 | the usual sense of the word while the term 726 | “indecomposable” is used in reference to composed 727 | products. This question is shown to have an 728 | affirmative answer in both situations, and thus yield 729 | unique factorization theorems (multiplicative and 730 | additive) for Γ. These theorems are then used to 731 | prove corresponding unique factorization theorems for 732 | all subfields of Γ. Next, it is shown that there are 733 | no irreducibles f in GF[q,x] which can be decomposed 734 | as f=f1Ōg1=f2∗g2 (except for trivial 735 | decompositions). A special inversion formula is then 736 | derived and using this inversion formula, the authors 737 | determine the numbers of irreducibles of degree n 738 | which are indecomposable with respect to (i) composed 739 | multiplication Ō, (ii) composed addition ∗, and 740 | (iii) both the composed products Ō and ∗ 741 | simultaneously. These numbers are given in terms of 742 | the well-known number of irreducibles of degree n 743 | over GF(q). A final section contains some discussion 744 | and several observations about the more general 745 | composed product.}, 746 | doi = {10.1016/0012-365X(87)90135-X}, 747 | } 748 | 749 | @Article{brent+kung, 750 | author = {Brent, Richard P. and Kung, Hsiang Te}, 751 | title = {Fast Algorithms for Manipulating Formal Power Series}, 752 | journal = {Journal of the ACM}, 753 | year = {1978}, 754 | volume = {25}, 755 | number = {4}, 756 | pages = {581--595}, 757 | issn = {0004-5411}, 758 | abstract = {Note: {OCR} errors may be found in this Reference 759 | List extracted from the full text article. {ACM} has 760 | opted to expose the complete List rather than only 761 | correct and linked references.}, 762 | address = {New York, NY, USA}, 763 | doi = {10.1145/322092.322099}, 764 | publisher = {ACM}, 765 | } 766 | 767 | @Article{brieulle2018computing, 768 | author = {Brieulle, Ludovic and De Feo, Luca and Doliskani, Javad and Flori, Jean-Pierre and Schost, {\'E}ric}, 769 | title = {Computing isomorphisms and embeddings of finite fields}, 770 | journal = {Mathematics of Computation}, 771 | year = {2019}, 772 | number = {88}, 773 | pages = {1391--1426}, 774 | doi = {10.1090/mcom/3363}, 775 | } 776 | 777 | @Article{Broeker2009, 778 | author = {Bröker, Reinier and Lauter, Kristin}, 779 | title = {Modular Polynomials for Genus 2}, 780 | journal = {LMS Journal of Computation and Mathematics}, 781 | year = {2009}, 782 | volume = {12}, 783 | pages = {326--339}, 784 | doi = {10.1112/S1461157000001546}, 785 | publisher = {Cambridge University Press}, 786 | } 787 | 788 | @Article{broker-ss, 789 | author = {Br{\"o}ker, Reinier}, 790 | title = {Constructing supersingular elliptic curves}, 791 | journal = {Journal of Combinatorics and Number Theory}, 792 | year = {2009}, 793 | volume = {1}, 794 | number = {3}, 795 | pages = {269--273}, 796 | issn = {1942-5600}, 797 | } 798 | 799 | @Article{Brooks2017, 800 | author = {Brooks, Ernest Hunter and Jetchev, Dimitar and Wesolowski, Benjamin}, 801 | title = {Isogeny graphs of ordinary abelian varieties}, 802 | journal = {Research in Number Theory}, 803 | year = {2017}, 804 | volume = {3}, 805 | number = {1}, 806 | pages = {28}, 807 | month = nov, 808 | issn = {2363-9555}, 809 | abstract = {Fix a prime number {\$}{\$}{\backslash}ell {\$}{\$} 810 | ℓ . Graphs of isogenies of degree a power of 811 | {\$}{\$}{\backslash}ell {\$}{\$} ℓ are 812 | well-understood for elliptic curves, but not for 813 | higher-dimensional abelian varieties. We study the 814 | case of absolutely simple ordinary abelian varieties 815 | over a finite field. We analyse graphs of so-called 816 | {\$}{\$}{\backslash}mathfrak l{\$}{\$} l -isogenies, 817 | resolving that, in arbitrary dimension, their 818 | structure is similar, but not identical, to the 819 | ``volcanoes'' occurring as graphs of isogenies of 820 | elliptic curves. Specializing to the case of 821 | principally polarizable abelian surfaces, we then 822 | exploit this structure to describe graphs of a 823 | particular class of isogenies known as 824 | {\$}{\$}({\backslash}ell , {\backslash}ell ){\$}{\$} 825 | ( ℓ , ℓ ) -isogenies: those whose kernels are 826 | maximal isotropic subgroups of the 827 | {\$}{\$}{\backslash}ell {\$}{\$} ℓ -torsion for the 828 | Weil pairing. We use these two results to write an 829 | algorithm giving a path of computable isogenies from 830 | an arbitrary absolutely simple ordinary abelian 831 | surface towards one with maximal endomorphism ring, 832 | which has immediate consequences for the CM-method in 833 | genus 2, for computing explicit isogenies, and for 834 | the random self-reducibility of the discrete 835 | logarithm problem in genus 2 cryptography.}, 836 | doi = {10.1007/s40993-017-0087-5}, 837 | } 838 | 839 | @Article{BruinierOS16, 840 | author = {Jan Hendrik Bruinier and Ken Ono and Andrew V. Sutherland}, 841 | title = {Class polynomials for nonholomorphic modular functions}, 842 | journal = {Journal of Number Theory}, 843 | year = {2016}, 844 | volume = {161}, 845 | pages = {204--229}, 846 | issn = {0022-314X}, 847 | doi = {10.1016/j.jnt.2015.07.002}, 848 | } 849 | 850 | @Article{Buchmann1988, 851 | author = {Buchmann, Johannes and Williams, Hugh C.}, 852 | title = {A key-exchange system based on imaginary quadratic fields}, 853 | journal = {Journal of Cryptology}, 854 | year = {1988}, 855 | volume = {1}, 856 | number = {2}, 857 | pages = {107--118}, 858 | month = jun, 859 | issn = {1432-1378}, 860 | abstract = {We describe another key-exchange system which, while 861 | based on the general idea of the well-known scheme of 862 | Diffie and Hellman, seems to be more secure than that 863 | technique. The new system is based on the arithmetic 864 | of an imaginary quadratic field, and makes use, 865 | specifically, of the properties of the class group of 866 | such a field.}, 867 | doi = {10.1007/BF02351719}, 868 | } 869 | 870 | @Book{burgisser+clausen-shokrollahi, 871 | title = {Algebraic Complexity Theory}, 872 | publisher = {Springer}, 873 | year = {1997}, 874 | author = {B\"{u}rgisser, Peter and Clausen, Michael and Shokrollahi, M. Amin}, 875 | month = feb, 876 | isbn = {3540605827}, 877 | abstract = {This is the first book to present an up-to-date and 878 | self-contained account of Algebraic Complexity Theory 879 | that is both comprehensive and unified. Requiring of 880 | the reader only some basic algebra and offering over 881 | 350 exercises, it is well-suited as a textbook for 882 | beginners at graduate level. With its extensive 883 | bibliography covering about 500 research papers, this 884 | text is also an ideal reference book for the 885 | professional researcher. The subdivision of the 886 | contents into 21 more or less independent chapters 887 | enables readers to familiarize themselves quickly 888 | with a specific topic, and facilitates the use of 889 | this book as a basis for complementary courses in 890 | other areas such as computer algebra.}, 891 | howpublished = {Hardcover}, 892 | } 893 | 894 | @InProceedings{canetti, 895 | author = {Canetti, Ran and Krawczyk, Hugo}, 896 | title = {Analysis of Key-Exchange Protocols and Their Use for Building Secure Channels}, 897 | booktitle = {EUROCRYPT}, 898 | year = {2001}, 899 | editor = {Birgit Pfitzmann}, 900 | volume = {2045}, 901 | series = {Lecture Notes in Computer Science}, 902 | pages = {453--474}, 903 | publisher = {Springer}, 904 | isbn = {3-540-42070-3}, 905 | } 906 | 907 | @InProceedings{canny+kaltofen+yagati89, 908 | author = {Canny, John F. and Kaltofen, Eric and Yagati, Lakshman N.}, 909 | title = {Solving systems of nonlinear polynomial equations faster}, 910 | booktitle = {ISSAC '89: Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation}, 911 | year = {1989}, 912 | pages = {121--128}, 913 | address = {New York, NY, USA}, 914 | publisher = {ACM}, 915 | abstract = {Note: {OCR} errors may be found in this Reference 916 | List extracted from the full text article. {ACM} has 917 | opted to expose the complete List rather than only 918 | correct and linked references.}, 919 | doi = {10.1145/74540.74556}, 920 | isbn = {0-89791-325-6}, 921 | } 922 | 923 | @Article{cantor+kaltofen91, 924 | author = {Cantor, David G. and Kaltofen, Erich}, 925 | title = {On fast multiplication of polynomials over arbitrary algebras}, 926 | journal = {Acta Informatica}, 927 | year = {1991}, 928 | volume = {28}, 929 | number = {7}, 930 | pages = {693--701}, 931 | month = jul, 932 | issn = {0001-5903}, 933 | doi = {10.1007/BF01178683}, 934 | publisher = {Springer}, 935 | } 936 | 937 | @Article{cantor1981, 938 | author = {Cantor, David G and Zassenhaus, Hans}, 939 | title = {A {N}ew {A}lgorithm for {F}actoring {P}olynomials over {F}inite {F}ields}, 940 | journal = {Mathematics of Computation}, 941 | year = {1981}, 942 | pages = {587--592}, 943 | publisher = {JSTOR}, 944 | } 945 | 946 | @Article{cantor89, 947 | author = {Cantor, David G.}, 948 | title = {On arithmetical algorithms over finite fields}, 949 | journal = {Journal of Combinatiorial Theory}, 950 | year = {1989}, 951 | volume = {50}, 952 | number = {2}, 953 | pages = {285--300}, 954 | issn = {0097-3165}, 955 | address = {Orlando, FL, USA}, 956 | doi = {10.1016/0097-3165(89)90020-4}, 957 | publisher = {Academic Press, Inc.}, 958 | series = {Series A}, 959 | } 960 | 961 | @Article{castryck+hubrechts13, 962 | author = {Castryck, Wouter and Hubrechts, Hendrik}, 963 | title = {The distribution of the number of points modulo an integer on elliptic curves over finite fields}, 964 | journal = {The Ramanujan Journal}, 965 | year = {2013}, 966 | volume = {30}, 967 | number = {2}, 968 | pages = {223--242}, 969 | publisher = {Springer}, 970 | } 971 | 972 | @Misc{cervino04, 973 | author = {Cervi\~{n}o, Juan M.}, 974 | title = {On the Correspondence between Supersingular Elliptic Curves and maximal quaternionic Orders}, 975 | month = apr, 976 | year = {2004}, 977 | abstract = {We present a deterministic and explicit algorithm to 978 | compute the endomorphism rings of supersingular 979 | elliptic curves. As an example we compute the 980 | endomorphism rings of all supersingular elliptic 981 | curves defined over characteristic p=29,...,97.}, 982 | url = {http://arxiv.org/abs/math/0404538}, 983 | } 984 | 985 | @Misc{charlap1991enumeration, 986 | author = {Charlap, Leonard S and Coley, Raymond and Robbins, David P}, 987 | title = {Enumeration of rational points on elliptic curves over finite fields}, 988 | year = {1991}, 989 | note = {Preprint}, 990 | publisher = {Draft}, 991 | } 992 | 993 | @Article{chung1989diameters, 994 | author = {Chung, Fan R.K.}, 995 | title = {Diameters and eigenvalues}, 996 | journal = {Journal of the American Mathematical Society}, 997 | year = {1989}, 998 | volume = {2}, 999 | number = {2}, 1000 | pages = {187--196}, 1001 | } 1002 | 1003 | @Article{Ciet2005, 1004 | author = {Ciet, Mathieu and Joye, Marc}, 1005 | title = {Elliptic Curve Cryptosystems in the Presence of Permanent and Transient Faults}, 1006 | journal = {Designs, Codes and Cryptography}, 1007 | year = {2005}, 1008 | volume = {36}, 1009 | number = {1}, 1010 | pages = {33--43}, 1011 | month = jul, 1012 | issn = {1573-7586}, 1013 | doi = {10.1007/s10623-003-1160-8}, 1014 | } 1015 | 1016 | @Article{CL08, 1017 | author = {Couveignes, Jean-Marc and Lercier, Reynald}, 1018 | title = {Galois invariant smoothness basis}, 1019 | journal = {Series on Number Theory and Its Applications}, 1020 | year = {2008}, 1021 | volume = {5}, 1022 | pages = {142--167}, 1023 | month = may, 1024 | note = {World Scientific}, 1025 | } 1026 | 1027 | @Book{Cohen1993, 1028 | title = {A Course in Computational Algebraic Number Theory}, 1029 | publisher = {Springer-Verlag New York, Inc.}, 1030 | year = {1993}, 1031 | author = {Cohen, Henri}, 1032 | address = {New York, NY, USA}, 1033 | isbn = {0-387-55640-0}, 1034 | } 1035 | 1036 | @Book{Conway:ONAG2000, 1037 | title = {On Numbers and Games}, 1038 | publisher = {AK Peters, Ltd.}, 1039 | year = {2000}, 1040 | author = {Conway, John H.}, 1041 | edition = {2nd edition}, 1042 | month = dec, 1043 | isbn = {1568811276}, 1044 | abstract = {{ONAG, as the book is known, is one of those rare 1045 | publications that sprang to life in a moment of 1046 | creative energy and has remained influential for over 1047 | a quarter of a century. Still in high demand, it is 1048 | being republished with some adjustments and 1049 | corrections. The original motivation for writing the 1050 | book was an attempt to understand the relation 1051 | between the theories of transfinite numbers and 1052 | mathematical games. By defining numbers as the 1053 | strengths of positions in certain games, the author 1054 | arrives at a new class, the surreal numbers (so named 1055 | by Donald Knuth) that includes at the same time the 1056 | real numbers and the ordinal numbers.

This new 1057 | edition ends with an epilogue that sets the stage for 1058 | further research on surreal numbers. The book is a 1059 | must-have for all readers with a serious interest in 1060 | the mathematical foundations of game strategies.}}, 1061 | howpublished = {Hardcover}, 1062 | } 1063 | 1064 | @Article{coppersmith+winograd, 1065 | author = {Coppersmith, Don and Winograd, Shmuel}, 1066 | title = {Matrix multiplication via arithmetic progressions}, 1067 | journal = {Journal of Symbolic Computation}, 1068 | year = {1990}, 1069 | volume = {9}, 1070 | number = {3}, 1071 | pages = {251--280}, 1072 | issn = {07477171}, 1073 | abstract = {We present a new method for accelerating matrix 1074 | multiplication asymptotically. Thiswork builds on 1075 | recent ideas of Volker Strassen, by using a basic 1076 | trilinear form which is not a matrix product. We make 1077 | novel use of the {Salem-Spencer} Theorem, which gives 1078 | a fairly dense set of integers with no three-term 1079 | arithmetic progression. Our resulting matrix exponent 1080 | is 2.376.}, 1081 | address = {Duluth, MN, USA}, 1082 | doi = {10.1016/S0747-7171(08)80013-2}, 1083 | publisher = {Academic Press, Inc.}, 1084 | } 1085 | 1086 | @Article{cosset2015computing, 1087 | author = {Cosset, Romain and Robert, Damien}, 1088 | title = {Computing $(\ell, \ell)$-isogenies in polynomial time on Jacobians of genus 2 curves}, 1089 | journal = {Mathematics of Computation}, 1090 | year = {2015}, 1091 | volume = {84}, 1092 | number = {294}, 1093 | pages = {1953--1975}, 1094 | } 1095 | 1096 | @Article{CostelloSmith2017, 1097 | author = {Costello, Craig and Smith, Benjamin}, 1098 | title = {Montgomery curves and their arithmetic}, 1099 | journal = {Journal of Cryptographic Engineering}, 1100 | year = {2017}, 1101 | doi = {10.1007/s13389-017-0157-6}, 1102 | publisher = {Springer}, 1103 | series = {Special issue on Montgomery arithmetic}, 1104 | url = {https://hal.inria.fr/hal-01483768}, 1105 | } 1106 | 1107 | @Article{couveignes+lercier11, 1108 | author = {Couveignes, Jean-Marc and Lercier, Reynald}, 1109 | title = {Fast construction of irreducible polynomials over finite fields}, 1110 | journal = {Israel Journal of Mathematics}, 1111 | year = {2013}, 1112 | volume = {194}, 1113 | number = {1}, 1114 | pages = {77--105}, 1115 | publisher = {Springer}, 1116 | } 1117 | 1118 | @InProceedings{couveignes+morain94, 1119 | author = {Couveignes, Jean-Marc and Morain, François}, 1120 | title = {Schoof's algorithm and isogeny cycles}, 1121 | booktitle = {ANTS-I: Proceedings of the First International Symposium on Algorithmic Number Theory}, 1122 | year = {1994}, 1123 | volume = {877}, 1124 | series = {Lecture Notes in Computer Science}, 1125 | pages = {43--58}, 1126 | address = {London, UK}, 1127 | publisher = {Springer}, 1128 | isbn = {3-540-58691-1}, 1129 | } 1130 | 1131 | @Article{couveignes00, 1132 | author = {Couveignes, Jean-Marc}, 1133 | title = {Isomorphisms between {A}rtin-{S}chreier towers}, 1134 | journal = {Mathematics of Computation}, 1135 | year = {2000}, 1136 | volume = {69}, 1137 | number = {232}, 1138 | pages = {1625--1631}, 1139 | issn = {0025-5718}, 1140 | address = {Boston, MA, USA}, 1141 | doi = {10.1090/S0025-5718-00-01193-5}, 1142 | publisher = {American Mathematical Society}, 1143 | } 1144 | 1145 | @PhdThesis{couveignes94, 1146 | author = {Couveignes, Jean-Marc}, 1147 | title = {{Quelques calculs en th{\'{e}}orie des nombres}}, 1148 | school = {Universit\'{e} de Bordeaux}, 1149 | year = {1994}, 1150 | } 1151 | 1152 | @InProceedings{couveignes96, 1153 | author = {Couveignes, Jean-Marc}, 1154 | title = {Computing $\ell$-isogenies using the $p$-Torsion}, 1155 | booktitle = {ANTS-II: Proceedings of the Second International Symposium on Algorithmic Number Theory}, 1156 | year = {1996}, 1157 | pages = {59--65}, 1158 | address = {London, UK}, 1159 | publisher = {Springer-Verlag}, 1160 | isbn = {3-540-61581-4}, 1161 | } 1162 | 1163 | @Book{Cox-Little-OShea:UAG2005, 1164 | title = {Using Algebraic Geometry}, 1165 | publisher = {Springer-Verlag}, 1166 | year = {2005}, 1167 | author = {Cox, David A. and Little, John and O'Shea, Donal}, 1168 | isbn = {0387207066}, 1169 | abstract = {In recent years, the discovery of new algorithms for 1170 | dealing with polynomial equations, coupled with their 1171 | implementation on fast inexpensive computers, has 1172 | sparked a minor revolution in the study and practice 1173 | of algebraic geometry. These algorithmic methods have 1174 | also given rise to some exciting new applications of 1175 | algebraic geometry. This book illustrates the many 1176 | uses of algebraic geometry, highlighting some of the 1177 | more recent applications of Gr\"{o}bner bases and 1178 | resultants. In order to do this, the authors provide 1179 | an introduction to some algebraic objects and 1180 | techniques which are more advanced than one typically 1181 | encounters in a first course, but nonetheless of 1182 | great utility. The book is written for nonspecialists 1183 | and for readers with a diverse range of backgrounds. 1184 | It assumes knowledge of the material covered in a 1185 | standard undergraduate course in abstract algebra, 1186 | and it would help to have some previous exposure to 1187 | Gr\"{o}bner bases. The book does not assume the 1188 | reader is familiar with more advanced concepts such 1189 | as modules. For this new edition the authors added 1190 | two new sections and a new chapter, updated the 1191 | references and made numerous minor improvements 1192 | throughout the text.}, 1193 | howpublished = {Hardcover}, 1194 | } 1195 | 1196 | @Misc{cryptoeprint:1999:007, 1197 | author = {Abdalla, Michel and Bellare, Mihir and Rogaway, Phillip}, 1198 | title = {{DHAES}: An Encryption Scheme Based on the {D}iffie--{H}ellman Problem}, 1199 | howpublished = {Cryptology ePrint Archive, Report 1999/007}, 1200 | year = {1999}, 1201 | url = {https://eprint.iacr.org/1999/007}, 1202 | } 1203 | 1204 | @Proceedings{DBLP:conf/ants/2006, 1205 | title = {Algorithmic Number Theory, 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006, Proceedings}, 1206 | year = {2006}, 1207 | editor = {Florian Hess and Sebastian Pauli and Michael E. Pohst}, 1208 | volume = {4076}, 1209 | series = {Lecture Notes in Computer Science}, 1210 | publisher = {Springer}, 1211 | isbn = {3-540-36075-1}, 1212 | biburl = {http://dblp.uni-trier.de/rec/bib/conf/ants/2006}, 1213 | } 1214 | 1215 | @Proceedings{DBLP:conf/eurocrypt/2001, 1216 | title = {Advances in Cryptology - EUROCRYPT 2001, International Conference on the Theory and Application of Cryptographic Techniques, Innsbruck, Austria, May 6-10, 2001, Proceeding}, 1217 | year = {2001}, 1218 | editor = {Birgit Pfitzmann}, 1219 | volume = {2045}, 1220 | series = {Lecture Notes in Computer Science}, 1221 | publisher = {Springer}, 1222 | isbn = {3-540-42070-3}, 1223 | booktitle = {EUROCRYPT}, 1224 | } 1225 | 1226 | @Proceedings{DBLP:conf/pkc/2000, 1227 | title = {Public Key Cryptography, Third International Workshop on Practice and Theory in Public Key Cryptography, {PKC} 2000, Melbourne, Victoria, Australia, January 18-20, 2000, Proceedings}, 1228 | year = {2000}, 1229 | editor = {Hideki Imai and Yuliang Zheng}, 1230 | volume = {1751}, 1231 | series = {Lecture Notes in Computer Science}, 1232 | publisher = {Springer}, 1233 | isbn = {3-540-66967-1}, 1234 | biburl = {http://dblp.uni-trier.de/rec/bib/conf/pkc/2000}, 1235 | } 1236 | 1237 | @Proceedings{DBLP:conf/pqcrypto/2011, 1238 | title = {Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, Taipei, Taiwan, November 29 - December 2, 2011. Proceedings}, 1239 | year = {2011}, 1240 | editor = {Bo-Yin Yang}, 1241 | volume = {7071}, 1242 | series = {Lecture Notes in Computer Science}, 1243 | publisher = {Springer}, 1244 | isbn = {978-3-642-25404-8}, 1245 | booktitle = {PQCrypto}, 1246 | } 1247 | 1248 | @InProceedings{DeDoSc13, 1249 | author = {De Feo, Luca and Doliskani, Javad and Schost, \'{E}ric}, 1250 | title = {Fast Algorithms for $\ell$-adic Towers over Finite Fields}, 1251 | booktitle = {Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation}, 1252 | year = {2013}, 1253 | series = {ISSAC '13}, 1254 | pages = {165--172}, 1255 | address = {New York, NY, USA}, 1256 | publisher = {ACM}, 1257 | doi = {10.1145/2465506.2465956}, 1258 | isbn = {978-1-4503-2059-7}, 1259 | } 1260 | 1261 | @InProceedings{DeDoSc2014, 1262 | author = {De Feo, Luca and Doliskani, Javad and Schost, \'{E}ric}, 1263 | title = {Fast Arithmetic for the Algebraic Closure of Finite Fields}, 1264 | booktitle = {Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation}, 1265 | year = {2014}, 1266 | series = {ISSAC '14}, 1267 | pages = {122--129}, 1268 | address = {New York, NY, USA}, 1269 | publisher = {ACM}, 1270 | doi = {10.1145/2608628.2608672}, 1271 | isbn = {978-1-4503-2501-1}, 1272 | } 1273 | 1274 | @Article{defeo2016explicit, 1275 | author = {De Feo, Luca and Hugounenq, Cyril and Pl{\^u}t, J{\'e}r{\^o}me and Schost, {\'E}ric}, 1276 | title = {Explicit isogenies in quadratic time in any characteristic}, 1277 | journal = {LMS Journal of Computation and Mathematics}, 1278 | year = {2016}, 1279 | volume = {19}, 1280 | number = {A}, 1281 | pages = {267--282}, 1282 | doi = {10.1112/S146115701600036X}, 1283 | publisher = {London Mathematical Society}, 1284 | } 1285 | 1286 | @Article{deuring41, 1287 | author = {Deuring, Max}, 1288 | title = {Die {T}ypen der {M}ultiplikatorenringe elliptischer {F}unktionenk\"{o}rper}, 1289 | journal = {Abhandlungen aus dem Mathematischen Seminar der Universit\"{a}t Hamburg}, 1290 | year = {1941}, 1291 | volume = {14}, 1292 | number = {1}, 1293 | pages = {197--272}, 1294 | month = dec, 1295 | issn = {0025-5858}, 1296 | doi = {10.1007/BF02940746}, 1297 | publisher = {Springer Berlin / Heidelberg}, 1298 | } 1299 | 1300 | @InProceedings{df+schost09, 1301 | author = {De Feo, Luca and Schost, \'{E}ric}, 1302 | title = {Fast arithmetics in {A}rtin-{S}chreier towers over finite fields}, 1303 | booktitle = {ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation}, 1304 | year = {2009}, 1305 | pages = {127--134}, 1306 | address = {New York, NY, USA}, 1307 | publisher = {ACM}, 1308 | abstract = {An {Artin-Schreier} tower over the finite field F p 1309 | is a tower of field extensions generated by 1310 | polynomials of the form X p - X -α. Following Cantor 1311 | and Couveignes, we give algorithms with quasi-linear 1312 | time complexity for arithmetic operations in such 1313 | towers. As an application, we present an 1314 | implementation of Couveignes' algorithm for computing 1315 | isogenies between elliptic curves using the p 1316 | -torsion.}, 1317 | doi = {10.1145/1576702.1576722}, 1318 | isbn = {978-1-60558-609-0}, 1319 | } 1320 | 1321 | @Article{df+schost12, 1322 | author = {De Feo, Luca and Schost, {\'E}ric}, 1323 | title = {Fast arithmetics in {A}rtin-{S}chreier towers over finite fields}, 1324 | journal = {Journal of Symbolic Computation}, 1325 | year = {2012}, 1326 | volume = {47}, 1327 | number = {7}, 1328 | pages = {771--792}, 1329 | issn = {07477171}, 1330 | abstract = {An {Artin-Schreier} tower over the finite field Fp is 1331 | a tower of field extensions generated by polynomials 1332 | of the form {Xp−X}−α. Following Cantor and 1333 | Couveignes, we give algorithms with quasi-linear time 1334 | complexity for arithmetic operations in such towers. 1335 | As an application, we present an implementation of 1336 | Couveignes' algorithm for computing isogenies between 1337 | elliptic curves using the p-torsion.}, 1338 | doi = {10.1016/j.jsc.2011.12.008}, 1339 | } 1340 | 1341 | @PhdThesis{df+thesis, 1342 | author = {De Feo, Luca}, 1343 | title = {{A}lgorithmes {R}apides pour les {T}ours de {C}orps {F}inis et les {I}sog{\'{e}}nies}, 1344 | school = {Ecole Polytechnique X}, 1345 | year = {2010}, 1346 | month = dec, 1347 | abstract = {{D}ans cette th{\`{e}}se nous appliquons des 1348 | techniques provenant du calcul formel et de la 1349 | th{\'{e}}orie des langages afin d'am{\'{e}}liorer 1350 | les op{\'{e}}rations {\'{e}}l{\'{e}}mentaires dans 1351 | certaines tours de corps finis. {N}ous appliquons 1352 | notre construction au probl{\`{e}}me du calcul 1353 | d'isog{\'{e}}nies entre courbes elliptiques et 1354 | obtenons une variante plus rapide ({\`{a}} la fois en 1355 | th{\'{e}}orie et en pratique) de l'algorithme de 1356 | {C}ouveignes. {L}e document est divis{\'{e}} en 1357 | quatre parties. {D}ans la partie {I} nous faisons des 1358 | rappels d'alg{\`{e}}bre et de th{\'{e}}orie de la 1359 | complexit{\'{e}}. {L}a partie {II} traite du principe 1360 | de transposition : nous g{\'{e}}n{\'{e}}ralisons des 1361 | id{\'{e}}es de {B}ostan, {S}chost et {L}ecerf et nous 1362 | montrons qu'il est possible de transposer 1363 | automatiquement des programmes sans pertes en 1364 | complexit{\'{e}}-temps et avec une petite perte en 1365 | complexit{\'{e}}-espace. {L}a partie {III} combine 1366 | les r{\'{e}}sultats sur le principe de transposition 1367 | avec des techniques classiques en th{\'{e}}orie de 1368 | l'{\'{e}}limination ; nous appliquons ces id{\'{e}}es 1369 | pour obtenir des algorithmes asymptotiquement 1370 | optimaux pour l'arithm{\'{e}}tique des tours 1371 | d'{A}rtin-{S}chreier de corps finis. {N}ous 1372 | d{\'{e}}crivons aussi une implantation de ces 1373 | algorithmes. {E}nfin, dans la partie {IV} nous 1374 | utilisons les r{\'{e}}sultats pr{\'{e}}c{\'{e}}dents 1375 | afin d'acc{\'{e}}l{\'{e}}rer l'algorithme de 1376 | {C}ouveignes et de comparer le r{\'{e}}sultat avec 1377 | les autres algorithmes pour le calcul 1378 | d'isog{\'{e}}nies qui font l'{\'{e}}tat de l'art. 1379 | {N}ous pr{\'{e}}sentons aussi une nouvelle 1380 | g{\'{e}}n{\'{e}}ralisation de l'algorithme de 1381 | {C}ouveignes qui calcule des isog{\'{e}}nies de 1382 | degr{\'{e}} inconnu.}, 1383 | url = {http://tel.archives-ouvertes.fr/tel-00547034/en/}, 1384 | } 1385 | 1386 | @Article{df10, 1387 | author = {De Feo, Luca}, 1388 | title = {Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic}, 1389 | journal = {Journal of Number Theory}, 1390 | year = {2011}, 1391 | volume = {131}, 1392 | number = {5}, 1393 | pages = {873--893}, 1394 | month = may, 1395 | issn = {0022-314X}, 1396 | abstract = {The problem of computing an explicit isogeny between 1397 | two given elliptic curves over , originally motivated 1398 | by point counting, has recently awaken new interest 1399 | in the cryptology community thanks to the works of 1400 | Teske and Rostovtsev \& Stolbunov. While the large 1401 | characteristic case is well understood, only 1402 | suboptimal algorithms are known in small 1403 | characteristic; they are due to Couveignes, Lercier, 1404 | Lercier \& Joux and Lercier \& Sirvent. In this paper 1405 | we discuss the differences between them and run some 1406 | comparative experiments. We also present the first 1407 | complete implementation of Couveignes' second 1408 | algorithm and present improvements that make it the 1409 | algorithm having the best asymptotic complexity in 1410 | the degree of the isogeny.}, 1411 | doi = {10.1016/j.jnt.2010.07.003}, 1412 | } 1413 | 1414 | @Article{doi:10.1137/S0097539702403773, 1415 | author = {Ronald Cramer and Victor Shoup}, 1416 | title = {Design and Analysis of Practical Public-Key Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack}, 1417 | journal = {SIAM Journal on Computing}, 1418 | year = {2003}, 1419 | volume = {33}, 1420 | number = {1}, 1421 | pages = {167--226}, 1422 | doi = {10.1137/S0097539702403773}, 1423 | } 1424 | 1425 | @Article{doliskanischost2011, 1426 | author = {Doliskani, Javad and Schost, {\'E}ric}, 1427 | title = {Taking roots over high extensions of finite fields}, 1428 | journal = {Mathematics of Computation}, 1429 | year = {2014}, 1430 | volume = {83}, 1431 | number = {285}, 1432 | pages = {435--446}, 1433 | } 1434 | 1435 | @Article{DoSc12, 1436 | author = {Doliskani, Javad and Schost, \'Eric}, 1437 | title = {Computing in degree $2^k$-extensions of finite fields of odd characteristic}, 1438 | journal = {Designs, Codes and Cryptography}, 1439 | year = {2015}, 1440 | volume = {74}, 1441 | number = {3}, 1442 | pages = {559--569}, 1443 | } 1444 | 1445 | @Book{DSV, 1446 | title = {Elementary number theory, group theory, and {R}amanujan graphs}, 1447 | publisher = {Cambridge University Press}, 1448 | year = {2003}, 1449 | author = {Davidoff, Giuliana and Sarnak, Peter and Valette, Alain}, 1450 | volume = {55}, 1451 | series = {London Mathematical Society Student Texts}, 1452 | address = {Cambridge}, 1453 | doi = {10.1017/CBO9780511615825}, 1454 | } 1455 | 1456 | @PhdThesis{dupont2006moyenne, 1457 | author = {Dupont, R{\'e}gis}, 1458 | title = {Moyenne arithm{\'e}tico-g{\'e}om{\'e}trique, suites de {B}orchardt et applications}, 1459 | year = {2006}, 1460 | institution = {École Polytechnique}, 1461 | journal = {{\'E}cole polytechnique, Palaiseau}, 1462 | } 1463 | 1464 | @Electronic{Echidna, 1465 | author = {David R. Kohel}, 1466 | year = {2018}, 1467 | title = {Echidna databases}, 1468 | url = {http://iml.univ-mrs.fr/~kohel/dbs/}, 1469 | } 1470 | 1471 | @InProceedings{ECM20, 1472 | author = {Paul Zimmermann and Bruce Dodson}, 1473 | title = {20 Years of {ECM}}, 1474 | booktitle = {Algorithmic Number Theory, 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006, Proceedings}, 1475 | year = {2006}, 1476 | editor = {Florian Hess and Sebastian Pauli and Michael E. Pohst}, 1477 | volume = {4076}, 1478 | series = {Lecture Notes in Computer Science}, 1479 | pages = {525--542}, 1480 | publisher = {Springer}, 1481 | biburl = {http://dblp.uni-trier.de/rec/bib/conf/ants/ZimmermannD06}, 1482 | doi = {10.1007/11792086\_37}, 1483 | isbn = {3-540-36075-1}, 1484 | } 1485 | 1486 | @Misc{efd, 1487 | author = {Bernstein, Daniel J. and Lange, Tanja}, 1488 | title = {{Explicit-Formulas Database}}, 1489 | year = {2007}, 1490 | url = {http://www.hyperelliptic.org/EFD/index.html}, 1491 | } 1492 | 1493 | @Article{elgamal, 1494 | author = {ElGamal, Taher}, 1495 | title = {A public key cryptosystem and a signature scheme based on discrete logarithms}, 1496 | journal = {IEEE transactions on information theory}, 1497 | year = {1985}, 1498 | volume = {31}, 1499 | number = {4}, 1500 | pages = {469--472}, 1501 | publisher = {Institute of Electrical and Electronics Engineers}, 1502 | } 1503 | 1504 | @Unpublished{elkies92, 1505 | author = {Elkies, Noam D.}, 1506 | title = {Explicit isogenies}, 1507 | year = {1992}, 1508 | howpublished = {Manuscript, Boston MA}, 1509 | } 1510 | 1511 | @InProceedings{elkies98, 1512 | author = {Elkies, Noam D.}, 1513 | title = {Elliptic and modular curves over finite fields and related computational issues}, 1514 | booktitle = {Computational perspectives on number theory (Chicago, IL, 1995)}, 1515 | year = {1998}, 1516 | volume = {7}, 1517 | series = {Studies in Advanced Mathematics}, 1518 | pages = {21--76}, 1519 | address = {Providence, RI}, 1520 | publisher = {AMS International Press}, 1521 | url = {http://www.ams.org/mathscinet-getitem?mr=1486831}, 1522 | } 1523 | 1524 | @InProceedings{enge+morain03, 1525 | author = {Enge, Andreas and Morain, Fran\c{c}ois}, 1526 | title = {Fast decomposition of polynomials with known Galois group}, 1527 | booktitle = {AAECC'03: Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes}, 1528 | year = {2003}, 1529 | pages = {254--264}, 1530 | address = {Berlin, Heidelberg}, 1531 | publisher = {Springer-Verlag}, 1532 | abstract = {Let {f(X}) be a separable polynomial with 1533 | coefficients in a field K , generating a field 1534 | extension {M/K} . If this extension is Galois with a 1535 | solvable automorphism group, then the equation {f(X}) 1536 | = 0 can be solved by radicals. The first step of the 1537 | solution consists of splitting the extension {M/K} 1538 | into intermediate fields. Such computations are 1539 | classical, and we explain how fast polynomial 1540 | arithmetic can be used to speed up the process. 1541 | Moreover, we extend the algorithms to a more general 1542 | case of extensions that are no longer Galois. 1543 | Numerical examples are provided, including results 1544 | obtained with our implementation for Hilbert class 1545 | fields of imaginary quadratic fields.}, 1546 | isbn = {3-540-40111-3}, 1547 | } 1548 | 1549 | @Article{enge09, 1550 | author = {Enge, Andreas}, 1551 | title = {Computing modular polynomials in quasi-linear time}, 1552 | journal = {Mathematics of Computation}, 1553 | year = {2009}, 1554 | volume = {78}, 1555 | number = {267}, 1556 | pages = {1809--1824}, 1557 | } 1558 | 1559 | @Article{feige+fiat+shamir88, 1560 | author = {Feige, Uriel and Fiat, Amos and Shamir, Adi}, 1561 | title = {Zero-knowledge proofs of identity}, 1562 | journal = {Journal of Cryptology}, 1563 | year = {1988}, 1564 | volume = {1}, 1565 | number = {2}, 1566 | pages = {77--94}, 1567 | month = jun, 1568 | issn = {0933-2790}, 1569 | abstract = {In this paper we extend the notion of interactive 1570 | proofs of assertions to interactive proofs of 1571 | knowledge. This leads to the definition of 1572 | unrestricted input zero-knowledge proofs of knowledge 1573 | in which the prover demonstrates possession of 1574 | knowledge without revealing any computational 1575 | information whatsoever (not even the one bit revealed 1576 | in zero-knowledge proofs of assertions). We show the 1577 | relevance of these notions to identification schemes, 1578 | in which parties prove their identity by 1579 | demonstrating their knowledge rather than by proving 1580 | the validity of assertions. We describe a novel 1581 | scheme which is provably secure if factoring is 1582 | difficult and whose practical implementations are 1583 | about two orders of magnitude faster than RSA-based 1584 | identification schemes. The advantages of thinking in 1585 | terms of proofs of knowledge rather than proofs of 1586 | assertions are demonstrated in two efficient variants 1587 | of the scheme: unrestricted input zero-knowledge 1588 | proofs of knowledge are used in the construction of a 1589 | scheme which needs no directory; a version of the 1590 | scheme based on parallel interactive proofs (which 1591 | are not known to be zero knowledge) is proved secure 1592 | by observing that the identification protocols are 1593 | proofs of knowledge.}, 1594 | doi = {10.1007/BF02351717}, 1595 | publisher = {Springer New York}, 1596 | } 1597 | 1598 | @Article{feisel1999normal, 1599 | author = {Feisel, Sandra and von zur Gathen, Joachim and Shokrollahi, M. Amin}, 1600 | title = {Normal bases via general Gauss periods}, 1601 | journal = {Mathematics of Computation}, 1602 | year = {1999}, 1603 | volume = {68}, 1604 | number = {225}, 1605 | pages = {271--290}, 1606 | publisher = {American Mathematical Society}, 1607 | } 1608 | 1609 | @Article{ffisom-long, 1610 | author = {Brieulle, Ludovic and De Feo, Luca and Doliskani, Javad and Flori, Jean-Pierre and Schost, {\'E}ric}, 1611 | title = {Computing isomorphisms and embeddings of finite fields (extended version)}, 1612 | journal = {arXiv preprint arXiv:1705.01221}, 1613 | year = {2017}, 1614 | url = {https://arxiv.org/abs/1705.01221}, 1615 | } 1616 | 1617 | @InProceedings{Fieker:2017:NCA:3087604.3087611, 1618 | author = {Fieker, Claus and Hart, William and Hofmann, Tommy and Johansson, Fredrik}, 1619 | title = {{Nemo/Hecke}: Computer Algebra and Number Theory Packages for the {Julia} Programming Language}, 1620 | booktitle = {Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation}, 1621 | year = {2017}, 1622 | series = {ISSAC '17}, 1623 | pages = {157--164}, 1624 | address = {New York, NY, USA}, 1625 | publisher = {ACM}, 1626 | doi = {10.1145/3087604.3087611}, 1627 | isbn = {978-1-4503-5064-8}, 1628 | url = {http://nemocas.org/}, 1629 | } 1630 | 1631 | @Manual{flint, 1632 | title = {{FLINT}: {F}ast {L}ibrary for {N}umber {T}heory}, 1633 | author = {Hart, William and Johansson, Fredrik and Pancratz, Sebastian}, 1634 | year = {2013}, 1635 | note = {Version 2.4.0}, 1636 | shorthand = {Flint}, 1637 | url = {http://flintlib.org}, 1638 | } 1639 | 1640 | @InProceedings{fouquet+morain02, 1641 | author = {Fouquet, Mireille and Morain, Fran\c{c}ois}, 1642 | title = {Isogeny Volcanoes and the {SEA} Algorithm}, 1643 | booktitle = {Algorithmic Number Theory Symposium}, 1644 | year = {2002}, 1645 | editor = {Fieker, Claus and Kohel, David R.}, 1646 | volume = {2369}, 1647 | series = {Lecture Notes in Computer Science}, 1648 | pages = {47--62}, 1649 | address = {Berlin, Heidelberg}, 1650 | publisher = {Springer Berlin / Heidelberg}, 1651 | abstract = {Recently, Kohel gave algorithms to compute the 1652 | conductor of the endomorphism ring of an ordinary 1653 | elliptic curve, given the cardinality of the curve. 1654 | Using his work, we give a complete description of the 1655 | structure of curves related via rational ℓ-degree 1656 | isogenies, a structure we call a volcano. We explain 1657 | how we can travel through this structure using 1658 | modular polynomials. The computation of the structure 1659 | is possible without knowing the cardinality of the 1660 | curve, and that as a result, we deduce information on 1661 | the cardinality.}, 1662 | chapter = {23}, 1663 | doi = {10.1007/3-540-45455-1_23}, 1664 | isbn = {978-3-540-43863-2}, 1665 | } 1666 | 1667 | @Book{galbraith2012mathematics, 1668 | title = {Mathematics of public key cryptography}, 1669 | publisher = {Cambridge University Press}, 1670 | year = {2012}, 1671 | author = {Galbraith, Steven D.}, 1672 | note = {\url{https://www.math.auckland.ac.nz/~sgal018/crypto-book/crypto-book.html}}, 1673 | } 1674 | 1675 | @InProceedings{gallant+lambert+vanstone01, 1676 | author = {Gallant, Robert P. and Lambert, Robert J. and Vanstone, Scott A.}, 1677 | title = {Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms}, 1678 | booktitle = {CRYPTO '01: Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology}, 1679 | year = {2001}, 1680 | pages = {190--200}, 1681 | address = {London, UK}, 1682 | publisher = {Springer-Verlag}, 1683 | abstract = {The fundamental operation in elliptic curve 1684 | cryptographic schemes is that of point multiplication 1685 | of an elliptic curve point by an integer. This paper 1686 | describes a new method for accelerating this 1687 | operation on classes of elliptic curves that have 1688 | efficiently-computable endomorphisms. One advantage 1689 | of the new method is that it is applicable to a 1690 | larger class of curves than previous such methods.}, 1691 | isbn = {3-540-42456-3}, 1692 | url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.2004}, 1693 | } 1694 | 1695 | @Manual{GAP4, 1696 | title = {{GAP -- Groups, Algorithms, and Programming, Version 4.9.2}}, 1697 | organization = {The GAP~Group}, 1698 | year = {2018}, 1699 | key = {GAP}, 1700 | shorthand = {GAP}, 1701 | url = {https://www.gap-system.org}, 1702 | } 1703 | 1704 | @PhdThesis{gaudry2000algorithmique, 1705 | author = {Gaudry, Pierrick}, 1706 | title = {Algorithmique des courbes hyperelliptiques et applications {\`a} la cryptologie}, 1707 | school = {Ecole Polytechnique}, 1708 | year = {2000}, 1709 | institution = {École Polytechnique}, 1710 | url = {https://hal-polytechnique.archives-ouvertes.fr/tel-00514848/}, 1711 | } 1712 | 1713 | @Book{gauss1986disquisitiones, 1714 | title = {Disquisitiones Arithmeticae}, 1715 | publisher = {Springer-Verlag}, 1716 | year = {1986}, 1717 | author = {Gauss, Carl Friedrich}, 1718 | editor = {Waterhouse, William C.}, 1719 | isbn = {9783540962540}, 1720 | url = {https://books.google.fr/books?id=Y-49PgAACAAJ}, 1721 | } 1722 | 1723 | @Article{giusti+lecerf+salvy01, 1724 | author = {Giusti, Marc and Lecerf, Gr\'{e}goire and Salvy, Bruno}, 1725 | title = {A {G}r\"{o}bner free alternative for polynomial system solving}, 1726 | journal = {Journal of Complexity}, 1727 | year = {2001}, 1728 | volume = {17}, 1729 | number = {1}, 1730 | pages = {154--211}, 1731 | month = mar, 1732 | issn = {0885-064X}, 1733 | abstract = {Given a system of polynomial equations and 1734 | inequations with coefficients in the field of 1735 | rational numbers, we show how to compute a geometric 1736 | resolution of the set of common roots of the system 1737 | over the field of complex numbers. A geometric 1738 | resolution consists of a primitive element of the 1739 | algebraic extension defined by the set of roots, its 1740 | minimal polynomial, and the parametrizations of the 1741 | coordinates. Such a representation of the solutions 1742 | has a long history which goes back to Leopold 1743 | Kronecker and has been revisited many times in 1744 | computer algebra. We introduce a new generation of 1745 | probabilistic algorithms where all the computations 1746 | use only univariate or bivariate polynomials. We give 1747 | a new codification of the set of solutions of a 1748 | positive dimensional algebraic variety relying on a 1749 | new global version of Newton's iterator. Roughly 1750 | speaking the complexity of our algorithm is 1751 | polynomial in some kind of degree of the system, in 1752 | its height, and linear in the complexity of 1753 | evaluation of the system. We present our 1754 | implementation in the Magma system which is called 1755 | Kronecker in homage to his method for solving systems 1756 | of polynomial equations. We show that the theoretical 1757 | complexity of our algorithm is well reflected in 1758 | practice and we exhibit some cases for which our 1759 | program is more efficient than the other available 1760 | software.}, 1761 | address = {Orlando, FL, USA}, 1762 | doi = {10.1006/jcom.2000.0571}, 1763 | publisher = {Academic Press, Inc.}, 1764 | } 1765 | 1766 | @Manual{givaro, 1767 | title = {Givaro -- {C++} library for arithmetic and algebraic computations}, 1768 | organization = {{The LinBox Team}}, 1769 | key = {Givaro}, 1770 | shorthand = {Giv}, 1771 | url = {https://github.com/linbox-team/givaro}, 1772 | } 1773 | 1774 | @Misc{GMP-ECM, 1775 | author = {Paul Zimmermann and others}, 1776 | title = {{GMP-ECM} software}, 1777 | year = {2018}, 1778 | url = {http://ecm.gforge.inria.fr/}, 1779 | } 1780 | 1781 | @Article{goldreich+micali+widgerson91, 1782 | author = {Goldreich, Oded and Micali, Silvio and Wigderson, Avi}, 1783 | title = {Proofs that yield nothing but their validity or all languages in {NP} have zero-knowledge proof systems}, 1784 | journal = {Journal of the Association for Computing Machinery}, 1785 | year = {1991}, 1786 | volume = {38}, 1787 | number = {3}, 1788 | pages = {690--728}, 1789 | month = jul, 1790 | issn = {0004-5411}, 1791 | abstract = {An abstract is not available.}, 1792 | address = {New York, NY, USA}, 1793 | doi = {10.1145/116825.116852}, 1794 | publisher = {ACM}, 1795 | } 1796 | 1797 | @InBook{Goldreich2011, 1798 | pages = {451--464}, 1799 | title = {Basic Facts about Expander Graphs}, 1800 | publisher = {Springer Berlin Heidelberg}, 1801 | year = {2011}, 1802 | author = {Goldreich, Oded}, 1803 | editor = {Goldreich, Oded}, 1804 | address = {Berlin, Heidelberg}, 1805 | isbn = {978-3-642-22670-0}, 1806 | abstract = {In this survey we review basic facts regarding 1807 | expander graphs that are most relevant to the theory 1808 | of computation.}, 1809 | booktitle = {Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation: In Collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman}, 1810 | doi = {10.1007/978-3-642-22670-0_30}, 1811 | } 1812 | 1813 | @Article{hanrot+quercia+zimmermann, 1814 | author = {Hanrot, Guillaume and Quercia, Michel and Zimmermann, Paul}, 1815 | title = {The Middle Product Algorithm {I}}, 1816 | journal = {Applicable Algebra in Engineering, Communication and Computing}, 1817 | year = {2004}, 1818 | volume = {14}, 1819 | number = {6}, 1820 | pages = {415--438}, 1821 | issn = {0938-1279}, 1822 | abstract = {We present new algorithms for the inverse, division, 1823 | and square root of power series. The key trick is a 1824 | new algorithm – {MiddleProduct} or, for short, {MP} 1825 | – computing the n middle coefficients of a (2 1826 | n-1)× n full product in the same number of 1827 | multiplications as a full n× n product. This 1828 | improves previous work of Brent, Mulders, Karp and 1829 | Markstein, Burnikel and Ziegler. These results apply 1830 | both to series and polynomials.}, 1831 | doi = {10.1007/s00200-003-0144-2}, 1832 | } 1833 | 1834 | @InProceedings{Hart2010, 1835 | author = {William B. Hart}, 1836 | title = {Fast Library for Number Theory: An Introduction}, 1837 | booktitle = {Proceedings of the Third International Congress on Mathematical Software}, 1838 | year = {2010}, 1839 | series = {ICMS'10}, 1840 | pages = {88--91}, 1841 | address = {Berlin, Heidelberg}, 1842 | publisher = {Springer-Verlag}, 1843 | location = {Kobe, Japan}, 1844 | numpages = {4}, 1845 | url = {http://flintlib.org/}, 1846 | } 1847 | 1848 | @Article{harvey09, 1849 | author = {Harvey, David}, 1850 | title = {Faster polynomial multiplication via multipoint {K}ronecker substitution}, 1851 | journal = {Journal of Symbolic Computation}, 1852 | year = {2009}, 1853 | volume = {44}, 1854 | number = {10}, 1855 | pages = {1502--1510}, 1856 | month = oct, 1857 | issn = {07477171}, 1858 | abstract = {We present several new algorithms for dense 1859 | polynomial multiplication in {inlMMLBox} based on the 1860 | Kronecker substitution method. Instead of reducing to 1861 | a single integer multiplication, we reduce to several 1862 | smaller multiplications. We describe an 1863 | implementation of multiplication in {inlMMLBox} for a 1864 | word-sized modulus n based on these methods, and 1865 | compare its performance to that of {NTL} and Magma.}, 1866 | doi = {10.1016/j.jsc.2009.05.004}, 1867 | } 1868 | 1869 | @InProceedings{heath+loehr99, 1870 | author = {Heath, Lenwood S. and Loehr, Nicholas A.}, 1871 | title = {New algorithms for generating {C}onway polynomials over finite fields}, 1872 | booktitle = {Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms}, 1873 | year = {1999}, 1874 | series = {SODA '99}, 1875 | pages = {429--437}, 1876 | address = {Philadelphia, PA, USA}, 1877 | publisher = {Society for Industrial and Applied Mathematics}, 1878 | abstract = {An abstract is not available.}, 1879 | isbn = {0-89871-434-6}, 1880 | } 1881 | 1882 | @InProceedings{heath1992zero, 1883 | author = {Heath-Brown, David R.}, 1884 | title = {Zero-free regions for {Dirichlet} {L-functions}, and the least prime in an arithmetic progression}, 1885 | booktitle = {Proceedings of the London Mathematical Society}, 1886 | year = {1992}, 1887 | volume = {64}, 1888 | number = {2}, 1889 | pages = {265--338}, 1890 | } 1891 | 1892 | @PhdThesis{hugounenq:tel-01635463, 1893 | author = {Hugounenq, Cyril}, 1894 | title = {Volcanoes and isogeny computing}, 1895 | school = {{Universit{\'e} Paris-Saclay}}, 1896 | year = {2017}, 1897 | month = sep, 1898 | number = {2017SACLV050}, 1899 | url = {https://tel.archives-ouvertes.fr/tel-01635463}, 1900 | } 1901 | 1902 | @Article{ionica+joux13, 1903 | author = {Ionica, Sorina and Joux, Antoine}, 1904 | title = {Pairing the volcano}, 1905 | journal = {Mathematics of Computation}, 1906 | year = {2013}, 1907 | volume = {82}, 1908 | number = {281}, 1909 | pages = {581--603}, 1910 | } 1911 | 1912 | @Article{ionica2014isogeny, 1913 | author = {Ionica, Sorina and Thom{\'e}, Emmanuel}, 1914 | title = {Isogeny graphs with maximal real multiplication}, 1915 | journal = {arXiv preprint arXiv:1407.6672}, 1916 | year = {2014}, 1917 | } 1918 | 1919 | @Inprocedings{joux, 1920 | author = {Joux, Antoine}, 1921 | title = {The {W}eil and {T}ate pairings as building blocks for public key cryptosystems}, 1922 | year = {2002}, 1923 | address = {Berlin}, 1924 | booktitle = {Algorithmic number theory}, 1925 | doi = {10.1007/3-540-45455-1_3}, 1926 | pages = {20--32}, 1927 | publisher = {Springer}, 1928 | series = {Lecture Notes in Computer Science}, 1929 | volume = {2369}, 1930 | } 1931 | 1932 | @InProceedings{kaltofen+shoup97, 1933 | author = {Kaltofen, Erich and Shoup, Victor}, 1934 | title = {Fast polynomial factorization over high algebraic extensions of finite fields}, 1935 | booktitle = {ISSAC '97: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation}, 1936 | year = {1997}, 1937 | pages = {184--188}, 1938 | address = {New York, NY, USA}, 1939 | publisher = {ACM}, 1940 | abstract = {Note: {OCR} errors may be found in this Reference 1941 | List extracted from the full text article. {ACM} has 1942 | opted to expose the complete List rather than only 1943 | correct and linked references.}, 1944 | doi = {10.1145/258726.258777}, 1945 | isbn = {0-89791-875-4}, 1946 | } 1947 | 1948 | @Article{kaltofen+shoup98, 1949 | author = {Kaltofen, Erich and Shoup, Victor}, 1950 | title = {Subquadratic-time factoring of polynomials over finite fields}, 1951 | journal = {Mathematics of Computation}, 1952 | year = {1998}, 1953 | volume = {67}, 1954 | number = {223}, 1955 | pages = {1179--1197}, 1956 | issn = {0025-5718}, 1957 | address = {Boston, MA, USA}, 1958 | doi = {10.1090/S0025-5718-98-00944-2}, 1959 | publisher = {American Mathematical Society}, 1960 | } 1961 | 1962 | @Article{kaltofen87, 1963 | author = {Kaltofen, Erich}, 1964 | title = {Computer algebra algorithms}, 1965 | journal = {Annual Review in Computer Science}, 1966 | year = {1987}, 1967 | volume = {2}, 1968 | pages = {91--118}, 1969 | address = {Palo Alto, California}, 1970 | publisher = {Annual Reviews Inc.}, 1971 | url = {http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf}, 1972 | } 1973 | 1974 | @Article{kedlaya01, 1975 | author = {Kedlaya, Kiran S.}, 1976 | title = {Counting points on hyperelliptic curves using {M}onsky-{W}ashnitzer cohomology}, 1977 | journal = {Journal of the Ramanujan Mathematical Society}, 1978 | year = {2001}, 1979 | volume = {16}, 1980 | number = {4}, 1981 | pages = {323--338}, 1982 | abstract = {In this paper an algorithm for counting points on 1983 | hyperelliptic curvesover finite fields of odd 1984 | characteristic is developed.

The approach is to 1985 | compute in the Monsky-Washnitzer cohomology,described 1986 | in sections 2 and 3, of an affine curve endowed with 1987 | anaction of Frobenius which can be \$p\$-adically 1988 | approximated efficientlyusing certain power series. 1989 | The running time of the algorithm is onthe order of 1990 | \$g^{4+\epsilon}n^{3+\epsilon}\$, where \$n\$ is the 1991 | degreeof the finite field and \$g\$ is the genus of 1992 | the curve (assuming thatthe curve has a rational 1993 | Weierstrass point).}, 1994 | } 1995 | 1996 | @InCollection{kedlaya04, 1997 | author = {Kedlaya, Kiran S.}, 1998 | title = {Computing zeta functions via \(p\)-adic cohomology}, 1999 | booktitle = {Algorithmic number theory}, 2000 | publisher = {Springer}, 2001 | year = {2004}, 2002 | volume = {3076}, 2003 | series = {Lecture Notes in Comput. Sci.}, 2004 | pages = {1--17}, 2005 | address = {Berlin}, 2006 | } 2007 | 2008 | @Article{KeUm11, 2009 | author = {Kedlaya, Kiran S and Umans, Christopher}, 2010 | title = {Fast polynomial factorization and modular composition}, 2011 | journal = {SIAM Journal on Computing}, 2012 | year = {2011}, 2013 | volume = {40}, 2014 | number = {6}, 2015 | pages = {1767--1802}, 2016 | publisher = {SIAM}, 2017 | } 2018 | 2019 | @Article{kitaev1995hsp, 2020 | author = {Kitaev, Alexey Yuri}, 2021 | title = {Quantum measurements and the Abelian stabilizer problem}, 2022 | journal = {arXiv preprint quant-ph/9511026}, 2023 | year = {1995}, 2024 | url = {https://arxiv.org/abs/quant-ph/9511026}, 2025 | } 2026 | 2027 | @InProceedings{kocher+jaffe+jun99, 2028 | author = {Kocher, Paul and Jaffe, Joshua and Jun, Benjamin}, 2029 | title = {{Differential Power Analysis}}, 2030 | booktitle = {Advances in Cryptology --- CRYPTO' 99}, 2031 | year = {1999}, 2032 | volume = {1666}, 2033 | series = {Lecture Notes in Computer Science}, 2034 | pages = {388--397}, 2035 | address = {Berlin, Heidelberg}, 2036 | month = dec, 2037 | publisher = {Springer Berlin Heidelberg}, 2038 | abstract = {{Cryptosystem designers frequently assume that 2039 | secrets will be manipulated in closed, reliable 2040 | computing environments. Unfortunately, actual 2041 | computers and microchips leak information about the 2042 | operations they process. This paper examines specific 2043 | methods for analyzing power consumption measurements 2044 | to find secret keys from tamper resistant devices. We 2045 | also discuss approaches for building cryptosystems 2046 | that can operate securely in existing hardware that 2047 | leaks information.}}, 2048 | chapter = {25}, 2049 | doi = {10.1007/3-540-48405-1_25}, 2050 | isbn = {978-3-540-66347-8}, 2051 | issn = {0302-9743}, 2052 | } 2053 | 2054 | @Article{Kummer1846, 2055 | author = {Kummer, Ernst Eduard}, 2056 | title = {{\"Uber die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung entstehen}}, 2057 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2058 | year = {1846}, 2059 | volume = {30}, 2060 | pages = {107--116}, 2061 | language = {German}, 2062 | url = {http://eudml.org/doc/147278}, 2063 | } 2064 | 2065 | @Article{Kummer1847a, 2066 | author = {Kummer, Ernst Eduard}, 2067 | title = {{\"Uber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren}}, 2068 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2069 | year = {1847}, 2070 | volume = {35}, 2071 | pages = {327--367}, 2072 | language = {German}, 2073 | url = {http://eudml.org/doc/147394}, 2074 | } 2075 | 2076 | @Article{Kummer1847b, 2077 | author = {Kummer, Ernst Eduard}, 2078 | title = {Sur les nombres complexes qui sont form\'es avec les nombres entiers r\'eels et les racines de l'unit\'e}, 2079 | journal = {Journal de Math\'ematiques Pures et Appliqu\'ees}, 2080 | year = {1847}, 2081 | pages = {185--212}, 2082 | language = {French}, 2083 | url = {http://eudml.org/doc/235075}, 2084 | } 2085 | 2086 | @Article{Kummer1847c, 2087 | author = {Kummer, Ernst Eduard}, 2088 | title = {{Zur Theorie der complexen Zahlen}}, 2089 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2090 | year = {1847}, 2091 | volume = {35}, 2092 | pages = {319--326}, 2093 | language = {German}, 2094 | url = {http://eudml.org/doc/147393}, 2095 | } 2096 | 2097 | @Article{Kummer1851, 2098 | author = {Kummer, Ernst Eduard}, 2099 | title = {M\'emoire sur la th\'eorie des nombres complexes compos\'es de racines de l'unit\'e et de nombres entiers}, 2100 | journal = {Journal de Math\'ematiques Pures et Appliqu\'ees}, 2101 | year = {1851}, 2102 | pages = {377--498}, 2103 | language = {French}, 2104 | url = {http://eudml.org/doc/235621}, 2105 | } 2106 | 2107 | @Article{Kummer1855, 2108 | author = {Kummer, Ernst Eduard}, 2109 | title = {{\"Uber eine besondere Art, aus complexen Einheiten gebildeter Ausdr\"ucke}}, 2110 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2111 | year = {1855}, 2112 | volume = {50}, 2113 | pages = {212--232}, 2114 | language = {German}, 2115 | url = {http://eudml.org/doc/147605}, 2116 | } 2117 | 2118 | @Article{Kummer1857, 2119 | author = {Kummer, Ernst Eduard}, 2120 | title = {{\"Uber die den Gau\ss{}schen Perioden der Kreistheilung entsprechenden Congruenzwurzeln}}, 2121 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2122 | year = {1857}, 2123 | volume = {53}, 2124 | pages = {142--148}, 2125 | language = {German}, 2126 | url = {http://eudml.org/doc/147659}, 2127 | } 2128 | 2129 | @Book{Kunz86, 2130 | title = {K\"ahler differentials}, 2131 | publisher = {Friedrich Vieweg \& Sohn}, 2132 | year = {1986}, 2133 | author = {Kunz, Ernst}, 2134 | } 2135 | 2136 | @InProceedings{lairez-vaccon, 2137 | author = {Lairez, Pierre and Vaccon, Tristan}, 2138 | title = {On $p$-adic Differential Equations with Separation of Variables}, 2139 | booktitle = {Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation}, 2140 | year = {2016}, 2141 | series = {ISSAC '16}, 2142 | pages = {319--323}, 2143 | address = {New York, NY, USA}, 2144 | publisher = {ACM}, 2145 | doi = {10.1145/2930889.2930912}, 2146 | isbn = {978-1-4503-4380-0}, 2147 | } 2148 | 2149 | @Book{lang, 2150 | title = {Algebra}, 2151 | publisher = {Springer}, 2152 | year = {2002}, 2153 | author = {Lang, Serge}, 2154 | edition = {3rd edition}, 2155 | month = jan, 2156 | isbn = {038795385X}, 2157 | abstract = {{"Lang's Algebra changed the way graduate algebra is 2158 | taught, retaining classical topics but introducing 2159 | language and ways of thinking from category theory 2160 | and homological algebra. It has affected all 2161 | subsequent graduate-level algebra books." 2162 | \ NOTICES OF THE AMS "The author has an 2163 | impressive knack for presenting the important and 2164 | interesting ideas of algebra in just the right way, 2165 | and he never gets bogged down in the dry formalism 2166 | which pervades some parts of 2167 | algebra."\ MATHEMATICAL REVIEWS This book is 2168 | intended as a basic text for a one-year course in 2169 | algebra at the graduate level, or as a useful 2170 | reference for mathematicians and professionals who 2171 | use higher-level algebra. It successfully addresses 2172 | the basic concepts of algebra. \ For the revised 2173 | third edition, the author has added exercises and 2174 | made numerous corrections to the text.}}, 2175 | howpublished = {Hardcover}, 2176 | } 2177 | 2178 | @Article{lauder04, 2179 | author = {Lauder, Alan G. B.}, 2180 | title = {Computing zeta functions of {A}rtin-{S}chreier curves over finite fields {II}}, 2181 | journal = {Journal of Complexity}, 2182 | year = {2004}, 2183 | volume = {20}, 2184 | number = {2-3}, 2185 | pages = {331--349}, 2186 | month = jun, 2187 | issn = {0885064X}, 2188 | abstract = {We describe a method which may be used to compute the 2189 | zeta function of an arbitrary {Artin-Schreier} cover 2190 | of the projective line over a finite field. 2191 | Specifically, for covers defined by equations of the 2192 | form Z p − Z = f ( X ) we present, and give the 2193 | complexity analysis of, an algorithm for the case in 2194 | which f ( X ) is a rational function whose poles all 2195 | have order 1. However, we only prove the correctness 2196 | of this algorithm when the field characteristic is at 2197 | least 5. The algorithm is based upon a cohomological 2198 | formula for the L -function of an additive character 2199 | sum. One consequence is a practical method of finding 2200 | the order of the group of rational points on the 2201 | Jacobian of a hyperelliptic curve in characteristic 2202 | 2.}, 2203 | doi = {10.1016/j.jco.2003.08.009}, 2204 | } 2205 | 2206 | @Article{LEBRETON2015230, 2207 | author = {Romain Lebreton}, 2208 | title = {Relaxed {H}ensel lifting of triangular sets}, 2209 | journal = {Journal of Symbolic Computation}, 2210 | year = {2015}, 2211 | volume = {68}, 2212 | pages = {230--258}, 2213 | issn = {0747-7171}, 2214 | note = {Effective Methods in Algebraic Geometry}, 2215 | abstract = {In this paper, we present a new lifting algorithm for 2216 | triangular sets over general p-adic rings. Our 2217 | contribution is to give, for any p-adic triangular 2218 | set, a shifted algorithm of which the triangular set 2219 | is a fixed point. Then we can apply the relaxed 2220 | recursive p-adic framework and deduce a relaxed 2221 | lifting algorithm for this triangular set. We compare 2222 | our algorithm to the existing technique and report on 2223 | implementations inside the C++ library Geomsolvex of 2224 | Mathemagix (van der Hoeven et al., 2002). Our new 2225 | relaxed algorithm is competitive and compare 2226 | favorably on some examples.}, 2227 | doi = {10.1016/j.jsc.2014.09.012}, 2228 | } 2229 | 2230 | @InProceedings{LeGall14, 2231 | author = {Le Gall, Fran\c{c}ois}, 2232 | title = {Powers of Tensors and Fast Matrix Multiplication}, 2233 | booktitle = {ISSAC'14}, 2234 | year = {2014}, 2235 | pages = {296--303}, 2236 | publisher = {ACM}, 2237 | } 2238 | 2239 | @InProceedings{LeMeSc13, 2240 | author = {Lebreton, Romain and Mehrabi, Esmaeil and Schost, \'Eric}, 2241 | title = {On the Complexity of Solving Bivariate Systems: The Case of Non-singular Solutions}, 2242 | booktitle = {ISSAC'13}, 2243 | year = {2013}, 2244 | pages = {251--258}, 2245 | publisher = {ACM}, 2246 | } 2247 | 2248 | @Electronic{lenstra+desmit08-stdmodels, 2249 | author = {Lenstra, Hendrik W. and de Smit, Bart}, 2250 | year = {2008}, 2251 | title = {Standard models for finite fields: the definition}, 2252 | url = {http://www.math.leidenuniv.nl/~desmit/papers/standard_models.pdf}, 2253 | pages = {1--4}, 2254 | } 2255 | 2256 | @Article{LENSTRA1977389, 2257 | author = {Lenstra, Hendrik W.}, 2258 | title = {On the algebraic closure of two}, 2259 | journal = {Indagationes Mathematicae (Proceedings)}, 2260 | year = {1977}, 2261 | volume = {80}, 2262 | number = {5}, 2263 | pages = {389--396}, 2264 | issn = {1385-7258}, 2265 | doi = {10.1016/1385-7258(77)90053-1}, 2266 | } 2267 | 2268 | @Article{lenstra87, 2269 | author = {Lenstra, Hendrik W.}, 2270 | title = {Factoring integers with elliptic curves}, 2271 | journal = {Annals of Mathematics}, 2272 | year = {1987}, 2273 | volume = {126}, 2274 | pages = {649--673}, 2275 | } 2276 | 2277 | @Article{LenstraJr91, 2278 | author = {Lenstra, Hendrik W.}, 2279 | title = {Finding isomorphisms between finite fields}, 2280 | journal = {Mathematics of Computation}, 2281 | year = {1991}, 2282 | volume = {56}, 2283 | number = {193}, 2284 | pages = {329--347}, 2285 | } 2286 | 2287 | @Article{lercier+sirvent08, 2288 | author = {Lercier, Reynald and Sirvent, Thomas}, 2289 | title = {On {E}lkies subgroups of \(\ell\)-torsion points in elliptic curves defined over a finite field}, 2290 | journal = {Journal de th\'{e}orie des nombres de Bordeaux}, 2291 | year = {2008}, 2292 | volume = {20}, 2293 | number = {3}, 2294 | pages = {783--797}, 2295 | url = {http://perso.univ-rennes1.fr/reynald.lercier/file/LS08.pdf}, 2296 | } 2297 | 2298 | @InProceedings{lercier96, 2299 | author = {Lercier, Reynald}, 2300 | title = {Computing Isogenies in {GF(2,n)}}, 2301 | booktitle = {ANTS-II: Proceedings of the Second International Symposium on Algorithmic Number Theory}, 2302 | year = {1996}, 2303 | pages = {197--212}, 2304 | address = {London, UK}, 2305 | publisher = {Springer-Verlag}, 2306 | isbn = {3-540-61581-4}, 2307 | } 2308 | 2309 | @InProceedings{li+moreno+schost07, 2310 | author = {Li, Xin and Moreno Maza, Marc and Schost, \'{E}ric}, 2311 | title = {Fast Arithmetic for Triangular Sets: From Theory to Practice}, 2312 | booktitle = {Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation}, 2313 | year = {2007}, 2314 | series = {ISSAC '07}, 2315 | pages = {269--276}, 2316 | address = {New York, NY, USA}, 2317 | publisher = {ACM}, 2318 | abstract = {We study arithmetic operations for triangular 2319 | families of polynomials, concentrating on 2320 | multiplication in dimension zero. By a suitable 2321 | extension of fast univariate Euclidean division, we 2322 | obtain theoretical and practical improvements over a 2323 | direct recursive approach; for a family of special 2324 | cases, we reach quasi-linear complexity. The main 2325 | outcome we have in mind is the acceleration of 2326 | higher-level algorithms, by interfacing our low-level 2327 | implementation with languages such as {AXIOM} or 2328 | Maple We show the potential for huge speed-ups, by 2329 | comparing two {AXIOM} implementations of van Hoeij 2330 | and Monagan's modular {GCD} algorithm.}, 2331 | doi = {10.1145/1277548.1277585}, 2332 | isbn = {978-1-59593-743-8}, 2333 | } 2334 | 2335 | @Article{littlewood1928class, 2336 | author = {Littlewood, John E}, 2337 | title = {On the Class-Number of the Corpus $P(\sqrt{k})$}, 2338 | journal = {Proceedings of the London Mathematical Society}, 2339 | year = {1928}, 2340 | volume = {2}, 2341 | number = {1}, 2342 | pages = {358--372}, 2343 | publisher = {Wiley Online Library}, 2344 | } 2345 | 2346 | @InCollection{lo, 2347 | author = {Lagarias, Jeffrey C. and Odlyzko, Andrew M.}, 2348 | title = {Effective versions of the {C}hebotarev density theorem}, 2349 | booktitle = {Algebraic number fields: {$L$}-functions and {G}alois properties}, 2350 | publisher = {Academic Press}, 2351 | year = {1977}, 2352 | pages = {409--464}, 2353 | address = {London}, 2354 | } 2355 | 2356 | @Article{longa+sica14, 2357 | author = {Longa, Patrick and Sica, Francesco}, 2358 | title = {Four-Dimensional {G}allant--{L}ambert--{V}anstone Scalar Multiplication}, 2359 | journal = {Journal of Cryptology}, 2360 | year = {2014}, 2361 | volume = {27}, 2362 | number = {2}, 2363 | pages = {248--283}, 2364 | issn = {1432-1378}, 2365 | doi = {10.1007/s00145-012-9144-3}, 2366 | } 2367 | 2368 | @Book{Lub, 2369 | title = {Discrete groups, expanding graphs and invariant measures}, 2370 | publisher = {Birkh\"{a}user Verlag}, 2371 | year = {1994}, 2372 | author = {Lubotzky, Alexander}, 2373 | volume = {125}, 2374 | series = {Progress in Mathematics}, 2375 | address = {Basel}, 2376 | isbn = {978-3-0346-0332-4}, 2377 | doi = {10.1007/978-3-0346-0332-4}, 2378 | } 2379 | 2380 | @Article{lubicz_robert_2012, 2381 | author = {Lubicz, David and Robert, Damien}, 2382 | title = {Computing isogenies between abelian varieties}, 2383 | journal = {Compositio Mathematica}, 2384 | year = {2012}, 2385 | volume = {148}, 2386 | number = {5}, 2387 | pages = {1483--1515}, 2388 | doi = {10.1112/S0010437X12000243}, 2389 | publisher = {London Mathematical Society}, 2390 | } 2391 | 2392 | @Article{lubicz_robert_2015, 2393 | author = {Lubicz, David and Robert, Damien}, 2394 | title = {Computing separable isogenies in quasi-optimal time}, 2395 | journal = {LMS Journal of Computation and Mathematics}, 2396 | year = {2015}, 2397 | volume = {18}, 2398 | number = {1}, 2399 | pages = {198--216}, 2400 | doi = {10.1112/S146115701400045X}, 2401 | publisher = {London Mathematical Society}, 2402 | } 2403 | 2404 | @Article{LubPS, 2405 | author = {Lubotzky, Alexander and Phillips, Ralph and Sarnak, Peter}, 2406 | title = {Ramanujan graphs}, 2407 | journal = {Combinatorica}, 2408 | year = {1988}, 2409 | volume = {8}, 2410 | number = {3}, 2411 | doi = {10.1007/BF02126799}, 2412 | } 2413 | 2414 | @Electronic{Luebeck, 2415 | author = {Lübeck, Frank}, 2416 | year = {2008}, 2417 | title = {Conway polynomials for finite fields}, 2418 | url = {http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/}, 2419 | } 2420 | 2421 | @Article{MAGMA, 2422 | author = {Bosma, Wieb and Cannon, John and Playoust, Catherine}, 2423 | title = {The {MAGMA} algebra system {I}: the user language}, 2424 | journal = {Journal of Symbolic Computation}, 2425 | year = {1997}, 2426 | volume = {24}, 2427 | number = {3-4}, 2428 | pages = {235--265}, 2429 | issn = {0747-7171}, 2430 | address = {Duluth, MN, USA}, 2431 | doi = {10.1006/jsco.1996.0125}, 2432 | publisher = {Academic Press, Inc.}, 2433 | } 2434 | 2435 | @InProceedings{mauer+menezes+teske01, 2436 | author = {Maurer, Markus and Menezes, Alfred and Teske, Edlyn}, 2437 | title = {Analysis of the {GHS} {W}eil Descent Attack on the {ECDLP} over Characteristic Two Finite Fields of Composite Degree}, 2438 | booktitle = {INDOCRYPT '01: Proceedings of the Second International Conference on Cryptology in India}, 2439 | year = {2001}, 2440 | pages = {195--213}, 2441 | address = {Berlin}, 2442 | publisher = {Springer-Verlag}, 2443 | isbn = {3-540-43010-5}, 2444 | } 2445 | 2446 | @MastersThesis{memoire, 2447 | author = {Kieffer, Jean}, 2448 | title = {\'Etude et accélération du protocole d'échange de clés de Couveignes--Rostovtsev--Stolbunov}, 2449 | school = {Inria Saclay \& Université Paris VI}, 2450 | year = {2017}, 2451 | } 2452 | 2453 | @Article{Mihailescu2010825, 2454 | author = {Preda Mihailescu and Victor Vuletescu}, 2455 | title = {Elliptic Gauss sums and applications to point counting}, 2456 | journal = {Journal of Symbolic Computation}, 2457 | year = {2010}, 2458 | volume = {45}, 2459 | number = {8}, 2460 | pages = {825--836}, 2461 | issn = {0747-7171}, 2462 | doi = {10.1016/j.jsc.2010.01.004}, 2463 | } 2464 | 2465 | @Article{milio_2015, 2466 | author = {Milio, Enea}, 2467 | title = {A quasi-linear time algorithm for computing modular polynomials in dimension 2}, 2468 | journal = {LMS Journal of Computation and Mathematics}, 2469 | year = {2015}, 2470 | volume = {18}, 2471 | number = {1}, 2472 | pages = {603--632}, 2473 | doi = {10.1112/S1461157015000170}, 2474 | publisher = {London Mathematical Society}, 2475 | } 2476 | 2477 | @Unpublished{milio:hal-01520262, 2478 | author = {Milio, Enea and Robert, Damien}, 2479 | title = {Modular polynomials on {H}ilbert surfaces}, 2480 | note = {Working paper or preprint}, 2481 | month = sep, 2482 | year = {2017}, 2483 | url = {https://hal.archives-ouvertes.fr/hal-01520262}, 2484 | } 2485 | 2486 | @Article{MIRET200867, 2487 | author = {Miret, Josep M. and Moreno, Ramiro and Sadornil, Daniel and Tena, Juan and Valls, Magda}, 2488 | title = {Computing the height of volcanoes of ℓ-isogenies of elliptic curves over finite fields}, 2489 | journal = {Applied Mathematics and Computation}, 2490 | year = {2008}, 2491 | volume = {196}, 2492 | number = {1}, 2493 | pages = {67--76}, 2494 | issn = {0096-3003}, 2495 | abstract = {The structure of the volcano of ℓ-isogenies, 2496 | ℓ-prime, of elliptic curves over finite fields has 2497 | been extensively studied over recent years. Previous 2498 | works present some results and algorithms concerning 2499 | the height of such volcanoes in the case of isogenies 2500 | whose kernels are generated by a rational point. The 2501 | main goal of this paper is to extend such works to 2502 | the case of ℓ-isogenies whose kernels are defined 2503 | by a rational subgroup. In particular, the height of 2504 | such volcanoes is completely characterized and can be 2505 | computationally obtained.}, 2506 | doi = {10.1016/j.amc.2007.05.037}, 2507 | } 2508 | 2509 | @Article{MiretMRV05, 2510 | author = {Josep M. Miret and Ramiro Moreno and Ana Rio and Magda Valls}, 2511 | title = {Determining the 2-{S}ylow subgroup of an elliptic curve over a finite field}, 2512 | journal = {Mathematics of Computation}, 2513 | year = {2005}, 2514 | volume = {74}, 2515 | number = {249}, 2516 | pages = {411--427}, 2517 | doi = {10.1090/S0025-5718-04-01640-0}, 2518 | } 2519 | 2520 | @Article{MiretMSTV06, 2521 | author = {Josep M. Miret and Ramiro Moreno and Daniel Sadornil and Juan Tena and Magda Valls}, 2522 | title = {An algorithm to compute volcanoes of 2-isogenies of elliptic curves over finite fields}, 2523 | journal = {Applied Mathematics and Computation}, 2524 | year = {2006}, 2525 | volume = {176}, 2526 | number = {2}, 2527 | pages = {739--750}, 2528 | doi = {10.1016/j.amc.2005.10.020}, 2529 | } 2530 | 2531 | @Article{moenck76, 2532 | author = {Moenck, Robert T.}, 2533 | title = {Another polynomial homomorphism}, 2534 | journal = {Acta Informatica}, 2535 | year = {1976}, 2536 | volume = {6}, 2537 | number = {2}, 2538 | pages = {153--169}, 2539 | month = jun, 2540 | issn = {0001-5903}, 2541 | abstract = {The current proposals for applying the so called fast 2542 | {O(N} {logaN}) algorithms to multivariate polynomials 2543 | is that the univariate methods be applied 2544 | recursively, much in the way more conventional 2545 | algorithms are used. Since the size of the problems 2546 | is rather large for which a fastrd algorithm is more 2547 | efficient than a classical one, the recursive 2548 | approach compounds this size completely out of any 2549 | practical range .}, 2550 | doi = {10.1007/BF00268498}, 2551 | publisher = {Springer Berlin / Heidelberg}, 2552 | } 2553 | 2554 | @Article{monico2007, 2555 | author = {Maze, G\'erard and Monico, Chris and Rosenthal, Joachim}, 2556 | title = {Public key cryptography based on semigroup actions}, 2557 | journal = {Advances in Mathematics of Communications}, 2558 | year = {2007}, 2559 | volume = {1}, 2560 | number = {4}, 2561 | pages = {489--507}, 2562 | issn = {1930-5346}, 2563 | doi = {10.3934/amc.2007.1.489}, 2564 | } 2565 | 2566 | @Article{montgomery, 2567 | author = {Montgomery, Peter L.}, 2568 | title = {Speeding the Pollard and Elliptic Curve Methods of Factorization}, 2569 | journal = {Mathematics of Computation}, 2570 | year = {1987}, 2571 | volume = {48}, 2572 | number = {177}, 2573 | pages = {243--264}, 2574 | issn = {00255718}, 2575 | abstract = {Since 1974, several algorithms have been developed 2576 | that attempt to factor a large number \$N\$ by doing 2577 | extensive computations modulo \$N\$ and occasionally 2578 | taking {GCDs} with \$N\$. These began with Pollard's 2579 | \$p - 1\$ and Monte Carlo methods. More recently, 2580 | Williams published a \$p + 1\$ method, and Lenstra 2581 | discovered an elliptic curve method ({ECM}). We 2582 | present ways to speed all of these. One improvement 2583 | uses two tables during the second phases of \$p \pm 2584 | 1\$ and {ECM}, looking for a match. Polynomial 2585 | preconditioning lets us search a fixed table of size 2586 | \$n\$ with \$n/2 + o(n)\$ multiplications. A 2587 | parametrization of elliptic curves lets Step 1 of 2588 | {ECM} compute the \$x\$-coordinate of \${nP}\$ from 2589 | that of \$P\$ in about \$9.3 \log\_2 n\$ 2590 | multiplications for arbitrary \$P\$.}, 2591 | doi = {10.2307/2007888}, 2592 | publisher = {American Mathematical Society}, 2593 | } 2594 | 2595 | @Article{MOODY20125249, 2596 | author = {Moody, Dustin}, 2597 | title = {Computing isogeny volcanoes of composite degree}, 2598 | journal = {Applied Mathematics and Computation}, 2599 | year = {2012}, 2600 | volume = {218}, 2601 | number = {9}, 2602 | pages = {5249--5258}, 2603 | issn = {0096-3003}, 2604 | abstract = {Isogeny volcanoes are an interesting structure that 2605 | have had several recent applications. An isogeny 2606 | volcano is a connected component of a larger graph 2607 | called a cordillera. In this paper, we further 2608 | explore properties of how to compute volcanoes given 2609 | that we have already computed one of a different 2610 | degree. This allows us to compute volcanoes of 2611 | composite degree more efficiently than a direct 2612 | construction using modular polynomials.}, 2613 | doi = {10.1016/j.amc.2011.11.008}, 2614 | } 2615 | 2616 | @Article{morain95, 2617 | author = {Morain, Fran\c{c}ois}, 2618 | title = {Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques}, 2619 | journal = {Journal de Th\'eorie des Nombres Bordeaux}, 2620 | year = {1995}, 2621 | volume = {7}, 2622 | number = {1}, 2623 | pages = {255--282}, 2624 | issn = {1246-7405}, 2625 | note = {Les Dix-huiti\`emes Journ\'ees Arithm\'etiques (Bordeaux, 1993)}, 2626 | url = {http://jtnb.cedram.org/item?id=JTNB_1995__7_1_255_0}, 2627 | } 2628 | 2629 | @Book{mullen2013handbook, 2630 | title = {Handbook of finite fields}, 2631 | publisher = {CRC Press}, 2632 | year = {2013}, 2633 | author = {Mullen, Gary L. and Panario, Daniel}, 2634 | } 2635 | 2636 | @InProceedings{narayanan2016fast, 2637 | author = {Narayanan, Anand Kumar}, 2638 | title = {Fast Computation of Isomorphisms Between Finite Fields Using Elliptic Curves}, 2639 | booktitle = {{International Workshop on the Arithmetic of Finite Fields, WAIFI 2018}}, 2640 | year = {2018}, 2641 | series = {Lecture Notes in Computer Science}, 2642 | publisher = {Springer Berlin / Heidelberg}, 2643 | } 2644 | 2645 | @Electronic{Nickel1988, 2646 | author = {Nickel, Werner}, 2647 | year = {1988}, 2648 | title = {{E}ndliche {K}örper in dem gruppentheoretischen {P}rogrammsystem {GAP}}, 2649 | url = {https://www2.mathematik.tu-darmstadt.de/~nickel/}, 2650 | institution = {RWTH Aachen}, 2651 | type = {mathesis}, 2652 | } 2653 | 2654 | @Electronic{NIST2016, 2655 | author = {{National Institute of Standards and Technology}}, 2656 | year = {2016}, 2657 | title = {Announcing Request for Nominations for Public-Key Post-Quantum Cryptographic Algorithms}, 2658 | organization = {National Institute of Standards and Technology}, 2659 | url = {https://www.federalregister.gov/documents/2016/12/20/2016-30615/announcing-request-for-nominations-for-public-key-post-quantum-cryptographic-algorithms}, 2660 | editor = {The Federal Register}, 2661 | } 2662 | 2663 | @Article{Noether1932, 2664 | author = {Noether, Emmy}, 2665 | title = {{Normalbasis bei K\"orpern ohne h\"ohere Verzweigung}}, 2666 | journal = {Journal f\"ur die reine und angewandte Mathematik}, 2667 | year = {1932}, 2668 | volume = {167}, 2669 | pages = {147--152}, 2670 | url = {http://eudml.org/doc/149800}, 2671 | } 2672 | 2673 | @Misc{oeis, 2674 | author = {{OEIS Foundation Inc.}}, 2675 | title = {The On-Line Encyclopedia of Integer Sequences}, 2676 | howpublished = {\url{http://oeis.org/A130715}}, 2677 | year = {2012}, 2678 | } 2679 | 2680 | @InProceedings{OKS00, 2681 | author = {Katsuyuki Okeya and Hiroyuki Kurumatani and Kouichi Sakurai}, 2682 | title = {Elliptic Curves with the {Montgomery}-Form and Their Cryptographic Applications}, 2683 | booktitle = {Public Key Cryptography --- {PKC} 2000}, 2684 | year = {2000}, 2685 | editor = {Hideki Imai and Yuliang Zheng}, 2686 | volume = {1751}, 2687 | series = {Lecture Notes in Computer Science}, 2688 | pages = {238--257}, 2689 | publisher = {Springer}, 2690 | biburl = {http://dblp.uni-trier.de/rec/bib/conf/pkc/OkeyaS00}, 2691 | doi = {10.1007/978-3-540-46588-1\_17}, 2692 | isbn = {3-540-66967-1}, 2693 | } 2694 | 2695 | @Manual{Pari, 2696 | title = {{PARI/GP, version {\texttt{2.8.0}}}}, 2697 | organization = {{The PARI Group}}, 2698 | address = {Bordeaux}, 2699 | year = {2016}, 2700 | key = {PARI}, 2701 | shorthand = {PARI}, 2702 | url = {https://pari.math.u-bordeaux.fr/}, 2703 | } 2704 | 2705 | @InProceedings{pascal+schost06, 2706 | author = {Pascal, Cyril and Schost, \'{E}ric}, 2707 | title = {Change of order for bivariate triangular sets}, 2708 | booktitle = {ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation}, 2709 | year = {2006}, 2710 | pages = {277--284}, 2711 | address = {New York, NY, USA}, 2712 | publisher = {ACM}, 2713 | abstract = {Changing the order of variables in bivariate 2714 | triangular sets has applications in Trager's 2715 | factorization algorithm, or in rational function 2716 | integration. We discuss the complexity of this 2717 | question, using baby steps / giant steps techniques 2718 | and trace formulas, obtaining subquadratic 2719 | estimates.}, 2720 | doi = {10.1145/1145768.1145814}, 2721 | isbn = {1-59593-276-3}, 2722 | } 2723 | 2724 | @Article{paterson_stockmeyer, 2725 | author = {Michael S. Paterson and Larry J. Stockmeyer}, 2726 | title = {On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials}, 2727 | journal = {SIAM Journal on Computing}, 2728 | year = {1973}, 2729 | volume = {2}, 2730 | number = {1}, 2731 | pages = {60--66}, 2732 | doi = {10.1137/0202007}, 2733 | } 2734 | 2735 | @Article{pila90, 2736 | author = {Pila, Jonathan}, 2737 | title = {Frobenius Maps of {A}belian Varieties and Finding Roots of Unity in Finite Fields}, 2738 | journal = {Mathematics of Computation}, 2739 | year = {1990}, 2740 | volume = {55}, 2741 | number = {192}, 2742 | pages = {745--763}, 2743 | issn = {00255718}, 2744 | abstract = {We give a generalization to Abelian varieties over 2745 | finite fields of the algorithm of Schoof for elliptic 2746 | curves. Schoof showed that for an elliptic curve 2747 | \$E\$ over \$\mathbf{F}\_q\$, given by a Weierstrass 2748 | equation, one can compute the number of 2749 | \$\mathbf{F}\_q\$-rational points of \$E\$ in time 2750 | \$O((\log q)^9)\$. Our result is the following. Let 2751 | \$A\$ be an Abelian variety over \$\mathbf{F}\_q\$. 2752 | Then one can compute the characteristic polynomial of 2753 | the Frobenius endomorphism of \$A\$ in time \$O((\log 2754 | q)^\Delta)\$, where \$\Delta\$ and the implied 2755 | constant depend only on the dimension of the 2756 | embedding space of \$A\$, the number of equations 2757 | defining \$A\$ and the addition law, and their 2758 | degrees. The method, generalizing that of Schoof, is 2759 | to use the machinery developed by Weil to prove the 2760 | Riemann hypothesis for Abelian varieties. By means of 2761 | this theory, the calculation is reduced to 2762 | ideal-theoretic computations in a ring of polynomials 2763 | in several variables over \$\mathbf{F}\_q\$. As 2764 | applications we show how to count the rational points 2765 | on the reductions modulo primes \$p\$ of a fixed 2766 | curve over \$\mathbf {Q}\$ in time polynomial in 2767 | \$\log p\$; we show also that, for a fixed prime 2768 | \$l\$, we can compute the \$l\$th roots of unity 2769 | \$\operatorname{mod} p\$, when they exist, in 2770 | polynomial time in \$\log p\$. This generalizes 2771 | Schoof's application of his algorithm to find square 2772 | roots of a fixed integer \$x \operatorname{mod} p\$.}, 2773 | doi = {10.2307/2008445}, 2774 | publisher = {American Mathematical Society}, 2775 | } 2776 | 2777 | @InProceedings{Pinch, 2778 | author = {Richard G. E. Pinch}, 2779 | title = {Recognising Elements Of Finite Fields}, 2780 | booktitle = {Cryptography and Coding II}, 2781 | year = {1992}, 2782 | pages = {193--197}, 2783 | publisher = {Oxford University Press}, 2784 | } 2785 | 2786 | @InProceedings{pointcheval95-pp, 2787 | author = {Pointcheval, David}, 2788 | title = {A New Identification Scheme Based on the Perceptrons Problem}, 2789 | booktitle = {Advances in Cryptology --- EUROCRYPT '95}, 2790 | year = {1995}, 2791 | volume = {921}, 2792 | series = {Lecture Notes in Computer Science}, 2793 | pages = {319--328}, 2794 | address = {Berlin, Heidelberg}, 2795 | publisher = {Springer Berlin / Heidelberg}, 2796 | abstract = {Identification is a useful cryptographic tool. Since 2797 | zero-know- ledge theory appeared [ 3 ], several 2798 | interactive identification schemes have been proposed 2799 | (in particular Fiat-Shamir [ 2 ] and its variants [ 8 2800 | , 5 , 4 ], Schnorr [ 9 ]). These identifications are 2801 | based on number theoretical prob- lems. More 2802 | recently, new schemes appeared with the peculiarity 2803 | that they are more efficient from the computational 2804 | point of view and that their security is based on NP 2805 | -complete problems: PKP (Permuted Ker- nels Problem) 2806 | [ 10 ], SD (Syndrome Decoding) [ 12 ] and CLE 2807 | (Constrained Linear Equations) [ 13 ]. We present a 2808 | new NP -complete linear problem which comes from 2809 | learn- ing machines: the Perceptrons Problem. We have 2810 | some constraints, m vectors X i of {−1, +1} n , and 2811 | we want to find a vector V of {−1, +1} n such that 2812 | X i · V ≥ 0 for all i . Next, we provide some 2813 | zero-knowledge interactive identification protocols 2814 | based on this problem, with an evaluation of their 2815 | security. Eventually, those protocols are well suited 2816 | for smart card applications.}, 2817 | chapter = {26}, 2818 | doi = {10.1007/3-540-49264-X\_26}, 2819 | isbn = {978-3-540-59409-3}, 2820 | url = {http://dx.doi.org/10.1007/3-540-49264-X\_26}, 2821 | } 2822 | 2823 | @Article{PoSc13a, 2824 | author = {Adrian Poteaux and \'Eric Schost}, 2825 | title = {Modular Composition Modulo Triangular Sets and Applications}, 2826 | journal = {Computational Complexity}, 2827 | year = {2013}, 2828 | volume = {22}, 2829 | number = {3}, 2830 | pages = {463--516}, 2831 | publisher = {Springer Basel}, 2832 | } 2833 | 2834 | @Article{PoSc13b, 2835 | author = {Adrien Poteaux and \'Eric Schost}, 2836 | title = {On the complexity of computing with zero-dimensional triangular sets}, 2837 | journal = {Journal of Symbolic Computation}, 2838 | year = {2013}, 2839 | volume = {50}, 2840 | pages = {110--138}, 2841 | } 2842 | 2843 | @InProceedings{quis, 2844 | author = {Petit, Christophe and Lauter, Kristin and Quisquater, Jean-Jacques}, 2845 | title = {Full Cryptanalysis of {LPS} and {M}orgenstern Hash Functions}, 2846 | booktitle = {Proceedings of the 6th international conference on Security and Cryptography for Networks}, 2847 | year = {2008}, 2848 | series = {SCN '08}, 2849 | address = {Berlin, Heidelberg}, 2850 | publisher = {Springer-Verlag}, 2851 | doi = {10.1007/978-3-540-85855-3_18}, 2852 | } 2853 | 2854 | @Unpublished{rains2008, 2855 | author = {Eric M. Rains}, 2856 | title = {Efficient Computation of Isomorphisms Between Finite Fields}, 2857 | note = {Personal communication}, 2858 | year = {1996}, 2859 | } 2860 | 2861 | @Misc{Roe2013, 2862 | author = {Roe, David and Flori, Jean-Pierre and Bruin, Peter}, 2863 | title = {Implement pseudo-{C}onway polynomials}, 2864 | howpublished = {Trac ticket \#14958}, 2865 | month = oct, 2866 | year = {2013}, 2867 | url = {https://trac.sagemath.org/ticket/14958}, 2868 | } 2869 | 2870 | @Article{rouiller99, 2871 | author = {Rouillier, Fabrice}, 2872 | title = {Solving Zero-Dimensional Systems Through the Rational Univariate Representation}, 2873 | journal = {Applicable Algebra in Engineering, Communication and Computing}, 2874 | year = {1999}, 2875 | volume = {9}, 2876 | number = {5}, 2877 | pages = {433--461}, 2878 | month = may, 2879 | issn = {0938-1279}, 2880 | abstract = {Abstract.\ \  This paper is devoted to the 2881 | resolution of zero-dimensional systems in {K[X} 1, 2882 | …X n ], where K is a field of characteristic zero 2883 | (or strictly positive under some conditions). We 2884 | follow the definition used in {MMM95} and basically 2885 | due to Kronecker for solving zero-dimensional 2886 | systems: A system is solved if each root is 2887 | represented in such way as to allow the performance 2888 | of any arithmetical operations over the arithmetical 2889 | expressions of its coordinates. We propose new 2890 | definitions for solving zero-dimensional systems in 2891 | this sense by introducing the Univariate 2892 | Representation of their roots. We show by this way 2893 | that the solutions of any zero-dimensional system of 2894 | polynomials can be expressed through a special kind 2895 | of univariate representation (Rational Univariate 2896 | Representation): where (f,g,g 1, …,g n ) are 2897 | polynomials of {K[X} 1, …, X n ]. A special feature 2898 | of our Rational Univariate Representation is that we 2899 | dont loose geometrical information contained in the 2900 | initial system. Moreover we propose different 2901 | efficient algorithms for the computation of the 2902 | Rational Univariate Representation, and we make a 2903 | comparison with standard known tools.}, 2904 | doi = {10.1007/s002000050114}, 2905 | } 2906 | 2907 | @Manual{Sage, 2908 | title = {{SageMath, the Sage Mathematics Software System (Version 8.0)}}, 2909 | organization = {{The Sage Developers}}, 2910 | year = {2018}, 2911 | key = {SageMath}, 2912 | shorthand = {Sage}, 2913 | url = {https://www.sagemath.org}, 2914 | } 2915 | 2916 | @Book{Sarnak, 2917 | title = {Some applications of modular forms}, 2918 | publisher = {Cambridge University Press}, 2919 | year = {1990}, 2920 | author = {Sarnak, Peter}, 2921 | volume = {99}, 2922 | series = {Cambridge Tracts in Mathematics}, 2923 | address = {Cambridge}, 2924 | } 2925 | 2926 | @Article{satoh00, 2927 | author = {Satoh, Takakazu}, 2928 | title = {The canonical lift of an ordinary elliptic curve over a finite field and its point counting}, 2929 | journal = {Journal of the Ramanujan Mathematical Society}, 2930 | year = {2000}, 2931 | volume = {15}, 2932 | number = {4}, 2933 | pages = {247--270}, 2934 | publisher = {The Ramanujan Mathematical Society}, 2935 | } 2936 | 2937 | @Article{schoof85, 2938 | author = {Schoof, Ren\'{e}}, 2939 | title = {Elliptic Curves Over Finite Fields and the Computation of Square Roots mod \(p\)}, 2940 | journal = {Mathematics of Computation}, 2941 | year = {1985}, 2942 | volume = {44}, 2943 | number = {170}, 2944 | pages = {483--494}, 2945 | issn = {00255718}, 2946 | abstract = {In this paper we present a deterministic algorithm to 2947 | compute the number of \$\mathbf{F}\_q\$-points of an 2948 | elliptic curve that is defined over a finite field 2949 | \$\mathbf{F}\_q\$ and which is given by a Weierstrass 2950 | equation. The algorithm takes \$O(\log^9 q)\$ 2951 | elementary operations. As an application we give an 2952 | algorithm to compute square roots 2953 | \$\operatorname{mod} p\$. For fixed \$x \in 2954 | \mathbf{Z}\$, it takes \$O(\log^9 p)\$ elementary 2955 | operations to compute \$\sqrt x \operatorname{mod} 2956 | p\$.}, 2957 | doi = {10.2307/2007968}, 2958 | publisher = {American Mathematical Society}, 2959 | } 2960 | 2961 | @Article{schoof95, 2962 | author = {Schoof, Ren\'{e}}, 2963 | title = {Counting points on elliptic curves over finite fields}, 2964 | journal = {Journal de Th\'{e}orie des Nombres de Bordeaux}, 2965 | year = {1995}, 2966 | volume = {7}, 2967 | number = {1}, 2968 | pages = {219--254}, 2969 | abstract = {Les Dix-huiti\`{e}mes Journ\'{e}es Arithm\'{e}tiques 2970 | (Bordeaux, 1993)}, 2971 | url = {http://www.ams.org/mathscinet-getitem?mr=1413578}, 2972 | } 2973 | 2974 | @Book{Serre.Arith, 2975 | title = {{Cours d'arithmétique}}, 2976 | publisher = {Presses Universitaires de France}, 2977 | year = {1970}, 2978 | author = {Serre, Jean-Pierre}, 2979 | address = {Paris}, 2980 | } 2981 | 2982 | @InProceedings{shamir89-pkp, 2983 | author = {Shamir, Adi}, 2984 | title = {An efficient identification scheme based on permuted kernels (extended abstract)}, 2985 | booktitle = {Proceedings on Advances in cryptology}, 2986 | year = {1989}, 2987 | series = {CRYPTO '89}, 2988 | pages = {606--609}, 2989 | address = {New York, NY, USA}, 2990 | publisher = {Springer-Verlag New York, Inc.}, 2991 | abstract = {In 1985 Goldwasser Micali and Rackoff proposed a new 2992 | type of interactive proof system which reveals no 2993 | knowledge whatsoever about the assertion except its 2994 | validity. The practical significance of these proofs 2995 | was demonstrated in 1986 by Fiat and Shamir, who 2996 | showed how to use efficient zero knowledge proofs of 2997 | quadratic residuosity to establish user identities 2998 | and to digitally sign messages. In this paper we 2999 | propose a new zero knowledge identification scheme, 3000 | which is even faster than the Fiat-Shamir scheme, 3001 | using a small number of communicated bits, simple 3002 | 8-bit arithmetic operations, and compact public and 3003 | private keys. The security of the new scheme depends 3004 | on an NP-complete algebraic problem rather than on 3005 | factoring, and thus it widens the basis of public key 3006 | cryptography, which has become dangerously dependent 3007 | on the difficulty of a single problem.}, 3008 | isbn = {0-387-97317-6}, 3009 | } 3010 | 3011 | @Article{Shoup_1990, 3012 | author = {Victor Shoup}, 3013 | title = {New algorithms for finding irreducible polynomials over finite fields}, 3014 | journal = {Mathematics of Computation}, 3015 | year = {1990}, 3016 | volume = {54}, 3017 | number = {189}, 3018 | pages = {435--435}, 3019 | month = jan, 3020 | doi = {10.1090/s0025-5718-1990-0993933-0}, 3021 | publisher = {American Mathematical Society ({AMS})}, 3022 | } 3023 | 3024 | @Manual{shoup2003ntl, 3025 | title = {{NTL}: A library for doing number theory}, 3026 | author = {Shoup, Victor}, 3027 | shorthand = {NTL}, 3028 | url = {http://www.shoup.net/ntl}, 3029 | } 3030 | 3031 | @InProceedings{shoup93, 3032 | author = {Shoup, Victor}, 3033 | title = {Fast construction of irreducible polynomials over finite fields}, 3034 | booktitle = {SODA '93: Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms}, 3035 | year = {1993}, 3036 | pages = {484--492}, 3037 | address = {Philadelphia, PA, USA}, 3038 | publisher = {Society for Industrial and Applied Mathematics}, 3039 | abstract = {Note: {OCR} errors may be found in this Reference 3040 | List extracted from the full text article. {ACM} has 3041 | opted to expose the complete List rather than only 3042 | correct and linked references.}, 3043 | isbn = {0-89871-313-7}, 3044 | } 3045 | 3046 | @Article{shoup94, 3047 | author = {Shoup, Victor}, 3048 | title = {Fast construction of irreducible polynomials over finite fields}, 3049 | journal = {Journal of Symbolic Computation}, 3050 | year = {1994}, 3051 | volume = {17}, 3052 | number = {5}, 3053 | pages = {371--391}, 3054 | issn = {0747-7171}, 3055 | address = {Duluth, MN, USA}, 3056 | doi = {10.1006/jsco.1994.1025}, 3057 | publisher = {Academic Press, Inc.}, 3058 | } 3059 | 3060 | @Article{shoup95, 3061 | author = {Shoup, Victor}, 3062 | title = {A New Polynomial Factorization Algorithm and its Implementation}, 3063 | journal = {Journal of Symbolic Computation}, 3064 | year = {1995}, 3065 | volume = {20}, 3066 | number = {4}, 3067 | pages = {363--397}, 3068 | issn = {0747-7171}, 3069 | abstract = {We consider the problem of factoring univariate 3070 | polynomials over a finite field. We demonstrate that 3071 | the new baby step/giant step factoring method, 3072 | recently developed by Kaltofen and Shoup, can be made 3073 | into a very practical algorithm. We describe an 3074 | implementation of this algorithm, and present the 3075 | results of empirical tests comparing this new 3076 | algorithm with others. When factoring polynomials 3077 | modulo large primes, the algorithm allows much larger 3078 | polynomials to be factored using a reasonable amount 3079 | of time and space than was previously possible. For 3080 | example, this new software has been used to factor a 3081 | generic polynomial of degree 2048 modulo a 2048-bit 3082 | prime in under 12 days on a Sun SPARC-station 10, 3083 | using 68 MB of main memory.}, 3084 | address = {Duluth, MN, USA}, 3085 | doi = {10.1006/jsco.1995.1055}, 3086 | publisher = {Academic Press, Inc.}, 3087 | } 3088 | 3089 | @InProceedings{shoup99, 3090 | author = {Shoup, Victor}, 3091 | title = {Efficient Computation of Minimal Polynomials in Algebraic Extensions of Finite Fields}, 3092 | booktitle = {Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation}, 3093 | year = {1999}, 3094 | series = {ISSAC '99}, 3095 | pages = {53--58}, 3096 | address = {New York, NY, USA}, 3097 | publisher = {ACM}, 3098 | doi = {10.1145/309831.309859}, 3099 | isbn = {1-58113-073-2}, 3100 | } 3101 | 3102 | @Article{Shparlinski2014, 3103 | author = {Shparlinski, Igor E. and Sutherland, Andrew V.}, 3104 | title = {On the Distribution of {A}tkin and {E}lkies Primes}, 3105 | journal = {Foundations of Computational Mathematics}, 3106 | year = {2014}, 3107 | volume = {14}, 3108 | number = {2}, 3109 | pages = {285--297}, 3110 | month = apr, 3111 | issn = {1615-3383}, 3112 | abstract = {Given an elliptic curve {\$}{\$}E{\$}{\$} E over a 3113 | finite field {\$}{\$}{\backslash}mathbb 3114 | {\{}F{\}}{\_}q{\$}{\$} F q of {\$}{\$}q{\$}{\$} q 3115 | elements, we say that an odd prime 3116 | {\$}{\$}{\backslash}ell {\backslash}not 3117 | {\backslash}mid q{\$}{\$} ℓ ∤ q is an Elkies 3118 | prime for {\$}{\$}E{\$}{\$} E if {\$}{\$}t{\_}E^2 - 3119 | 4q{\$}{\$} t E 2 - 4 q is a square modulo  3120 | {\$}{\$}{\backslash}ell {\$}{\$} ℓ , where 3121 | {\$}{\$}t{\_}E = q+1 - 3122 | {\backslash}{\#}E({\backslash}mathbb 3123 | {\{}F{\}}{\_}q){\$}{\$} t E = q + 1 - {\#} E ( F q ) 3124 | and {\$}{\$}{\backslash}{\#}E({\backslash}mathbb 3125 | {\{}F{\}}{\_}q){\$}{\$} {\#} E ( F q ) is the number 3126 | of {\$}{\$}{\backslash}mathbb {\{}F{\}}{\_}q{\$}{\$} 3127 | F q -rational points on {\$}{\$}E{\$}{\$} E ; 3128 | otherwise, {\$}{\$}{\backslash}ell {\$}{\$} ℓ is 3129 | called an Atkin prime. We show that there are 3130 | asymptotically the same number of Atkin and Elkies 3131 | primes {\$}{\$}{\backslash}ell < L{\$}{\$} ℓ < L on 3132 | average over all curves {\$}{\$}E{\$}{\$} E over 3133 | {\$}{\$}{\backslash}mathbb {\{}F{\}}{\_}q{\$}{\$} F q 3134 | , provided that {\$}{\$}L {\backslash}ge 3135 | ({\backslash}log q)^{\backslash}varepsilon {\$}{\$} L 3136 | ≥ ( log q ) $\epsilon$ for any fixed 3137 | {\$}{\$}{\backslash}varepsilon >0{\$}{\$} $\epsilon$ 3138 | > 0 and a sufficiently large {\$}{\$}q{\$}{\$} q . We 3139 | use this result to design and analyze a fast 3140 | algorithm to generate random elliptic curves with 3141 | {\$}{\$}{\backslash}{\#}E({\backslash}mathbb 3142 | {\{}F{\}}{\_}p){\$}{\$} {\#} E ( F p ) prime, where 3143 | {\$}{\$}p{\$}{\$} p varies uniformly over primes in a 3144 | given interval {\$}{\$}[x,2x]{\$}{\$} [ x , 2 x ] .}, 3145 | doi = {10.1007/s10208-013-9181-9}, 3146 | } 3147 | 3148 | @Book{SL2, 3149 | title = {Arbres, amalgames, $SL_2$}, 3150 | publisher = {Société Mathématique de France}, 3151 | year = {1977}, 3152 | author = {Serre, Jean-Pierre}, 3153 | volume = {46}, 3154 | series = {Astérisque}, 3155 | address = {Paris}, 3156 | } 3157 | 3158 | @TechReport{solinas01, 3159 | author = {Solinas, Jerome A.}, 3160 | title = {Low-Weight Binary Representations for Pairs of Integers}, 3161 | institution = {National Security Agency, USA}, 3162 | year = {2001}, 3163 | } 3164 | 3165 | @InProceedings{stern94-CLE, 3166 | author = {Stern, Jacques}, 3167 | title = {Designing Identification Schemes with Keys of Short Size}, 3168 | booktitle = {Advances in Cryptology --- CRYPTO '94}, 3169 | year = {1994}, 3170 | volume = {839}, 3171 | series = {Lecture Notes in Computer Science}, 3172 | pages = {164--173}, 3173 | address = {Berlin, Heidelberg}, 3174 | publisher = {Springer Berlin / Heidelberg}, 3175 | abstract = {In the last few years, there have been several 3176 | attempts to build identification protocols that do 3177 | not rely on arithmetical operations with large 3178 | numbers but only use simple operations (see [ 10 , 8 3179 | ]). One was presented at the CRYPTO 89 rump session 3180 | ([ 8 ]) and depends on the so-called Permuted Kernel 3181 | problem (PKP). Another appeared in the CRYPTO 93 3182 | proceedings and is based on the syndrome decoding 3183 | problem (SD) form the theory of error correcting 3184 | codes ([ 11 ]). In this paper, we introduce a new 3185 | scheme of the same family with the distinctive 3186 | character that both the secret key and the public 3187 | identification key can be taken to be of short 3188 | length. By short, we basically mean the usual size of 3189 | conventional symmetric cryptosystems. As is known, 3190 | the possibility of using short keys has been a 3191 | challenge in public key cryptography and has 3192 | practical applications. Our scheme relies on a 3193 | combinatorial problem which we call Constrained 3194 | Linear Equations (CLE in short) and which consists of 3195 | solving a set of linear equations modulo some small 3196 | prime q , the unknowns being subject to belong to a 3197 | specific subset of the integers mod q . Thus, we 3198 | enlarge the set of tools that can be used in 3199 | cryptography.}, 3200 | chapter = {18}, 3201 | doi = {10.1007/3-540-48658-5\_18}, 3202 | isbn = {978-3-540-58333-2}, 3203 | url = {http://dx.doi.org/10.1007/3-540-48658-5\_18}, 3204 | } 3205 | 3206 | @InProceedings{stern94-SD, 3207 | author = {Stern, Jacques}, 3208 | title = {A new identification scheme based on syndrome decoding}, 3209 | booktitle = {Advances in Cryptology --- CRYPTO' 93}, 3210 | year = {1994}, 3211 | volume = {773}, 3212 | series = {Lecture Notes in Computer Science}, 3213 | pages = {13--21}, 3214 | address = {Berlin, Heidelberg}, 3215 | publisher = {Springer Berlin / Heidelberg}, 3216 | abstract = {Zero-knowledge proofs were introduced in 1985, in a 3217 | paper by Goldwasser, Micali and Rackoff ([ 6 ]). 3218 | Their practical significance was soon demonstrated in 3219 | the work of Fiat and Shamir ([ 4 ]), who turned 3220 | zero-knowledge proofs of quadratic residuosity into 3221 | efficient means of establishing user identities. 3222 | Still, as is almost always the case in public-key 3223 | cryptography, the Fiat-Shamir scheme relied on 3224 | arithmetic operations on large numbers. In 1989, 3225 | there were two attempts to build identification 3226 | protocols that only use simple operations (see [ 11 , 3227 | 10 ]). One appeared in the EUROCRYPT proceedings and 3228 | relies on the intractability of some coding problems, 3229 | the other was presented at the CRYPTO rump session 3230 | and depends on the so-called Permuted Kernel problem 3231 | (PKP). Unfortunately, the first of the schemes was 3232 | not really practical. In the present paper, we 3233 | propose a new identification scheme, based on 3234 | error-correcting codes, which is zero-knowledge and 3235 | is of practical value. Furthermore, we describe 3236 | several variants, including one which has an identity 3237 | based character. The security of our scheme depends 3238 | on the hardness of decoding a word of given syndrome 3239 | w.r.t. some binary linear error-correcting code.}, 3240 | chapter = {2}, 3241 | doi = {10.1007/3-540-48329-2\_2}, 3242 | isbn = {978-3-540-57766-9}, 3243 | url = {http://dx.doi.org/10.1007/3-540-48329-2\_2}, 3244 | } 3245 | 3246 | @Article{sutherland10:modpol, 3247 | author = {Br{\"o}ker, Reinier and Lauter, Kristin and Sutherland, Andrew}, 3248 | title = {Modular polynomials via isogeny volcanoes}, 3249 | journal = {Mathematics of Computation}, 3250 | year = {2012}, 3251 | volume = {81}, 3252 | number = {278}, 3253 | pages = {1201--1231}, 3254 | abstract = {We present a new algorithm to compute the classical 3255 | modular polynomial Phi\_nin the rings {Z[X},Y] and 3256 | ({Z/mZ})[{X,Y}], for a prime n and any positive 3257 | integer {m.Our} approach uses the graph of 3258 | n-isogenies to efficiently compute Phi\_n mod pfor 3259 | many primes p of a suitable form, and then applies 3260 | the Chinese {RemainderTheorem} ({CRT}). Under the 3261 | Generalized Riemann Hypothesis ({GRH}), we achieve 3262 | anexpected running time of O(n^3 (log n)^3 log log 3263 | n), and compute Phi\_n mod musing O(n^2 (log n)^2 + 3264 | n^2 log m) space. We have used the new algorithm 3265 | tocompute Phi\_n with n over 5000, and Phi\_n mod m 3266 | with n over 20000. We alsoconsider several modular 3267 | functions g for which Phi\_n^g is smaller than 3268 | Phi\_n,allowing us to handle n over 60000.}, 3269 | doi = {10.1090/S0025-5718-2011-02508-1}, 3270 | } 3271 | 3272 | @Article{Sutherland2012, 3273 | author = {Sutherland, Andrew V.}, 3274 | title = {Accelerating the {CM} method}, 3275 | journal = {LMS Journal of Computational Mathematics}, 3276 | year = {2012}, 3277 | volume = {15}, 3278 | pages = {172--204}, 3279 | issn = {1461-1570}, 3280 | doi = {10.1112/S1461157012001015}, 3281 | } 3282 | 3283 | @Article{sutherland2012constructing, 3284 | author = {Sutherland, Andrew V.}, 3285 | title = {Constructing elliptic curves over finite fields with prescribed torsion}, 3286 | journal = {Mathematics of Computation}, 3287 | year = {2012}, 3288 | volume = {81}, 3289 | pages = {1131--1147}, 3290 | } 3291 | 3292 | @Article{sutherland2013evaluation, 3293 | author = {Sutherland, Andrew}, 3294 | title = {On the evaluation of modular polynomials}, 3295 | journal = {The Open Book Series}, 3296 | year = {2013}, 3297 | volume = {1}, 3298 | number = {1}, 3299 | pages = {531--555}, 3300 | publisher = {Mathematical Sciences Publishers}, 3301 | } 3302 | 3303 | @InProceedings{sutherland2013isogeny, 3304 | author = {Sutherland, Andrew}, 3305 | title = {Isogeny volcanoes}, 3306 | booktitle = {ANTS X: Proceedings of the Algorithmic Number Theory 10th International Symposium}, 3307 | year = {2013}, 3308 | volume = {1}, 3309 | pages = {507--530}, 3310 | address = {Berkeley}, 3311 | publisher = {Mathematical Sciences Publishers}, 3312 | journal = {The Open Book Series}, 3313 | } 3314 | 3315 | @Misc{SutherlandDatabase, 3316 | author = {Andrew V. Sutherland}, 3317 | title = {Modular polynomials}, 3318 | year = {2018}, 3319 | url = {https://math.mit.edu/~drew/ClassicalModPolys.html}, 3320 | } 3321 | 3322 | @Electronic{tao2011expander, 3323 | author = {Terence Tao}, 3324 | year = {2011}, 3325 | title = {Expansion in groups of {Lie} type -- Basic theory of expander graphs}, 3326 | url = {https://terrytao.wordpress.com/2011/12/02/245b-notes-1-basic-theory-of-expander-graphs/}, 3327 | } 3328 | 3329 | @Article{Tate, 3330 | author = {Tate, John}, 3331 | title = {Endomorphisms of abelian varieties over finite fields}, 3332 | journal = {Inventiones mathematicae}, 3333 | year = {1966}, 3334 | volume = {2}, 3335 | number = {2}, 3336 | pages = {134--144}, 3337 | month = apr, 3338 | issn = {1432-1297}, 3339 | doi = {10.1007/BF01404549}, 3340 | } 3341 | 3342 | @Article{teske-ph, 3343 | author = {Teske, Edlyn}, 3344 | title = {The {P}ohlig-{H}ellman Method Generalized for Group Structure Computation}, 3345 | journal = {Journal of Symbolic Computation}, 3346 | year = {1999}, 3347 | volume = {27}, 3348 | number = {6}, 3349 | pages = {521--534}, 3350 | issn = {0747-7171}, 3351 | doi = {10.1006/jsco.1999.0279}, 3352 | } 3353 | 3354 | @InProceedings{tillich2008collisions, 3355 | author = {Tillich, Jean-Pierre and Z{\'e}mor, Gilles}, 3356 | title = {Collisions for the {LPS} expander graph hash function}, 3357 | booktitle = {Annual International Conference on the Theory and Applications of Cryptographic Techniques}, 3358 | year = {2008}, 3359 | pages = {254--269}, 3360 | organization = {Springer}, 3361 | } 3362 | 3363 | @InProceedings{twisted-edwards, 3364 | author = {Bernstein, Daniel and Birkner, Peter and Joye, Marc and Lange, Tanja and Peters, Christiane}, 3365 | title = {{Twisted Edwards Curves}}, 3366 | booktitle = {Progress in Cryptology --- AFRICACRYPT 2008}, 3367 | year = {2008}, 3368 | pages = {389--405}, 3369 | abstract = {{This paper introduces ” twisted Edwards curves,” 3370 | a generalization of the recently introduced Edwards 3371 | curves; shows that twisted Edwards curves include 3372 | more curves over finite fields, and in particular 3373 | every elliptic curve in Montgomery form; shows how to 3374 | cover even more curves via isogenies; presents fast 3375 | explicit formulas for twisted Edwards curves in 3376 | projective and inverted coordinates; and shows that 3377 | twisted Edwards curves save time for many curves that 3378 | were already expressible as Edwards curves.}}, 3379 | doi = {10.1007/978-3-540-68164-9_26}, 3380 | } 3381 | 3382 | @InProceedings{vanderHoeven:2004:TFT:1005285.1005327, 3383 | author = {van der Hoeven, Joris}, 3384 | title = {The Truncated {F}ourier Transform and Applications}, 3385 | booktitle = {Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation}, 3386 | year = {2004}, 3387 | series = {ISSAC '04}, 3388 | pages = {290--296}, 3389 | address = {New York, NY, USA}, 3390 | publisher = {ACM}, 3391 | doi = {10.1145/1005285.1005327}, 3392 | isbn = {1-58113-827-X}, 3393 | } 3394 | 3395 | @Electronic{Voight2018, 3396 | author = {Voight, John}, 3397 | year = {2018}, 3398 | title = {Quaternion Algebras}, 3399 | url = {https://math.dartmouth.edu/~jvoight/quat-book.pdf}, 3400 | } 3401 | 3402 | @InProceedings{vzgathen+shoup92, 3403 | author = {Joachim von zur Gathen and Shoup, Victor}, 3404 | title = {Computing {F}robenius maps and factoring polynomials}, 3405 | booktitle = {STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of computing}, 3406 | year = {1992}, 3407 | pages = {97--105}, 3408 | address = {New York, NY, USA}, 3409 | publisher = {ACM}, 3410 | abstract = {A new probabilistic algorithm for factoring 3411 | univariate polynomials over finite fields is 3412 | presented whose asymptotic running time improves upon 3413 | previous results. To factor a polynomial of degree n 3414 | over F q , the algorithm uses O (( n 2 + n log q 3415 | )•(log n ) 2 log log n ) arithmetic operations in F 3416 | q . The main technical innovation is a new way to 3417 | compute Frobenius and trace maps in the ring of 3418 | polynomials modulo the polynomial to be factored.}, 3419 | doi = {10.1145/129712.129722}, 3420 | isbn = {0-89791-511-9}, 3421 | } 3422 | 3423 | @Article{vzgathen+shoup92:journal, 3424 | author = {von zur Gathen, Joachim and Shoup, Victor}, 3425 | title = {Computing {F}robenius Maps and Factoring Polynomials}, 3426 | journal = {Computational Complexity}, 3427 | year = {1992}, 3428 | volume = {2}, 3429 | pages = {187--224}, 3430 | abstract = {A new probabilistic algorithm for factoring 3431 | univariate polynomials over finite fields is 3432 | presented. To factor a polynomial of degree n over F 3433 | q , the number of arithmetic operations in F q is 3434 | O((n 2 +n log q) \Delta (log n) 2 loglog n). The main 3435 | technical innovation is a new way to compute 3436 | Frobenius and trace maps in the ring of polynomials 3437 | modulo the polynomial to be factored. Subject 3438 | classifications. {68Q40}; {11Y16}, {12Y05}. 1. 3439 | Introduction We consider the problem of factoring a 3440 | univariate polynomial over a finite field. This 3441 | problem plays a central role in computational 3442 | algebra. Indeed, many of the efficient algorithms for 3443 | factoring univariate and multivariate polynomials 3444 | over finite fields, the field of rational numbers, 3445 | and finite extensions of the rationals solve as a 3446 | subproblem the problem of factoring univariate 3447 | polynomials over finite fields (Kaltofen 1990). This 3448 | problem also has important applications in number 3449 | theory (Buchmann 1990), coding theory (Berlekamp 3450 | 1968), and ...}, 3451 | } 3452 | 3453 | @Book{vzGG, 3454 | title = {Modern Computer Algebra}, 3455 | publisher = {Cambridge University Press}, 3456 | year = {1999}, 3457 | author = {von zur Gathen, Joachim and Gerhard, Jurgen}, 3458 | address = {New York, NY, USA}, 3459 | isbn = {0-521-64176-4}, 3460 | } 3461 | 3462 | @InProceedings{Williams12, 3463 | author = {Vassilevska Williams, Virginia}, 3464 | title = {Multiplying matrices faster than {C}oppersmith-{W}inograd}, 3465 | booktitle = {STOC'12}, 3466 | year = {2012}, 3467 | pages = {887--898}, 3468 | publisher = {ACM}, 3469 | } 3470 | 3471 | @Article{williams1982, 3472 | author = {Williams, Hugh C.}, 3473 | title = {A $p+1$ method of factoring}, 3474 | journal = {Mathematics of Computation}, 3475 | year = {1982}, 3476 | volume = {39}, 3477 | number = {159}, 3478 | pages = {225--234}, 3479 | } 3480 | 3481 | @InProceedings{zhang, 3482 | author = {Zhang, Shengyu}, 3483 | title = {{Promised and Distributed Quantum Search Computing and Combinatorics}}, 3484 | booktitle = {Proceedings of the Eleventh Annual International Conference on Computing and Combinatorics}, 3485 | year = {2005}, 3486 | volume = {3595}, 3487 | series = {Lecture Notes in Computer Science}, 3488 | pages = {430--439}, 3489 | address = {Berlin, Heidelberg}, 3490 | publisher = {Springer Berlin / Heidelberg}, 3491 | abstract = {{This paper gives a quantum algorithm to search in an 3492 | set S for a k -tuple satisfying some predefined 3493 | relation, with the promise that some components of a 3494 | desired k -tuple are in some subsets of S . In 3495 | particular when k =2, we show a tight bound of the 3496 | quantum query complexity for the Claw Finding 3497 | problem, improving previous upper and lower bounds by 3498 | Buhrman, Durr, Heiligman, Hoyer, Magniez, Santha and 3499 | de Wolf [7]. We also consider the distributed 3500 | scenario, where two parties each holds an n -element 3501 | set, and they want to decide whether the two sets 3502 | share a common element. We show a family of protocols 3503 | s . t . q ( P ) 3/2 . c ( P )= O ( n 2 log n ), where 3504 | q ( P ) and c ( P ) are the number of quantum queries 3505 | and the number of communication qubits that the 3506 | protocol P makes, respectively. This implies that we 3507 | can pay more for quantum queries to save on quantum 3508 | communication, and vice versa. To our knowledge, it 3509 | is the first result about the tradeoff between the 3510 | two resources.}}, 3511 | chapter = {44}, 3512 | doi = {10.1007/11533719_44}, 3513 | isbn = {978-3-540-28061-3}, 3514 | } 3515 | 3516 | @Article{gaudry+hess+smart02, 3517 | author = {Gaudry, Pierrick and Hess, Florian and Smart, Niegel}, 3518 | title = {Constructive and destructive facets of {W}eil descent on elliptic curves}, 3519 | journal = {Journal of Cryptology}, 3520 | year = {2002}, 3521 | volume = {15}, 3522 | number = {1}, 3523 | pages = {19-46-46}, 3524 | month = mar, 3525 | issn = {0933-2790}, 3526 | abstract = {In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic 2 of composite degree. We explain how this method can be used to construct hyperelliptic cryptosystems which could be as secure as cryptosystems based on the original elliptic curve. On the other hand, we show that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves. We examine the resulting higher genus curves in some detail and propose an additional check on elliptic curve systems defined over fields of characteristic 2 so as to make them immune from the methods in this paper.}, 3527 | citeulike-article-id = {7751788}, 3528 | citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00145-001-0011-x}, 3529 | citeulike-linkout-1 = {http://www.springerlink.com/content/hx8b621p8p4417qv}, 3530 | day = {1}, 3531 | doi = {10.1007/s00145-001-0011-x}, 3532 | keywords = {cryptography, ghs, hyperelliptic\_curves, weil\_descent}, 3533 | posted-at = {2010-09-01 15:45:05}, 3534 | publisher = {Springer New York}, 3535 | url = {http://dx.doi.org/10.1007/s00145-001-0011-x}, 3536 | } 3537 | 3538 | @Book{joux2009algorithmic, 3539 | title = {Algorithmic cryptanalysis}, 3540 | publisher = {CRC Press}, 3541 | year = {2009}, 3542 | author = {Joux, Antoine}, 3543 | } 3544 | 3545 | @Article{koblitz87, 3546 | author = {Koblitz, Neal}, 3547 | title = {Elliptic Curve Cryptosystems}, 3548 | journal = {Mathematics of Computation}, 3549 | year = {1987}, 3550 | volume = {48}, 3551 | number = {177}, 3552 | pages = {203-209}, 3553 | citeulike-article-id = {2405523}, 3554 | citeulike-linkout-0 = {http://www.jstor.org/stable/2007884}, 3555 | keywords = {cryptography}, 3556 | posted-at = {2010-07-13 18:02:46}, 3557 | url = {http://www.jstor.org/stable/2007884}, 3558 | } 3559 | 3560 | @PhdThesis{lercier-algorithmique, 3561 | author = {Lercier, Reynald}, 3562 | title = {Algorithmique des courbes elliptiques dans les corps finis}, 3563 | school = {LIX - CNRS}, 3564 | year = {1997}, 3565 | month = jun, 3566 | citeulike-article-id = {7300668}, 3567 | groups = {Isogenies}, 3568 | keywords = {lercier}, 3569 | posted-at = {2010-06-14 18:24:56}, 3570 | } 3571 | 3572 | @Electronic{sutherland10, 3573 | author = {Sutherland, Andrew V.}, 3574 | year = {2010}, 3575 | title = {Genus 1 point counting over prime fields}, 3576 | howpublished = {Last accessed July 16, 2010. \url{http://www-math.mit.edu/\~drew/SEArecords.html}}, 3577 | url = {#}, 3578 | citeulike-article-id = {7499834}, 3579 | citeulike-linkout-0 = {#}, 3580 | groups = {Isogenies}, 3581 | keywords = {cryptography, elliptic\_curve, schoof}, 3582 | posted-at = {2010-07-16 13:25:40}, 3583 | } 3584 | 3585 | @article{10.1016/j.dam.2007.12.010, 3586 | author = {Galbraith, Steven D. and Paterson, Kenneth G. and Smart, Nigel P.}, 3587 | title = {Pairings for Cryptographers}, 3588 | year = {2008}, 3589 | issue_date = {September, 2008}, 3590 | publisher = {Elsevier Science Publishers B. V.}, 3591 | address = {NLD}, 3592 | volume = {156}, 3593 | number = {16}, 3594 | issn = {0166-218X}, 3595 | doi = {10.1016/j.dam.2007.12.010}, 3596 | abstract = {Many research papers in pairing-based cryptography treat pairings as a ''black box''. These papers build cryptographic schemes making use of various properties of pairings. If this approach is taken, then it is easy for authors to make invalid assumptions concerning the properties of pairings. The cryptographic schemes developed may not be realizable in practice, or may not be as efficient as the authors assume. The aim of this paper is to outline, in as simple a fashion as possible, the basic choices that are available when using pairings in cryptography. For each choice, the main properties and efficiency issues are summarized. The paper is intended to be of use to non-specialists who are interested in using pairings to design cryptographic schemes.}, 3597 | journal = {Discrete Applied Mathematics}, 3598 | month = {sep}, 3599 | pages = {3113–3121}, 3600 | numpages = {9}, 3601 | keywords = {Pairings, Cryptography} 3602 | } 3603 | 3604 | @electronic{trevisan-graphs, 3605 | author = {Luca Trevisan}, 3606 | title = {Lecture Notes on Graph Partitioning, Expanders and Spectral Methods}, 3607 | url = {https://lucatrevisan.github.io/books/expanders-2016.pdf}, 3608 | year = {2017}, 3609 | } 3610 | 3611 | @electronic{sutherland-notes, 3612 | author = {Andrew Sutherland}, 3613 | title = {Lecture Notes on Elliptic Curves}, 3614 | url = {https://math.mit.edu/classes/18.783/2017/lectures.html}, 3615 | year = {2017}, 3616 | } 3617 | 3618 | @article{waterhouse69, 3619 | author = {Waterhouse, William C.}, 3620 | journal = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure}, 3621 | number = {4}, 3622 | pages = {521--560}, 3623 | title = {Abelian varieties over finite fields}, 3624 | volume = {2}, 3625 | year = {1969} 3626 | } 3627 | 3628 | @article{mordell61, 3629 | title = {Mathematical Notes: The congruence $(p-1/2)! \equiv \pm 1 \mod p$}, 3630 | volume = {68}, 3631 | ISSN = {1930-0972}, 3632 | DOI = {10.1080/00029890.1961.11989636}, 3633 | number = {2}, 3634 | journal = {The American Mathematical Monthly}, 3635 | publisher = {Informa UK Limited}, 3636 | author = {Mordell, Louis J.}, 3637 | year = {1961}, 3638 | month = {Feb}, 3639 | pages = {131--149} 3640 | } 3641 | 3642 | @article{pizer1980, 3643 | title = {An algorithm for computing modular forms on $\Gamma_0(N)$}, 3644 | volume = {64}, 3645 | ISSN = {0021-8693}, 3646 | DOI = {10.1016/0021-8693(80)90151-9}, 3647 | number = {2}, 3648 | journal = {Journal of Algebra}, 3649 | publisher = {Elsevier BV}, 3650 | author = {Pizer, Arnold}, 3651 | year = {1980}, 3652 | month = {Jun}, 3653 | pages = {340--390} 3654 | } 3655 | 3656 | @article{isogpoksurvey, 3657 | title = {Proving knowledge of isogenies: a survey}, 3658 | volume = {91}, 3659 | ISSN = {1573-7586}, 3660 | DOI = {10.1007/s10623-023-01243-3}, 3661 | number = {11}, 3662 | journal = {Designs, Codes and Cryptography}, 3663 | publisher = {Springer Science and Business Media LLC}, 3664 | author = {Beullens, Ward and De Feo, Luca and Galbraith, 3665 | Steven D. and Petit, Christophe}, 3666 | year = {2023}, 3667 | month = {Jun}, 3668 | pages = {3425--3456} 3669 | } 3670 | 3671 | @article{onuki2021, 3672 | title = {On oriented supersingular elliptic curves}, 3673 | volume = {69}, 3674 | ISSN = {1071-5797}, 3675 | DOI = {10.1016/j.ffa.2020.101777}, 3676 | journal = {Finite Fields and Their Applications}, 3677 | publisher = {Elsevier BV}, 3678 | author = {Onuki, Hiroshi}, 3679 | year = {2021}, 3680 | month = {Jan}, 3681 | pages = {101777} 3682 | } 3683 | --------------------------------------------------------------------------------