├── .gitignore ├── Classic ciphers.ipynb ├── DHKE.ipynb ├── DSA.ipynb ├── Elliptic Curves.ipynb ├── Hash Functions.ipynb ├── Information-Theoretic Security.ipynb ├── LICENSE ├── README.md ├── RSA.ipynb ├── Secret Sharing (gf).ipynb ├── Secret Sharing.ipynb ├── StreamCiphers.ipynb ├── Symmetric.ipynb ├── cat.txt ├── encrypted_cat.txt ├── requirements.txt ├── tux.png └── tux_gray.png /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | *$py.class 5 | 6 | # C extensions 7 | *.so 8 | 9 | # Distribution / packaging 10 | .Python 11 | build/ 12 | develop-eggs/ 13 | dist/ 14 | downloads/ 15 | eggs/ 16 | .eggs/ 17 | lib/ 18 | lib64/ 19 | parts/ 20 | sdist/ 21 | var/ 22 | wheels/ 23 | *.egg-info/ 24 | .installed.cfg 25 | *.egg 26 | MANIFEST 27 | 28 | # PyInstaller 29 | # Usually these files are written by a python script from a template 30 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 31 | *.manifest 32 | *.spec 33 | 34 | # Installer logs 35 | pip-log.txt 36 | pip-delete-this-directory.txt 37 | 38 | # Unit test / coverage reports 39 | htmlcov/ 40 | .tox/ 41 | .coverage 42 | .coverage.* 43 | .cache 44 | nosetests.xml 45 | coverage.xml 46 | *.cover 47 | .hypothesis/ 48 | .pytest_cache/ 49 | 50 | # Translations 51 | *.mo 52 | *.pot 53 | 54 | # Django stuff: 55 | *.log 56 | local_settings.py 57 | db.sqlite3 58 | 59 | # Flask stuff: 60 | instance/ 61 | .webassets-cache 62 | 63 | # Scrapy stuff: 64 | .scrapy 65 | 66 | # Sphinx documentation 67 | docs/_build/ 68 | 69 | # PyBuilder 70 | target/ 71 | 72 | # Jupyter Notebook 73 | .ipynb_checkpoints 74 | 75 | # pyenv 76 | .python-version 77 | 78 | # celery beat schedule file 79 | celerybeat-schedule 80 | 81 | # SageMath parsed files 82 | *.sage.py 83 | 84 | # Environments 85 | .env 86 | .venv 87 | env/ 88 | venv/ 89 | ENV/ 90 | env.bak/ 91 | venv.bak/ 92 | 93 | # Spyder project settings 94 | .spyderproject 95 | .spyproject 96 | 97 | # Rope project settings 98 | .ropeproject 99 | 100 | # mkdocs documentation 101 | /site 102 | 103 | # mypy 104 | .mypy_cache/ 105 | 106 | secret_notebooks/ 107 | progetti/ 108 | 109 | .vagrant* -------------------------------------------------------------------------------- /Classic ciphers.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "slideshow": { 7 | "slide_type": "slide" 8 | } 9 | }, 10 | "source": [ 11 | "## The Shift Cipher" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": {}, 17 | "source": [ 18 | "The shift cipher is one of the oldest encryption schemes also used by Caesar, who\n", 19 | "> wrote it in cipher, that is, by so changing the order of the letters of the alphabet, that not a word could be made out. If anyone wishes to decipher these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for A, and so with the others. (Suetonius)" 20 | ] 21 | }, 22 | { 23 | "cell_type": "markdown", 24 | "metadata": {}, 25 | "source": [ 26 | "Messages are strings of any size of uppercase letters.\n", 27 | "\n", 28 | "Let’s encode each letter as A=0,B=1, …,Z=25.\n", 29 | "\n", 30 | "The key is any integer $0 \\leq k \\leq 25$." 31 | ] 32 | }, 33 | { 34 | "cell_type": "code", 35 | "execution_count": null, 36 | "metadata": {}, 37 | "outputs": [], 38 | "source": [ 39 | "import math\n", 40 | "import collections\n", 41 | "import matplotlib.pyplot as plt\n", 42 | "from sympy.crypto.crypto import AZ\n", 43 | "from sympy.crypto.crypto import encipher_shift, decipher_shift\n", 44 | "from sympy.crypto.crypto import encipher_vigenere, decipher_vigenere" 45 | ] 46 | }, 47 | { 48 | "cell_type": "code", 49 | "execution_count": null, 50 | "metadata": {}, 51 | "outputs": [], 52 | "source": [ 53 | "# Let's define a ciphertext using the letters of the alphabet\n", 54 | "\n", 55 | "m = AZ(\"Go Navy! Beat Army!\")\n", 56 | "m" 57 | ] 58 | }, 59 | { 60 | "cell_type": "code", 61 | "execution_count": null, 62 | "metadata": {}, 63 | "outputs": [], 64 | "source": [ 65 | "# encrypt with k = 1\n", 66 | "k = 1\n", 67 | "c = encipher_shift(m, k)\n", 68 | "c" 69 | ] 70 | }, 71 | { 72 | "cell_type": "code", 73 | "execution_count": null, 74 | "metadata": { 75 | "slideshow": { 76 | "slide_type": "-" 77 | } 78 | }, 79 | "outputs": [], 80 | "source": [ 81 | "# now decrypt\n", 82 | "decipher_shift(c,k)" 83 | ] 84 | }, 85 | { 86 | "cell_type": "code", 87 | "execution_count": null, 88 | "metadata": { 89 | "slideshow": { 90 | "slide_type": "slide" 91 | } 92 | }, 93 | "outputs": [], 94 | "source": [ 95 | "# Now a bit more complicated\n", 96 | "# Let's find the decryption of a ciphertext\n", 97 | "c = AZ(\"LRZZOACZZQTDZYPESLEXLVPDFDHTDPC\")\n", 98 | "\n", 99 | "# We try all the keys\n", 100 | "for i in range(26):\n", 101 | "\t\tprint(i, decipher_shift(c, i))" 102 | ] 103 | }, 104 | { 105 | "cell_type": "code", 106 | "execution_count": null, 107 | "metadata": { 108 | "slideshow": { 109 | "slide_type": "slide" 110 | } 111 | }, 112 | "outputs": [], 113 | "source": [ 114 | "# k = 11 makes sense\n", 115 | "k = 11\n", 116 | "decipher_shift(c, k)" 117 | ] 118 | }, 119 | { 120 | "cell_type": "code", 121 | "execution_count": null, 122 | "metadata": {}, 123 | "outputs": [], 124 | "source": [ 125 | "# Can we make it automatic?\n", 126 | "# Let's use a longer text\n", 127 | "raven = \"\"\"Once upon a midnight dreary, while I pondered, weak and weary, \n", 128 | " Over many a quaint and curious volume of forgotten lore, \n", 129 | " While I nodded, nearly napping, suddenly there came a tapping,\n", 130 | " As of some one gently rapping, rapping at my chamber door.\n", 131 | " Tis some visiter, I muttered, tapping at my chamber door --\n", 132 | " Only this,and nothing more.\"\"\"\n", 133 | "m = AZ(raven)\n", 134 | "freq = collections.Counter(m)\n", 135 | "plt.bar(freq.keys(),freq.values());" 136 | ] 137 | }, 138 | { 139 | "cell_type": "markdown", 140 | "metadata": {}, 141 | "source": [ 142 | "# Entropy\n", 143 | "Given a letter $m$ taken from a set of letter $\\{m_1, \\dots, m_N\\}$ with probabilities $p(m_i)\\ 1\\leq i \\leq N$, the source entropy is\n", 144 | "\n", 145 | "$H(m) = -\\sum_{i=1}^N p_i \\log_2 p_i$\n", 146 | "\n", 147 | "Source entropy is a measure of how much a source's messages are uniformly distributed." 148 | ] 149 | }, 150 | { 151 | "cell_type": "code", 152 | "execution_count": null, 153 | "metadata": {}, 154 | "outputs": [], 155 | "source": [ 156 | "# Calculate entropy given the histogram\n", 157 | "def entropy(p):\n", 158 | " # normalization constant (probability must sum to 1)\n", 159 | " norm = sum(p)\n", 160 | " # definition of entropy\n", 161 | " return -sum([ pi/norm*math.log2(pi/norm) for pi in p ])" 162 | ] 163 | }, 164 | { 165 | "cell_type": "code", 166 | "execution_count": null, 167 | "metadata": { 168 | "slideshow": { 169 | "slide_type": "slide" 170 | } 171 | }, 172 | "outputs": [], 173 | "source": [ 174 | "# Calculate and plot the frequencies of letters in a message\n", 175 | "def plot_frequencies(msg):\n", 176 | " freq = collections.Counter(msg)\n", 177 | " plt.bar(freq.keys(),freq.values())\n", 178 | " plt.title(\"Entropy: {} bits\".format(entropy(freq.values())))\n", 179 | " plt.show()" 180 | ] 181 | }, 182 | { 183 | "cell_type": "code", 184 | "execution_count": null, 185 | "metadata": { 186 | "slideshow": { 187 | "slide_type": "slide" 188 | } 189 | }, 190 | "outputs": [], 191 | "source": [ 192 | "# Show the letter frequencies in the plaintext\n", 193 | "# Letter E is the most common\n", 194 | "plot_frequencies(m)" 195 | ] 196 | }, 197 | { 198 | "cell_type": "code", 199 | "execution_count": null, 200 | "metadata": { 201 | "slideshow": { 202 | "slide_type": "slide" 203 | } 204 | }, 205 | "outputs": [], 206 | "source": [ 207 | "# Let's decrypt a longer message using frequency analysis\n", 208 | "ciphertext = \"\"\"VUJLBWVUHTPKUPNOAKYLHYFDOPSLPWVUK\n", 209 | " LYLKDLHRHUKDLHYFVCLYTHUFHXBHPUAHU\n", 210 | " KJBYPVBZCVSBTLVMMVYNVAALUSVYLDOPS\n", 211 | " LPUVKKLKULHYSFUHWWPUNZBKKLUSFAOLY\n", 212 | " LJHTLHAHWWPUNHZVMZVTLVULNLUASFYHW\n", 213 | " WPUNYHWWPUNHATFJOHTILYKVVYAPZZVTL\n", 214 | " CPZPALYPTBAALYLKAHWWPUNHATFJOHTIL\n", 215 | " YKVVYVUSFAOPZHUKUVAOPUNTVY\"\"\"\n", 216 | "c = AZ(ciphertext)\n", 217 | "plot_frequencies(c)" 218 | ] 219 | }, 220 | { 221 | "cell_type": "code", 222 | "execution_count": null, 223 | "metadata": { 224 | "slideshow": { 225 | "slide_type": "slide" 226 | } 227 | }, 228 | "outputs": [], 229 | "source": [ 230 | "# A shift cipher does not change the shape of the histogram and neither its entropy\n", 231 | "\n", 232 | "# Maybe E is encrypted to L?\n", 233 | "# We calculate the shift\n", 234 | "shift = ord(\"L\")- ord(\"E\")\n", 235 | "shift" 236 | ] 237 | }, 238 | { 239 | "cell_type": "code", 240 | "execution_count": null, 241 | "metadata": { 242 | "slideshow": { 243 | "slide_type": "-" 244 | } 245 | }, 246 | "outputs": [], 247 | "source": [ 248 | "decipher_shift(c, 7)" 249 | ] 250 | }, 251 | { 252 | "cell_type": "code", 253 | "execution_count": null, 254 | "metadata": { 255 | "slideshow": { 256 | "slide_type": "-" 257 | } 258 | }, 259 | "outputs": [], 260 | "source": [ 261 | "# compare to m\n", 262 | "m" 263 | ] 264 | }, 265 | { 266 | "cell_type": "markdown", 267 | "metadata": { 268 | "slideshow": { 269 | "slide_type": "slide" 270 | } 271 | }, 272 | "source": [ 273 | "## Vigenère Cipher\n", 274 | "The Vigenère cipher is a method of encrypting alphabetic text by using a series of different Caesar ciphers based on the letters of a keyword.\n", 275 | "\n", 276 | "Let $m=m_0 \\| m_1 \\| \\dots \\| m_n$ be the plaintext, $c=c_0 \\| c_1 \\| \\dots \\| c_n$ be the ciphertext and $k=k_0 \\| k_1 \\| \\dots \\| k_m$ be the encryption key (which is repeated until it matches the length of the plaintext).\n", 277 | "\n", 278 | "Encryption works as follows:\n", 279 | "$c_i = Enc(m_i) = m_i + k_i \\mod 26$\n", 280 | "\n", 281 | "Decryption can be performed as:\n", 282 | "$m_i = Dec(c_i) = m_i - k_i \\mod 26$" 283 | ] 284 | }, 285 | { 286 | "cell_type": "code", 287 | "execution_count": null, 288 | "metadata": { 289 | "slideshow": { 290 | "slide_type": "slide" 291 | } 292 | }, 293 | "outputs": [], 294 | "source": [ 295 | "# We use a Vigenère cipher with a key of 14 letters\n", 296 | "\n", 297 | "k = AZ('ABCDEFGHIJKLMN')\n", 298 | "\n", 299 | "m = AZ(\"THECATINTHEHAT\")\n", 300 | "m" 301 | ] 302 | }, 303 | { 304 | "cell_type": "code", 305 | "execution_count": null, 306 | "metadata": { 307 | "slideshow": { 308 | "slide_type": "-" 309 | } 310 | }, 311 | "outputs": [], 312 | "source": [ 313 | "c = encipher_vigenere(m,k)\n", 314 | "c" 315 | ] 316 | }, 317 | { 318 | "cell_type": "code", 319 | "execution_count": null, 320 | "metadata": { 321 | "slideshow": { 322 | "slide_type": "-" 323 | } 324 | }, 325 | "outputs": [], 326 | "source": [ 327 | "decipher_vigenere(c,k)" 328 | ] 329 | }, 330 | { 331 | "cell_type": "markdown", 332 | "metadata": { 333 | "slideshow": { 334 | "slide_type": "slide" 335 | } 336 | }, 337 | "source": [ 338 | "A Vigenère cipher can be attacked with frequency analysis if we know the size of the key.\n", 339 | "\n", 340 | "If the key-length is $t$, then the ciphertext characters $c_1$, $c_{1+t}$, $c_{1+2t}$, $\\dots$ are encrypted using the same shift. Therefore, the frequencies of such characters are expected to be identical to the frequencies of standard English text." 341 | ] 342 | }, 343 | { 344 | "cell_type": "markdown", 345 | "metadata": { 346 | "slideshow": { 347 | "slide_type": "slide" 348 | } 349 | }, 350 | "source": [ 351 | "To find the key size we use the Index of Coincidence Method.\n", 352 | "\n", 353 | "We sample the ciphertext with period $t$ and calculate the frequencies.\n", 354 | "\n", 355 | "If the key length is $t$, then the letters in the sampled text have the same frequencies as in the plaintext language, otherwise the letters will occurr with roughly equal probability." 356 | ] 357 | }, 358 | { 359 | "cell_type": "code", 360 | "execution_count": null, 361 | "metadata": { 362 | "slideshow": { 363 | "slide_type": "slide" 364 | } 365 | }, 366 | "outputs": [], 367 | "source": [ 368 | "ct = AZ(\"\"\"\n", 369 | "YOVTZFJBUOYIPWRLHISZQHEHCJMFJHMLDGGYDZDRDLKECXAYGAJSPFOWYRLWALNZWHOZSAFIFMH\n", 370 | "YVGWBNUVAHMLIWLULTLBTYJGZNJZLPJSZWTRHUABILVVKTBCVWGLKGSCWVUHNAZFOHHJWKMLIWA\n", 371 | "DCVJMXLEKSXYVBSHAKZSNYFOBJCZVSSJVQSYTRVORPEGHFUUNSWFJMFJSPVCNUFLRWLREPZAKGA\n", 372 | "TYIGKNKZWOSKKGRFFZOCZSUMBGBILVJUDQGTBCEMNTDWRNHKWDZYGGGJPJLCUSRUSGLWGFJAYWK\n", 373 | "TYCVDQHZFZDZLUQNUTLZDHEVKNAYGIYJFEAJUKSGJYZWGTMDWFJOFMGJOFDRJCVFHXPELVJPIUC\n", 374 | "SZVIIJUTWGYOVKSJCVFHXORNSYLIJWKPVVVFCVLCWALJSIORNSILJLFTFVVAJFVLWBPCDBTARLH\n", 375 | "JTGLHTLOHCZUULVJTKGAJAYWMMHMWDWLJWBYLUDWYACWPZAYGFWVILCRHEQHMLPOWQSJWSRSVKG\n", 376 | "YLIJWGSVLVFUSSFTXLWGMLIWOKAVJDJYYSDXZFESNUKWZQLTLAFFSWTTBEVKMPTZKNSCJSIBTWA\n", 377 | "DWYSBYHJEHTAYWQTTDGBUSRUSXVDWWSAVDZJJKECWLTSZRTFJSQVXAQFSRFRKHIDSXZVPQNARTZ\n", 378 | "JAYSBRFFOBBOZUVBPCDDJYTWWALZFHMLTAFHBDKHFUTWGNKVLONSNAHMHNWBTAYABLTFJSYORFO\n", 379 | "SVIVWSHIQGZJTWGXPFFCKCVJMSHKMFFSTSIXLJSBILWXSHAJXFTTDQWSMRFQDPNSGSVKWRKVILV\n", 380 | "JKFUWQPKQOSKYMAFUZLMTMDQRNZGGGNAZGBRFKWBILIFSXZFXVJHILKFZVNSSZFUCSZGAQZVLKO\n", 381 | "XAFEOPLDWHMLAWGYVWEMHVDHOSPFFGNDRKSXWVUWFSCQTTUUGTFUZEOQZRFRBHJABIBCYSIIPEM\n", 382 | "UHIWBYZNAHMHXJSFAMSFNLKQCKWVLGBPKZHMLJWWXWVFHRVJLCKTPLWRLRFRSLMWFBHJKCMHGHM\n", 383 | "FZNZSSMVWRNUXSBIJRJSXZZFUYOVEHMPJHSHBCAOWVWUVFYRUHJYXJSBDZLVRFXJCBAYSBIPEEM\n", 384 | "RHEZCTKZVSWPMWRKYFEWYVEWCKTPHFNUTADFSJGIWJVKCKWCWOXBIWHTAYGGJDYGVFCVUVJYZKV\n", 385 | "JKRFOKMVUHNVEXCWHWSWYOWMZFUUKOLHTACZZUGUNUVWRMHIVZDIVSHYOVLFTBSDSTMVPDQHZFW\n", 386 | "SNKZSSHKMFJVILVJPELSSZZLMTMKZSLYRLWKPTSHNVELVZZUWFNCRTZJAYWFJPJKCRLKZWSNZFH\n", 387 | "MLLFGJSWAGMHEVGJSWKOHYZXWHPEYZTCVGTFIIMHJDYAQMNFWGIPIWQYSPLCYOVZSFYKGTMPDOV\n", 388 | "TORKVFKWJSVBVFHTJTSGNVELCYLJLHMLGSZYYPXFNLEVGMPGSBINFKGFTVJTNKVDWYFFXAJYVEO\n", 389 | "SPDSFWPVVSFYCQOSKNSGMHGHMYVWABIPEEMBPWWOIPJHCXPKACSUFLISJFFUJUZSZBPKZADVNFC\n", 390 | "GZVJJNUXEMUHILWFSZLMKVIVCRLJLWHWVLGXOVDCXAEGCUWFJHZUZLMTMGJCHBIABLAYGGJVWLV\n", 391 | "JTFKHFNIWSFICWYNUUOSMHUTWWKJYCQKWAGMHWABJKFYFFISAHXHJEOQSDGBPLPSBIHTSHYOZKZ\n", 392 | "FAKWFBHJSFJTRJYFICQZFYXWOSKSWOZAZXIQHEAAFSVFHNYVDMGSRUYFUUKOLHTACZZKGOSHJLC\n", 393 | "SPJZWSNUWUWLVABXWVSYNUXGTMPJABYLCDWLLEUSRFNATJDYGOYOVSFYDRKBTARDWYACWHNUTLI\n", 394 | "WLUOWYOJMDJYJLWYPFFAFKVXFJXLWBYHCDIXPFFHTAYWOSJZWBYWFHIQHIFCYPFFKMPTZFJNRJR\n", 395 | "JKRDZGSRUYHHKKOXDZLQMLJABIPJYINZVFCYAYSHXOVOOXLMWFXLIACZZLHCSAYAGUVZFHFUUAA\n", 396 | "JUKACSAYWAFAKWFFARDZKVIFCGLKLSWYVSGTUKZOSAYSHNAYSDULEKXZZKFCBAFTSWLDWAGLIWR\n", 397 | "USLLCYOZKKFZKZSHHKKBFTVOOXTPXOAVIAHJWVLOSKGDODTRLSNHCGBJMVVVNTRFRMLRLHJUUWR\n", 398 | "RLNZSWLMWFNDVFHFIFMHYOVZCZZVAHBHJWJJUNAHMKZXTNJLDHDAYSHNJFMZIWIWJJUKZWRMIGA\n", 399 | "KVCDCBPEYAJAYJCZNYLVJZKJSJAJGIWMIASSKJZWUSRKHJKZFHMPJEOSUVJTTYJWJJYRDMJHIKR\n", 400 | "ZYZFUBOZUVRFXWBJYRDHJTGWFFTVFHFUUUVFYRUHJYKZFTBXZHMLZFGYYLESSARDWYFFXHMLWAS\n", 401 | "SKZFHJTGWFFUTWVFKZTZZZYLCHVEXSXZZLSCWVJWJUTWRFYRVWHHCSZYLISHNVEXCWAYWKTYJWW\n", 402 | "LYVORFFSQRFFDGFJTFGRDTFJSNYIAHFICWATYVJSLHIVZJZJGTYOVXSJSZFUXVWGHMLIKWXBWXS\n", 403 | "WLUEMXLCXHTBJWWSAVEDJYRLSQHEYIFNVLCRFRLZJUXLVNLMWBTMWWFJKYWFULIKCSHCNWTSVFQ\n", 404 | "JTPHSYZFXQTBIKSBLIWAFKVLCKLVDHMLTZOSNVABRFUAGUVJAHNVEABTAFFZDUVYZJJKWRGBKAZ\n", 405 | "QBJWRYOVETTYGDIYVYGKJCVJWXAZDZWLKSWSLUKIKMZUWJUKJSLHIVHTYVKHWHZFAJMIGARHCLF\n", 406 | "JHKABLOZEOXPDSRJUFKQWBGDSTMDSZYYVSHNUXLVJYRTPNAJLVJTFFYJFFJSALELVJKFYKMLETM\n", 407 | "FJTARJUKGFYOIGILORXTJJKACSAYWMHHDWWSTPOODILLADKZKSFZVYFJDLHCSTVXCWDYSHIPJWO\n", 408 | "XLZKZNRVSZHVYGZFUUSHQLEYHMLMWBUSLLCBOFOOXUFOPJJFEWSNFDRFUUUCSZVIIJUKDMXVDWK\n", 409 | "MHKHSJCZKVJCVFDQBKGPJNRFHTLOHSWPVFQJAYWSKMVUHXVWEMNSCLSRWVJCSLEAUMAIWHZYEAB\n", 410 | "LOFESRBTZWSAFPWHHKWRKYFECSLFXADORMBYZRTCZAKGKSPWSBHPVVHMHKLVJJRLOAVZVSITPHF\n", 411 | "JZVFQJPJWWELUZWRDYWBNUYAGKYZYVYHKEMAPFDSSJVZSNUWDWHAVVOXSZYVYDFMBIBGGBRFYSB\n", 412 | "IDZLVMPJLSJAYLVJMLJMTMRVSRVEABXARFHQFGGGXLJKSITVAYSLNEMXLCXBTSFFUJYDQCWPXAB\n", 413 | "FSJGIQZVWAJKRLCSJVLCYHBWWYZWDWLOKXFTTDQPTKPSBIHDGFJAYSBKPVFRNZYEOQLMGZJUTWU\n", 414 | 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"TVBXCWAYWPJHJLKMPTZVFKSWSSAYWQFBJWCKZFEIHONJSYJYWRSLJKTTYZZOIHKDSSNKZTNYDDM\n", 558 | "WLJGZALULCUBKAHYVUWOYOYSRNIVWBFICWHTTVWHBPKZWYHKLVJTFESSAKZSWLTGIQKYSJJIVWB\n", 559 | "SVUGIGAFXWYZWSHJILLWYHGHSFYVVHMHKLVJJISTYFRFWRHCZOIIVWBFSRJAJKRLHMLMACQLEUS\n", 560 | "TMDQDWLMACZZRFUJYRFRKVIWPTYVLCUYVKSSAZLGJSWABRFGJSXLELATVUAHNZZEDTZJAPQLKGR\n", 561 | "JZTJWGLFJHTPDSUNUVLVJKVWDYOVTZNZJXIQZVFGJVWJSQPVXKMPTZHMLRTGJUTWCKAYWRJAVKH\n", 562 | "JKTJSFALJSTJTSGNVEWRNUDQPTZFEWYKZVBTADSYJPKKOUWVSFFUTWRZYZFUYOVFWLOKSBIAYMG\n", 563 | "KVIGBJUZYVYHKDSFZKKWSJVAHXPELFTKLUHNVEABYVKZSMVLKSNZFMBISPSBIAISBVBZDZDZCWD\n", 564 | "YHPWGQLGLSALEOWYOKZSGBIVSSVWEIWKVJIUVEEMXVLDHMLJWQTUUSBIAYWHMPIVRFFGSGXLUSB\n", 565 | "IZKAZQTPLCWTVFHTYTSAJUFLCSJVSUFPEAPWLRLVJKRKOKYVWAFUKZSRVEKHJYZFHJYIGFMHUXZ\n", 566 | "JKKZSUYVEWXLJXCWLMWFNZYGIQKSWVTSUAHSVDGFJTPZOUWZFSXZNSGXBGJSRLKZSLBZDHTMDQR\n", 567 | "FYBVSJKUAGYBITSITVTIYSZLHQLJGAJMVOWSXLAFNLJZOIIVWBRHUWPZAKZSXLYSRGLVFFJHUAZ\n", 568 | "DHEKKJYVVSALESGJHIUVMHUTSJUZFGYPKMHJKSMHTMTGIWZVFCYOZFUBHJLCGLUAGHVMWFJKZDC\n", 569 | "TRVVIUVEEMKBKMFJMVDWHPKQOXZVUIWLUMDTUKZSKVLJHMKRQCKAYWOXZRKGNURLWTURHOWAPGT\n", 570 | "YOVHCQPTWQFTVNSWFLFSCWVUHJKCQWSAFLVJOFMGJHEVDWVTWSILUSUFPELCRHBWFNNFJCZZZFJ\n", 571 | "JZKAUFAZGBTMKZSUYVEWXLJKSHBIWVTDVNSWPELVJPEKQWBKSPNSZLMTMDQDQHTWCKJFFQJHCES\n", 572 | "SAZXSQAEGSRIRJFFZJESSANZOYLMWFYOVGTKPTWFXIRVSRLRUQTTGSBDAYWANUKZSNYJWOWJYLV\n", 573 | "JFCWTYUFFCTRFJQTYEWFZUVPDQVIWRFACWBLAYXCWAYWHMPIVCWMFMFYOKAAJAYWMILJUSSKVVW\n", 574 | "SAFLVJJVDZFYZIINCVJSIUFLWSHDMGHSVEMMLRJHGLRLQFSDDMFZKZOYVWGBJDYGGQBDTSWZZFW\n", 575 | "SUFUSSJVAKFSBWRYOVUSQSRJTWVDWBIAFWBIPWGZILUEMFYDKIUVEEMGVJGAFUUJCFTVVSFZZDM\n", 576 | "YVRFRKYFLVJWFDWHLNWFJAYGFTBXZZDZRLWXMZWRFUUHFJWRJSIAFVSUHILHMLXDSJHKEMMLRJH\n", 577 | "BHJLCTZKJCSNKGPJYVKHWHZFSIPSMFSLULCXHPATGBKGBJDFJRGFNSMTMKJWZTGZOSKKGFJUUWF\n", 578 | "IVLTZDZLJSYOVAFFZJMFFUTWCKTPYINSKDSXZEWGXNVFHQLDWBNZRARFACSGYHJLVJWRJHDHJUS\n", 579 | "SKVVHMLJLSUZZVSQPXZHYVYSJJHCDODLUQCZYJMGUPTACSZZOWXOPGIFSCZSFSKZOSKRDWYACWA\n", 580 | "TYVUCZYKWGDIPLVJIPWUJUKDSRLELVNZKZWXPJSJJYPOSQSTGBXAIMQYLUZCZZVABYOVJOGPUVS\n", 581 | "XPIWHTZRQGTTVLVNUXWOXPCQWXJRJQJSPCBJDNZOYPLLHJYVVOYHCDWRHPKODHEWLHLCDSSACQK\n", 582 | "JSCUCSZKJIHAVVVTBJWHMLJWKFSCKOWLPGILVZFULLELZJTVFHMLJWKFSCKOWLJGZNKCQDZAKGU\n", 583 | "JAYWFFUUZSWLKZFTBXZHMLDWFJWYJSSGPGTGYRNOIVZJOUWVVVJHMAZDDZLVFJRFSBOZUVNOVDR\n", 584 | "NUDQVFUUMDTUKZOYCVJMUVILWTUFXHMLSJWHRNGFPIVZWSKNZWHOJLCTKKZSHVIHGJVWLVJDZXS\n", 585 | "TMDQPTZFEPZADSMLVUKVNLCVOSKUWZNCVJAJMIGAYOVXOSNJGTYOVSFHOWASSKEGGTVEWFMHULV\n", 586 | "JYVNSWIVJOYPFFCKTPTZTDJKISRZFHTZZDSSJVLVFUZOOXHEKKJYVVPDHMGWHLWJCRDZLVNUKZS\n", 587 | "YVDTPDHTJMFAWAFXADMTKSVVOSKSJCPLEDWPLKZSXVSTWSNFXOHOZDRFUULVJUHMWHRCQGBLCDW\n", 588 | "SNZFHTVEWZTUXDCZKRFRHVELWSBFMGXJIWORBKLSWSPSBTTRDCZZRFRNUYMAFURZCBSROONSZFU\n", 589 | "XOIASPORDTTMYGFWVISBIORDTTMKJWZTGZGZJYSGRPXZHMHMWOWPJWBTUCQCZAFXVJSCUCSQFAB\n", 590 | "YSPXFTTKZSYOIGOYZFXHMLUSASLUABYOVAFFNFFMFUUGTYOVVSRVEKHMHKWLZSKABYOVVORURLW\n", 591 | "TUFXADVNFHMVLYVYZZLWXMFDZDAFKDJHBKKTVEABLPJLOLNVJSIAFLVJVGHCXPKWKFSCXCWVEWW\n", 592 | "SZKSBYAYWDFYKQIUVELVJZKSWWZIWAFPEWRRVKACSSVKGYOIGILOVPHWLDAHDVWLSWYFJOSKFXO\n", 593 | "BLZFHMLEWLYHUGNJUJLCZARJAXDVJSYVCDWSNRLHMLNSZQPKXSQSSGRNSPLVJJFJDXLRDFJHUQU\n", 594 | "WLRLZDKVUODLUSBIJCGHYLUOWYOXGFJZKGCILIWQYIVXCWLKZSJFVKCKAYWGULTLOYVIKIUVEAH\n", 595 | "XOVSRBPKZFJKVPHJUUWRRVLLVFUUKCQPKSFDLPWCKMZJSXHKLVJOZVSTBJTSFZKOVTZVUFFMKZO\n", 596 | "IZVVIHLUESNUKGAZYUWFFUUOVTZVABKVIEWSNMGWHLYSRHVEKWLUVVAJAFLVJORFURHEAVFKNSZ\n", 597 | "QLULVJTFFGYLIMDBPKZWSAYWHTTS\n", 598 | "\"\"\")" 599 | ] 600 | }, 601 | { 602 | "cell_type": "code", 603 | "execution_count": null, 604 | "metadata": { 605 | "slideshow": { 606 | "slide_type": "slide" 607 | } 608 | }, 609 | "outputs": [], 610 | "source": [ 611 | "# Let's try a key length equal to 1\n", 612 | "t = 1\n", 613 | "\n", 614 | "# sample with step t\n", 615 | "p = ct[0::1]\n", 616 | "\n", 617 | "plot_frequencies(p)" 618 | ] 619 | }, 620 | { 621 | "cell_type": "markdown", 622 | "metadata": {}, 623 | "source": [ 624 | "That the entropy of the uniform distribution is\n", 625 | "$$\n", 626 | " -\\sum_{i=0}^{25} \\frac{1}{26} \\log_2\\frac{1}{26} = \\log_2(26) = 4.7 \n", 627 | "$$\n", 628 | "\n", 629 | "The histogram with $t = 1$ is close to uniform." 630 | ] 631 | }, 632 | { 633 | "cell_type": "code", 634 | "execution_count": null, 635 | "metadata": {}, 636 | "outputs": [], 637 | "source": [ 638 | "# We can test multiple key sizes\n", 639 | "e = []\n", 640 | "for t in range(1,10):\n", 641 | " p = ct[0::t]\n", 642 | " freq = collections.Counter(p)\n", 643 | " e.append(entropy(freq.values()))\n", 644 | "plt.bar(range(1,10),e)\n", 645 | "plt.show()" 646 | ] 647 | }, 648 | { 649 | "cell_type": "code", 650 | "execution_count": null, 651 | "metadata": { 652 | "slideshow": { 653 | "slide_type": "slide" 654 | } 655 | }, 656 | "outputs": [], 657 | "source": [ 658 | "# Probably the key size is 5\n", 659 | "t = 5\n", 660 | "\n", 661 | "p = ct[0::t]\n", 662 | "\n", 663 | "plot_frequencies(p)" 664 | ] 665 | }, 666 | { 667 | "cell_type": "code", 668 | "execution_count": null, 669 | "metadata": { 670 | "slideshow": { 671 | "slide_type": "slide" 672 | } 673 | }, 674 | "outputs": [], 675 | "source": [ 676 | "# The most common letter in the ciphertext (J) could be the letter E in the plaintext\n", 677 | "# The key would be\n", 678 | "ord(\"J\")-ord(\"E\")" 679 | ] 680 | }, 681 | { 682 | "cell_type": "code", 683 | "execution_count": null, 684 | "metadata": { 685 | "slideshow": { 686 | "slide_type": "slide" 687 | } 688 | }, 689 | "outputs": [], 690 | "source": [ 691 | "# repeating for all the letters we find the key \"FHRSO\"\n", 692 | "decipher_vigenere(ct,\"FHRSO\")" 693 | ] 694 | }, 695 | { 696 | "cell_type": "code", 697 | "execution_count": null, 698 | "metadata": { 699 | "slideshow": { 700 | "slide_type": "slide" 701 | } 702 | }, 703 | "outputs": [], 704 | "source": [ 705 | "# compare to the original plaintext\n", 706 | "m = \"\"\"\n", 707 | "THE BLACK CAT\n", 708 | "\n", 709 | " by Edgar Allan Poe\n", 710 | " (1843)\n", 711 | "\n", 712 | " FOR the most wild, yet most homely narrative which I am about to pen, I\n", 713 | "neither expect nor solicit belief. Mad indeed would I be to expect it, in a\n", 714 | "case where my very senses reject their own evidence. Yet, mad am I not --and\n", 715 | "very surely do I not dream. But to-morrow I die, and to-day I would\n", 716 | "unburthen my soul. My immediate purpose is to place before the world,\n", 717 | "plainly, succinctly, and without comment, a series of mere household events.\n", 718 | "In their consequences, these events have terrified --have tortured --have\n", 719 | "destroyed me. Yet I will not attempt to expound them. To me, they have\n", 720 | "presented little but Horror --to many they will seem less terrible than\n", 721 | "baroques. Hereafter, perhaps, some intellect may be found which will reduce\n", 722 | "my phantasm to the common-place --some intellect more calm, more logical,\n", 723 | "and far less excitable than my own, which will perceive, in the\n", 724 | "circumstances I detail with awe, nothing more than an ordinary succession of\n", 725 | "very natural causes and effects.\n", 726 | " From my infancy I was noted for the docility and humanity of my\n", 727 | "disposition. My tenderness of heart was even so conspicuous as to make me\n", 728 | "the jest of my companions. I was especially fond of animals, and was\n", 729 | "indulged by my parents with a great variety of pets. With these I spent most\n", 730 | "of my time, and never was so happy as when feeding and caressing them. This\n", 731 | "peculiar of character grew with my growth, and in my manhood, I derived from\n", 732 | "it one of my principal sources of pleasure. To those who have cherished an\n", 733 | "affection for a faithful and sagacious dog, I need hardly be at the trouble\n", 734 | "of explaining the nature or the intensity of the gratification thus\n", 735 | "derivable. There is something in the unselfish and self-sacrificing love of\n", 736 | "a brute, which goes directly to the heart of him who has had frequent\n", 737 | "occasion to test the paltry friendship and gossamer fidelity of mere Man.\n", 738 | " I married early, and was happy to find in my wife a disposition not\n", 739 | "uncongenial with my own. Observing my partiality for domestic pets, she lost\n", 740 | "no opportunity of procuring those of the most agreeable kind. We had birds,\n", 741 | "gold fish, a fine dog, rabbits, a small monkey, and a cat.\n", 742 | " This latter was a remarkably large and beautiful animal, entirely black,\n", 743 | "and sagacious to an astonishing degree. In speaking of his intelligence, my\n", 744 | "wife, who at heart was not a little tinctured with superstition, made\n", 745 | "frequent allusion to the ancient popular notion, which regarded all black\n", 746 | "cats as witches in disguise. Not that she was ever serious upon this point\n", 747 | "--and I mention the matter at all for no better reason than that it happens,\n", 748 | "just now, to be remembered.\n", 749 | " Pluto --this was the cat's name --was my favorite pet and playmate. I\n", 750 | "alone fed him, and he attended me wherever I went about the house. It was\n", 751 | "even with difficulty that I could prevent him from following me through the\n", 752 | "streets.\n", 753 | " Our friendship lasted, in this manner, for several years, during which my\n", 754 | "general temperament and character --through the instrumentality of the Fiend\n", 755 | "Intemperance --had (I blush to confess it) experienced a radical alteration\n", 756 | "for the worse. I grew, day by day, more moody, more irritable, more\n", 757 | "regardless of the feelings of others. I suffered myself to use intemperate\n", 758 | "language to my At length, I even offered her personal violence. My pets, of\n", 759 | "course, were made to feel the change in my disposition. I not only\n", 760 | "neglected, but ill-used them. For Pluto, however, I still retained\n", 761 | "sufficient regard to restrain me from maltreating him, as I made no scruple\n", 762 | "of maltreating the rabbits, the monkey, or even the dog, when by accident,\n", 763 | "or through affection, they came in my way. But my disease grew upon me --for\n", 764 | "what disease is like Alcohol! --and at length even Pluto, who was now\n", 765 | "becoming old, and consequently somewhat peevish --even Pluto began to\n", 766 | "experience the effects of my ill temper.\n", 767 | " One night, returning home, much intoxicated, from one of my haunts about\n", 768 | "town, I fancied that the cat avoided my presence. I seized him; when, in his\n", 769 | "fright at my violence, he inflicted a slight wound upon my hand with his\n", 770 | "teeth. The fury of a demon instantly possessed me. I knew myself no longer.\n", 771 | "My original soul seemed, at once, to take its flight from my body; and a\n", 772 | "more than fiendish malevolence, gin-nurtured, thrilled every fibre of my\n", 773 | "frame. I took from my waistcoat-pocket a pen-knife, opened it, grasped the\n", 774 | "poor beast by the throat, and deliberately cut one of its eyes from the\n", 775 | "socket! I blush, I burn, I shudder, while I pen the damnable atrocity.\n", 776 | " When reason returned with the morning --when I had slept off the fumes of\n", 777 | "the night's debauch --I experienced a sentiment half of horror, half of\n", 778 | "remorse, for the crime of which I had been guilty; but it was, at best, a\n", 779 | "feeble and equivocal feeling, and the soul remained untouched. I again\n", 780 | "plunged into excess, and soon drowned in wine all memory of the deed.\n", 781 | " In the meantime the cat slowly recovered. The socket of the lost eye\n", 782 | "presented, it is true, a frightful appearance, but he no longer appeared to\n", 783 | "suffer any pain. He went about the house as usual, but, as might be\n", 784 | "expected, fled in extreme terror at my approach. I had so much of my old\n", 785 | "heart left, as to be at first grieved by this evident dislike on the part of\n", 786 | "a creature which had once so loved me. But this feeling soon gave place to\n", 787 | "irritation. And then came, as if to my final and irrevocable overthrow, the\n", 788 | "spirit of PERVERSENESS. Of this spirit philosophy takes no account. Yet I am\n", 789 | "not more sure that my soul lives, than I am that perverseness is one of the\n", 790 | "primitive impulses of the human heart --one of the indivisible primary\n", 791 | "faculties, or sentiments, which give direction to the character of Man. Who\n", 792 | "has not, a hundred times, found himself committing a vile or a silly action,\n", 793 | "for no other reason than because he knows he should not? Have we not a\n", 794 | "perpetual inclination, in the teeth of our best judgment, to violate that\n", 795 | "which is Law, merely because we understand it to be such? This spirit of\n", 796 | "perverseness, I say, came to my final overthrow. It was this unfathomable\n", 797 | "longing of the soul to vex itself --to offer violence to its own nature --to\n", 798 | "do wrong for the wrong's sake only --that urged me to continue and finally\n", 799 | "to consummate the injury I had inflicted upon the unoffending brute. One\n", 800 | "morning, in cool blood, I slipped a noose about its neck and hung it to the\n", 801 | "limb of a tree; --hung it with the tears streaming from my eyes, and with\n", 802 | "the bitterest remorse at my heart; --hung it because I knew that it had\n", 803 | "loved me, and because I felt it had given me no reason of offence; --hung it\n", 804 | "because I knew that in so doing I was committing a sin --a deadly sin that\n", 805 | "would so jeopardize my immortal soul as to place it --if such a thing were\n", 806 | "possible --even beyond the reach of the infinite mercy of the Most Merciful\n", 807 | "and Most Terrible God.\n", 808 | " On the night of the day on which this cruel deed was done, I was aroused\n", 809 | "from sleep by the cry of fire. The curtains of my bed were in flames. The\n", 810 | "whole house was blazing. It was with great difficulty that my wife, a\n", 811 | "servant, and myself, made our escape from the conflagration. The destruction\n", 812 | "was complete. My entire worldly wealth was swallowed up, and I resigned\n", 813 | "myself thenceforward to despair.\n", 814 | " I am above the weakness of seeking to establish a sequence of cause and\n", 815 | "effect, between the disaster and the atrocity. But I am detailing a chain of\n", 816 | "facts --and wish not to leave even a possible link imperfect. On the day\n", 817 | "succeeding the fire, I visited the ruins. The walls, with one exception, had\n", 818 | "fallen in. This exception was found in a compartment wall, not very thick,\n", 819 | "which stood about the middle of the house, and against which had rested the\n", 820 | "head of my bed. The plastering had here, in great measure, resisted the\n", 821 | "action of the fire --a fact which I attributed to its having been recently\n", 822 | "spread. About this wall a dense crowd were collected, and many persons\n", 823 | "seemed to be examining a particular portion of it with every minute and\n", 824 | "eager attention. The words \"strange!\" \"singular!\" and other similar\n", 825 | "expressions, excited my curiosity. I approached and saw, as if graven in bas\n", 826 | "relief upon the white surface, the figure of a gigantic cat. The impression\n", 827 | "was given with an accuracy truly marvellous. There was a rope about the\n", 828 | "animal's neck.\n", 829 | " When I first beheld this apparition --for I could scarcely regard it as\n", 830 | "less --my wonder and my terror were extreme. But at length reflection came\n", 831 | "to my aid. The cat, I remembered, had been hung in a garden adjacent to the\n", 832 | "house. Upon the alarm of fire, this garden had been immediately filled by\n", 833 | "the crowd --by some one of whom the animal must have been cut from the tree\n", 834 | "and thrown, through an open window, into my chamber. This had probably been\n", 835 | "done with the view of arousing me from sleep. The falling of other walls had\n", 836 | "compressed the victim of my cruelty into the substance of the freshly-spread\n", 837 | "plaster; the lime of which, had then with the flames, and the ammonia from\n", 838 | "the carcass, accomplished the portraiture as I saw it.\n", 839 | " Although I thus readily accounted to my reason, if not altogether to my\n", 840 | "conscience, for the startling fact 'just detailed, it did not the less fall\n", 841 | "to make a deep impression upon my fancy. For months I could not rid myself\n", 842 | "of the phantasm of the cat; and, during this period, there came back into my\n", 843 | "spirit a half-sentiment that seemed, but was not, remorse. I went so far as\n", 844 | "to regret the loss of the animal, and to look about me, among the vile\n", 845 | "haunts which I now habitually frequented, for another pet of the same\n", 846 | "species, and of somewhat similar appearance, with which to supply its place.\n", 847 | "\n", 848 | " One night as I sat, half stupefied, in a den of more than infamy, my\n", 849 | "attention was suddenly drawn to some black object, reposing upon the head of\n", 850 | "one of the immense hogsheads of Gin, or of Rum, which constituted the chief\n", 851 | "furniture of the apartment. I had been looking steadily at the top of this\n", 852 | "hogshead for some minutes, and what now caused me surprise was the fact that\n", 853 | "I had not sooner perceived the object thereupon. I approached it, and\n", 854 | "touched it with my hand. It was a black cat --a very large one --fully as\n", 855 | "large as Pluto, and closely resembling him in every respect but one. Pluto\n", 856 | "had not a white hair upon any portion of his body; but this cat had a large,\n", 857 | "although indefinite splotch of white, covering nearly the whole region of\n", 858 | "the breast.\n", 859 | " Upon my touching him, he immediately arose, purred loudly, rubbed against\n", 860 | "my hand, and appeared delighted with my notice. This, then, was the very\n", 861 | "creature of which I was in search. I at once offered to purchase it of the\n", 862 | "landlord; but this person made no claim to it --knew nothing of it --had\n", 863 | "never seen it before.\n", 864 | " I continued my caresses, and, when I prepared to go home, the animal\n", 865 | "evinced a disposition to accompany me. I permitted it to do so; occasionally\n", 866 | "stooping and patting it as I proceeded. When it reached the house it\n", 867 | "domesticated itself at once, and became immediately a great favorite with my\n", 868 | "wife.\n", 869 | " For my own part, I soon found a dislike to it arising within me. This was\n", 870 | "just the reverse of what I had anticipated; but I know not how or why it was\n", 871 | "--its evident fondness for myself rather disgusted and annoyed. By slow\n", 872 | "degrees, these feelings of disgust and annoyance rose into the bitterness of\n", 873 | "hatred. I avoided the creature; a certain sense of shame, and the\n", 874 | "remembrance of my former deed of cruelty, preventing me from physically\n", 875 | "abusing it. I did not, for some weeks, strike, or otherwise violently ill\n", 876 | "use it; but gradually --very gradually --I came to look upon it with\n", 877 | "unutterable loathing, and to flee silently from its odious presence, as from\n", 878 | "the breath of a pestilence.\n", 879 | " What added, no doubt, to my hatred of the beast, was the discovery, on\n", 880 | "the morning after I brought it home, that, like Pluto, it also had been\n", 881 | "deprived of one of its eyes. This circumstance, however, only endeared it to\n", 882 | "my wife, who, as I have already said, possessed, in a high degree, that\n", 883 | "humanity of feeling which had once been my distinguishing trait, and the\n", 884 | "source of many of my simplest and purest pleasures.\n", 885 | " With my aversion to this cat, however, its partiality for myself seemed\n", 886 | "to increase. It followed my footsteps with a pertinacity which it would be\n", 887 | "difficult to make the reader comprehend. Whenever I sat, it would crouch\n", 888 | "beneath my chair, or spring upon my knees, covering me with its loathsome\n", 889 | "caresses. If I arose to walk it would get between my feet and thus nearly\n", 890 | "throw me down, or, fastening its long and sharp claws in my dress, clamber,\n", 891 | "in this manner, to my breast. At such times, although I longed to destroy it\n", 892 | "with a blow, I was yet withheld from so doing, partly it at by a memory of\n", 893 | "my former crime, but chiefly --let me confess it at once --by absolute dread\n", 894 | "of the beast.\n", 895 | " This dread was not exactly a dread of physical evil-and yet I should be\n", 896 | "at a loss how otherwise to define it. I am almost ashamed to own --yes, even\n", 897 | "in this felon's cell, I am almost ashamed to own --that the terror and\n", 898 | "horror with which the animal inspired me, had been heightened by one of the\n", 899 | "merest chimaeras it would be possible to conceive. My wife had called my\n", 900 | "attention, more than once, to the character of the mark of white hair, of\n", 901 | "which I have spoken, and which constituted the sole visible difference\n", 902 | "between the strange beast and the one I had y si destroyed. The reader will\n", 903 | "remember that this mark, although large, had been originally very\n", 904 | "indefinite; but, by slow degrees --degrees nearly imperceptible, and which\n", 905 | "for a long time my Reason struggled to reject as fanciful --it had, at\n", 906 | "length, assumed a rigorous distinctness of outline. It was now the\n", 907 | "representation of an object that I shudder to name --and for this, above\n", 908 | "all, I loathed, and dreaded, and would have rid myself of the monster had I\n", 909 | "dared --it was now, I say, the image of a hideous --of a ghastly thing --of\n", 910 | "the GALLOWS! --oh, mournful and terrible engine of Horror and of Crime --of\n", 911 | "Agony and of Death!\n", 912 | " And now was I indeed wretched beyond the wretchedness of mere Humanity.\n", 913 | "And a brute beast --whose fellow I had contemptuously destroyed --a brute\n", 914 | "beast to work out for me --for me a man, fashioned in the image of the High\n", 915 | "God --so much of insufferable wo! Alas! neither by day nor by night knew I\n", 916 | "the blessing of Rest any more! During the former the creature left me no\n", 917 | "moment alone; and, in the latter, I started, hourly, from dreams of\n", 918 | "unutterable fear, to find the hot breath of the thing upon my face, and its\n", 919 | "vast weight --an incarnate Night-Mare that I had no power to shake off\n", 920 | "--incumbent eternally upon my heart!\n", 921 | " Beneath the pressure of torments such as these, the feeble remnant of the\n", 922 | "good within me succumbed. Evil thoughts became my sole intimates --the\n", 923 | "darkest and most evil of thoughts. The moodiness of my usual temper\n", 924 | "increased to hatred of all things and of all mankind; while, from the\n", 925 | "sudden, frequent, and ungovernable outbursts of a fury to which I now\n", 926 | "blindly abandoned myself, my uncomplaining wife, alas! was the most usual\n", 927 | "and the most patient of sufferers.\n", 928 | " One day she accompanied me, upon some household errand, into the cellar\n", 929 | "of the old building which our poverty compelled us to inhabit. The cat\n", 930 | "followed me down the steep stairs, and, nearly throwing me headlong,\n", 931 | "exasperated me to madness. Uplifting an axe, and forgetting, in my wrath,\n", 932 | "the childish dread which had hitherto stayed my hand, I aimed a blow at the\n", 933 | "animal which, of course, would have proved instantly fatal had it descended\n", 934 | "as I wished. But this blow was arrested by the hand of my wife. Goaded, by\n", 935 | "the interference, into a rage more than demoniacal, I withdrew my arm from\n", 936 | "her grasp and buried the axe in her brain. She fell dead upon the spot,\n", 937 | "without a groan.\n", 938 | " This hideous murder accomplished, I set myself forthwith, and with entire\n", 939 | "deliberation, to the task of concealing the body. I knew that I could not\n", 940 | "remove it from the house, either by day or by night, without the risk of\n", 941 | "being observed by the neighbors. Many projects entered my mind. At one\n", 942 | "period I thought of cutting the corpse into minute fragments, and destroying\n", 943 | "them by fire. At another, I resolved to dig a grave for it in the floor of\n", 944 | "the cellar. Again, I deliberated about casting it in the well in the yard\n", 945 | "--about packing it in a box, as if merchandize, with the usual arrangements,\n", 946 | "and so getting a porter to take it from the house. Finally I hit upon what I\n", 947 | "considered a far better expedient than either of these. I determined to wall\n", 948 | "it up in the cellar --as the monks of the middle ages are recorded to have\n", 949 | "walled up their victims.\n", 950 | " For a purpose such as this the cellar was well adapted. Its walls were\n", 951 | "loosely constructed, and had lately been plastered throughout with a rough\n", 952 | "plaster, which the dampness of the atmosphere had prevented from hardening.\n", 953 | "Moreover, in one of the walls was a projection, caused by a false chimney,\n", 954 | "or fireplace, that had been filled up, and made to resemble the rest of the\n", 955 | "cellar. I made no doubt that I could readily displace the at this point,\n", 956 | "insert the corpse, and wall the whole up as before, so that no eye could\n", 957 | "detect anything suspicious.\n", 958 | " And in this calculation I was not deceived. By means of a crow-bar I\n", 959 | "easily dislodged the bricks, and, having carefully deposited the body\n", 960 | "against the inner wall, I propped it in that position, while, with little\n", 961 | "trouble, I re-laid the whole structure as it originally stood. Having\n", 962 | "procured mortar, sand, and hair, with every possible precaution, I prepared\n", 963 | "a plaster could not every poss be distinguished from the old, and with this\n", 964 | "I very carefully went over the new brick-work. When I had finished, I felt\n", 965 | "satisfied that all was right. The wall did not present the slightest\n", 966 | "appearance of having been disturbed. The rubbish on the floor was picked up\n", 967 | "with the minutest care. I looked around triumphantly, and said to myself\n", 968 | "--\"Here at least, then, my labor has not been in vain.\"\n", 969 | " My next step was to look for the beast which had been the cause of so\n", 970 | "much wretchedness; for I had, at length, firmly resolved to put it to death.\n", 971 | "Had I been able to meet with it, at the moment, there could have been no\n", 972 | "doubt of its fate; but it appeared that the crafty animal had been alarmed\n", 973 | "at the violence of my previous anger, and forebore to present itself in my\n", 974 | "present mood. It is impossible to describe, or to imagine, the deep, the\n", 975 | "blissful sense of relief which the absence of the detested creature\n", 976 | "occasioned in my bosom. It did not make its appearance during the night\n", 977 | "--and thus for one night at least, since its introduction into the house, I\n", 978 | "soundly and tranquilly slept; aye, slept even with the burden of murder upon\n", 979 | "my soul!\n", 980 | " The second and the third day passed, and still my tormentor came not.\n", 981 | "Once again I breathed as a free-man. The monster, in terror, had fled the\n", 982 | "premises forever! I should behold it no more! My happiness was supreme! The\n", 983 | "guilt of my dark deed disturbed me but little. Some few inquiries had been\n", 984 | "made, but these had been readily answered. Even a search had been instituted\n", 985 | "--but of course nothing was to be discovered. I looked upon my future\n", 986 | "felicity as secured.\n", 987 | " Upon the fourth day of the assassination, a party of the police came,\n", 988 | "very unexpectedly, into the house, and proceeded again to make rigorous\n", 989 | "investigation of the premises. Secure, however, in the inscrutability of my\n", 990 | "place of concealment, I felt no embarrassment whatever. The officers bade me\n", 991 | "accompany them in their search. They left no nook or corner unexplored. At\n", 992 | "length, for the third or fourth time, they descended into the cellar. I\n", 993 | "quivered not in a muscle. My heart beat calmly as that of one who slumbers\n", 994 | "in innocence. I walked the cellar from end to end. I folded my arms upon my\n", 995 | "bosom, and roamed easily to and fro. The police were thoroughly satisfied\n", 996 | "and prepared to depart. The glee at my heart was too strong to be\n", 997 | "restrained. I burned to say if but one word, by way of triumph, and to\n", 998 | "render doubly sure their assurance of my guiltlessness.\n", 999 | " \"Gentlemen,\" I said at last, as the party ascended the steps, \"I delight\n", 1000 | "to have allayed your suspicions. I wish you all health, and a little more\n", 1001 | "courtesy. By the bye, gentlemen, this --this is a very well constructed\n", 1002 | "house.\" (In the rabid desire to say something easily, I scarcely knew what I\n", 1003 | "uttered at all.) --\"I may say an excellently well constructed house. These\n", 1004 | "walls --are you going, gentlemen? --these walls are solidly put together\";\n", 1005 | "and here, through the mere phrenzy of bravado, I rapped heavily, with a cane\n", 1006 | "which I held in my hand, upon that very portion of the brick-work behind\n", 1007 | "which stood the corpse of the wife of my bosom.\n", 1008 | " But may God shield and deliver me from the fangs of the Arch-Fiend! No\n", 1009 | "sooner had the reverberation of my blows sunk into silence than I was\n", 1010 | "answered by a voice from within the tomb! --by a cry, at first muffled and\n", 1011 | "broken, like the sobbing of a child, and then quickly swelling into one\n", 1012 | "long, loud, and continuous scream, utterly anomalous and inhuman --a howl\n", 1013 | "--a wailing shriek, half of horror and half of triumph, such as might have\n", 1014 | "arisen only out of hell, conjointly from the throats of the damned in their\n", 1015 | "agony and of the demons that exult in the damnation.\n", 1016 | " Of my own thoughts it is folly to speak. Swooning, I staggered to the\n", 1017 | "opposite wall. For one instant the party upon the stairs remained\n", 1018 | "motionless, through extremity of terror and of awe. In the next, a dozen\n", 1019 | "stout arms were tolling at the wall. It fell bodily. The corpse, already\n", 1020 | "greatly decayed and clotted with gore, stood erect before the eyes of the\n", 1021 | "spectators. Upon its head, with red extended mouth and solitary eye of fire,\n", 1022 | "sat the hideous beast whose craft had seduced me into murder, and whose\n", 1023 | "informing voice had consigned me to the hangman. I had walled the monster up\n", 1024 | "within the tomb!\n", 1025 | "\"\"\"" 1026 | ] 1027 | }, 1028 | { 1029 | "cell_type": "code", 1030 | "execution_count": null, 1031 | "metadata": {}, 1032 | "outputs": [], 1033 | "source": [] 1034 | } 1035 | ], 1036 | "metadata": { 1037 | "kernelspec": { 1038 | "display_name": "Python 3 (ipykernel)", 1039 | "language": "python", 1040 | "name": "python3" 1041 | }, 1042 | "language_info": { 1043 | "codemirror_mode": { 1044 | "name": "ipython", 1045 | "version": 3 1046 | }, 1047 | "file_extension": ".py", 1048 | "mimetype": "text/x-python", 1049 | "name": "python", 1050 | "nbconvert_exporter": "python", 1051 | "pygments_lexer": "ipython3", 1052 | "version": "3.9.7" 1053 | } 1054 | }, 1055 | "nbformat": 4, 1056 | "nbformat_minor": 2 1057 | } 1058 | -------------------------------------------------------------------------------- /DHKE.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Diffie-Hellman Key Exchange" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": null, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "from cryptography.hazmat.primitives import hashes\n", 17 | "from cryptography.hazmat.primitives.asymmetric import dh\n", 18 | "from cryptography.hazmat.primitives.kdf.hkdf import HKDF\n", 19 | "from cryptography.hazmat.primitives.serialization import (\n", 20 | " Encoding, ParameterFormat, PublicFormat,\n", 21 | " load_pem_parameters, load_pem_public_key\n", 22 | ")" 23 | ] 24 | }, 25 | { 26 | "cell_type": "markdown", 27 | "metadata": {}, 28 | "source": [ 29 | "We play the role of Alice and generate the parameters ($p$, $g$, $q$)" 30 | ] 31 | }, 32 | { 33 | "cell_type": "code", 34 | "execution_count": null, 35 | "metadata": {}, 36 | "outputs": [], 37 | "source": [ 38 | "parameters = dh.generate_parameters(generator=2, key_size=1024)" 39 | ] 40 | }, 41 | { 42 | "cell_type": "markdown", 43 | "metadata": {}, 44 | "source": [ 45 | "We need to send the parameters to Bob, so we serialize the parameters into a string that can be sent over the Internet.\n", 46 | "Bob will load the parameters with the command `parameters = load_pem_parameters(b'...string...')`" 47 | ] 48 | }, 49 | { 50 | "cell_type": "code", 51 | "execution_count": null, 52 | "metadata": {}, 53 | "outputs": [], 54 | "source": [ 55 | "# If you are Bob, comment the following line\n", 56 | "parameters.parameter_bytes(Encoding.PEM,ParameterFormat.PKCS3)\n", 57 | "\n", 58 | "# If you are Bob, uncomment the following line and write the DH parameter string that you received from Alice\n", 59 | "# parameters = load_pem_parameters(b'...parameter string...')" 60 | ] 61 | }, 62 | { 63 | "cell_type": "code", 64 | "execution_count": null, 65 | "metadata": {}, 66 | "outputs": [], 67 | "source": [ 68 | "# Now generate the local keypair\n", 69 | "local_private_key = parameters.generate_private_key()\n", 70 | "local_public_key = local_private_key.public_key()" 71 | ] 72 | }, 73 | { 74 | "cell_type": "code", 75 | "execution_count": null, 76 | "metadata": {}, 77 | "outputs": [], 78 | "source": [ 79 | "# We need to send the local public key to the remote host\n", 80 | "# The remote host will load with the command\n", 81 | "# remote_public_key = load_pem_public_key(b'...string...')\n", 82 | "local_public_key.public_bytes(Encoding.PEM,PublicFormat.SubjectPublicKeyInfo)" 83 | ] 84 | }, 85 | { 86 | "cell_type": "code", 87 | "execution_count": null, 88 | "metadata": {}, 89 | "outputs": [], 90 | "source": [ 91 | "# If you have the remote public key, uncomment the following line\n", 92 | "# remote_public_key = load_pem_public_key(\"...string...\")\n", 93 | "\n", 94 | "# If you do not have the remote public key and are just testing, you can generate a test key with the following lines\n", 95 | "# If you have the remote public key, comment the following lines\n", 96 | "test_private_key = parameters.generate_private_key()\n", 97 | "remote_public_key = test_private_key.public_key()" 98 | ] 99 | }, 100 | { 101 | "cell_type": "code", 102 | "execution_count": null, 103 | "metadata": {}, 104 | "outputs": [], 105 | "source": [ 106 | "# This is the shared group element\n", 107 | "shared_secret = local_private_key.exchange(remote_public_key)" 108 | ] 109 | }, 110 | { 111 | "cell_type": "code", 112 | "execution_count": null, 113 | "metadata": {}, 114 | "outputs": [], 115 | "source": [ 116 | "# You can use derived_key to encrypt a message using any symmetric cipher\n", 117 | "# If possible prefer an authenticated\n", 118 | "derived_key = HKDF(\n", 119 | " algorithm=hashes.SHA256(),\n", 120 | " length=32,\n", 121 | " salt=None,\n", 122 | " info=b'some shared data',\n", 123 | ").derive(shared_secret)\n", 124 | "derived_key" 125 | ] 126 | }, 127 | { 128 | "cell_type": "markdown", 129 | "metadata": {}, 130 | "source": [ 131 | "## Laboratory" 132 | ] 133 | }, 134 | { 135 | "cell_type": "markdown", 136 | "metadata": {}, 137 | "source": [ 138 | "Pick a classmate, decide who's Alice and who's Bob. Generate a key and send a message.\n", 139 | "You can use email or chat to exchange the protocol messages." 140 | ] 141 | } 142 | ], 143 | "metadata": { 144 | "kernelspec": { 145 | "display_name": "Python 3 (ipykernel)", 146 | "language": "python", 147 | "name": "python3" 148 | }, 149 | "language_info": { 150 | "codemirror_mode": { 151 | "name": "ipython", 152 | "version": 3 153 | }, 154 | "file_extension": ".py", 155 | "mimetype": "text/x-python", 156 | "name": "python", 157 | "nbconvert_exporter": "python", 158 | "pygments_lexer": "ipython3", 159 | "version": "3.9.12" 160 | } 161 | }, 162 | "nbformat": 4, 163 | "nbformat_minor": 2 164 | } 165 | -------------------------------------------------------------------------------- /DSA.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Legacy DSA\n", 8 | "(it is a legacy algorithm because it does not use Elliptic Curves)" 9 | ] 10 | }, 11 | { 12 | "cell_type": "code", 13 | "execution_count": 133, 14 | "metadata": {}, 15 | "outputs": [], 16 | "source": [ 17 | "from sympy.ntheory import isprime\n", 18 | "\n", 19 | "from cryptography.hazmat.primitives import hashes\n", 20 | "from cryptography.hazmat.primitives.asymmetric import dsa\n", 21 | "from cryptography.hazmat.primitives.asymmetric import utils #decode_dss_signature" 22 | ] 23 | }, 24 | { 25 | "cell_type": "markdown", 26 | "metadata": {}, 27 | "source": [ 28 | "DSA works over a modular group $Z_p$, with $p$ prime.\n", 29 | "\n", 30 | "The private key is $x \\in [0,q)$.\n", 31 | "\n", 32 | "The public key is $y = g^x \\mod p$" 33 | ] 34 | }, 35 | { 36 | "cell_type": "code", 37 | "execution_count": 154, 38 | "metadata": {}, 39 | "outputs": [ 40 | { 41 | "name": "stdout", 42 | "output_type": "stream", 43 | "text": [ 44 | "Private key is: 961379271991097943513107905027321113521197974689\n", 45 | "Public key is : 48477392754393955024438228634301590594143208439654261402727738814431244365383291877571586040764824333019609272369268314232735617882729981394183302519262765414636339542136111572084592255503665718126044923788418771871782266374209405133883754338260355467090455976840997696455985413893208152786415108744976526421\n" 46 | ] 47 | } 48 | ], 49 | "source": [ 50 | "private_key = dsa.generate_private_key( key_size=1024 )\n", 51 | "public_key = private_key.public_key()\n", 52 | "chosen_hash = hashes.SHA1()\n", 53 | "\n", 54 | "print(\"Private key is: %d\" % private_key.private_numbers().x)\n", 55 | "print(\"Public key is : %s\" % public_key.public_numbers().y)" 56 | ] 57 | }, 58 | { 59 | "cell_type": "code", 60 | "execution_count": 155, 61 | "metadata": {}, 62 | "outputs": [], 63 | "source": [ 64 | "# parameters\n", 65 | "# The group is Zp\n", 66 | "p = private_key.private_numbers().public_numbers.parameter_numbers.p\n", 67 | "# The order is q\n", 68 | "q = private_key.private_numbers().public_numbers.parameter_numbers.q\n", 69 | "# The group generator is g\n", 70 | "g = private_key.private_numbers().public_numbers.parameter_numbers.g\n", 71 | "# The private key is x\n", 72 | "x = private_key.private_numbers().x\n", 73 | "# The public key is y\n", 74 | "y = private_key.private_numbers().public_numbers.y" 75 | ] 76 | }, 77 | { 78 | "cell_type": "code", 79 | "execution_count": 156, 80 | "metadata": {}, 81 | "outputs": [ 82 | { 83 | "data": { 84 | "text/plain": [ 85 | "True" 86 | ] 87 | }, 88 | "execution_count": 156, 89 | "metadata": {}, 90 | "output_type": "execute_result" 91 | } 92 | ], 93 | "source": [ 94 | "# Is q prime?\n", 95 | "isprime(q)" 96 | ] 97 | }, 98 | { 99 | "cell_type": "code", 100 | "execution_count": 157, 101 | "metadata": {}, 102 | "outputs": [ 103 | { 104 | "data": { 105 | "text/plain": [ 106 | "0" 107 | ] 108 | }, 109 | "execution_count": 157, 110 | "metadata": {}, 111 | "output_type": "execute_result" 112 | } 113 | ], 114 | "source": [ 115 | "# Does q divide p-1?\n", 116 | "(p-1) % q" 117 | ] 118 | }, 119 | { 120 | "cell_type": "code", 121 | "execution_count": 158, 122 | "metadata": {}, 123 | "outputs": [ 124 | { 125 | "data": { 126 | "text/plain": [ 127 | "True" 128 | ] 129 | }, 130 | "execution_count": 158, 131 | "metadata": {}, 132 | "output_type": "execute_result" 133 | } 134 | ], 135 | "source": [ 136 | "# is y = g^x mod p?\n", 137 | "pow(g,x,p) == y" 138 | ] 139 | }, 140 | { 141 | "cell_type": "markdown", 142 | "metadata": {}, 143 | "source": [ 144 | "## DSA signature\n", 145 | "To sign we need a random nonce $k \\in [1,q)$.\n", 146 | "\n", 147 | "The signature is the pair $(r,s)$ with\n", 148 | "$$ r = (g^k \\mod p) \\mod q $$\n", 149 | "$$ s = (k^{-1} (h(m) + xr) ) \\mod q$$" 150 | ] 151 | }, 152 | { 153 | "cell_type": "code", 154 | "execution_count": 159, 155 | "metadata": {}, 156 | "outputs": [ 157 | { 158 | "data": { 159 | "text/plain": [ 160 | "'302d021500a5f5b8c0a8947ffa138d628f07e180467ac4d77e021414f5e089dd521b0a171e07e6c412f52aadc6dc9b'" 161 | ] 162 | }, 163 | "execution_count": 159, 164 | "metadata": {}, 165 | "output_type": "execute_result" 166 | } 167 | ], 168 | "source": [ 169 | "message = b'message'\n", 170 | "sig = private_key.sign(message,chosen_hash)\n", 171 | "\n", 172 | "# print the signature as hex\n", 173 | "sig.hex()" 174 | ] 175 | }, 176 | { 177 | "cell_type": "code", 178 | "execution_count": 160, 179 | "metadata": {}, 180 | "outputs": [ 181 | { 182 | "data": { 183 | "text/plain": [ 184 | "(947463253978480775779753407201527889614952781694,\n", 185 | " 119663058055035479453519114597518445521616166043)" 186 | ] 187 | }, 188 | "execution_count": 160, 189 | "metadata": {}, 190 | "output_type": "execute_result" 191 | } 192 | ], 193 | "source": [ 194 | "(r,s)=utils.decode_dss_signature(sig)\n", 195 | "(r,s)" 196 | ] 197 | }, 198 | { 199 | "cell_type": "markdown", 200 | "metadata": {}, 201 | "source": [ 202 | "## Can we verify the signature?\n", 203 | "\n", 204 | "We need\n", 205 | "$$ u_1 = h(m) s^{-1} \\mod q $$\n", 206 | "$$ u_2 = r s^{-1} \\mod q $$\n", 207 | "The signature is verified if\n", 208 | "$$ (g^{u_1}y^{u_2} \\mod p) \\mod q= r $$" 209 | ] 210 | }, 211 | { 212 | "cell_type": "code", 213 | "execution_count": 161, 214 | "metadata": {}, 215 | "outputs": [], 216 | "source": [ 217 | "hasher = hashes.Hash(chosen_hash)\n", 218 | "hasher.update(message)\n", 219 | "digest = hasher.finalize()\n", 220 | "hm = int.from_bytes(digest,\"big\")" 221 | ] 222 | }, 223 | { 224 | "cell_type": "code", 225 | "execution_count": 166, 226 | "metadata": {}, 227 | "outputs": [ 228 | { 229 | "data": { 230 | "text/plain": [ 231 | "947463253978480775779753407201527889614952781694" 232 | ] 233 | }, 234 | "execution_count": 166, 235 | "metadata": {}, 236 | "output_type": "execute_result" 237 | } 238 | ], 239 | "source": [ 240 | "u1 = (hm*pow(s,-1,q)) % q\n", 241 | "u2 = (r*pow(s,-1,q)) % q\n", 242 | "( (pow(g,u1,p)*pow(y,u2,p)) % p) % q" 243 | ] 244 | }, 245 | { 246 | "cell_type": "code", 247 | "execution_count": 163, 248 | "metadata": {}, 249 | "outputs": [ 250 | { 251 | "data": { 252 | "text/plain": [ 253 | "947463253978480775779753407201527889614952781694" 254 | ] 255 | }, 256 | "execution_count": 163, 257 | "metadata": {}, 258 | "output_type": "execute_result" 259 | } 260 | ], 261 | "source": [ 262 | "r" 263 | ] 264 | }, 265 | { 266 | "cell_type": "code", 267 | "execution_count": 168, 268 | "metadata": {}, 269 | "outputs": [ 270 | { 271 | "name": "stdout", 272 | "output_type": "stream", 273 | "text": [ 274 | "OK\n" 275 | ] 276 | } 277 | ], 278 | "source": [ 279 | "# Now we use the library function for signature verification\n", 280 | "try:\n", 281 | " public_key.verify(sig,message,chosen_hash)\n", 282 | " print(\"OK\")\n", 283 | "except:\n", 284 | " print(\"KO\")" 285 | ] 286 | }, 287 | { 288 | "cell_type": "code", 289 | "execution_count": null, 290 | "metadata": {}, 291 | "outputs": [], 292 | "source": [] 293 | } 294 | ], 295 | "metadata": { 296 | "kernelspec": { 297 | "display_name": "Python 3 (ipykernel)", 298 | "language": "python", 299 | "name": "python3" 300 | }, 301 | "language_info": { 302 | "codemirror_mode": { 303 | "name": "ipython", 304 | "version": 3 305 | }, 306 | "file_extension": ".py", 307 | "mimetype": "text/x-python", 308 | "name": "python", 309 | "nbconvert_exporter": "python", 310 | "pygments_lexer": "ipython3", 311 | "version": "3.8.8" 312 | } 313 | }, 314 | "nbformat": 4, 315 | "nbformat_minor": 2 316 | } 317 | -------------------------------------------------------------------------------- /Elliptic Curves.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "## Elliptic Curves" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": 12, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "from cryptography.hazmat.primitives import hashes\n", 17 | "from cryptography.hazmat.primitives.asymmetric import ec\n", 18 | "from cryptography.hazmat.primitives.asymmetric.utils import decode_dss_signature\n", 19 | "from cryptography.hazmat.primitives.serialization import (\n", 20 | " Encoding, ParameterFormat, PublicFormat,\n", 21 | " load_pem_parameters, load_pem_public_key\n", 22 | ")" 23 | ] 24 | }, 25 | { 26 | "cell_type": "markdown", 27 | "metadata": {}, 28 | "source": [ 29 | "## ECDSA\n", 30 | "\n", 31 | "ECDSA is a Digital Signature Algorithm. " 32 | ] 33 | }, 34 | { 35 | "cell_type": "markdown", 36 | "metadata": {}, 37 | "source": [ 38 | "We use NIST curve SECP384r1. The key size is 384 bits.\n", 39 | "There is no need to generate parameters, because the standard already provides the curve equation, a generator and its order.\n", 40 | "\n", 41 | "We proceed with the generation of a private key and of the corresponding public key." 42 | ] 43 | }, 44 | { 45 | "cell_type": "code", 46 | "execution_count": 3, 47 | "metadata": {}, 48 | "outputs": [ 49 | { 50 | "name": "stdout", 51 | "output_type": "stream", 52 | "text": [ 53 | "Private key is: 33635704155717693405353620886293070477745946902130715807204371010321656843010542517516917047316355171352655336034740\n" 54 | ] 55 | } 56 | ], 57 | "source": [ 58 | "private_key = ec.generate_private_key( ec.SECP384R1() )\n", 59 | "print(\"Private key is: %d\" % private_key.private_numbers().private_value)" 60 | ] 61 | }, 62 | { 63 | "cell_type": "code", 64 | "execution_count": 4, 65 | "metadata": {}, 66 | "outputs": [], 67 | "source": [ 68 | "# Alternatively we can generate a key from an integer number (eg 1234) by using\n", 69 | "# private_key = ec.derive_private_key( 1234, ec.SECP384R1() )" 70 | ] 71 | }, 72 | { 73 | "cell_type": "code", 74 | "execution_count": 5, 75 | "metadata": {}, 76 | "outputs": [ 77 | { 78 | "name": "stdout", 79 | "output_type": "stream", 80 | "text": [ 81 | "Public key is : (x = 21263400085053286231367251025669544792039179980223644345493395480206074336308964713630426595512819598350299752861949, y = 5441330656277392332567791299355424278499640228179736378409126419609263799624126099855761639354640175647738023353457)\n" 82 | ] 83 | } 84 | ], 85 | "source": [ 86 | "public_key = private_key.public_key()\n", 87 | "print(\"Public key is : (x = %s, y = %s)\" % (public_key.public_numbers().x,public_key.public_numbers().y))" 88 | ] 89 | }, 90 | { 91 | "cell_type": "markdown", 92 | "metadata": {}, 93 | "source": [ 94 | "Remember that the private key is an integer, while the public key is a point. Now we can sign a message.\n", 95 | "The signature will be encoded using the DER rules, so it will be a string of bits." 96 | ] 97 | }, 98 | { 99 | "cell_type": "code", 100 | "execution_count": 6, 101 | "metadata": {}, 102 | "outputs": [ 103 | { 104 | "name": "stdout", 105 | "output_type": "stream", 106 | "text": [ 107 | "3065023100891e87c0d49e94d640d4425c6fcd8a8f38737589ca448b2ca0830002ed2bab341f427a7d96d752c62232c70783398f0a0230582f0ce0f0d05610bdeef2d8fdc10c8d523aa88cca3037f17a27fd524e03eda028a2ff9c6f28c40cd9570eceb6435e7c\n" 108 | ] 109 | } 110 | ], 111 | "source": [ 112 | "data = b\"this is some data I'd like to sign\"\n", 113 | "signature = private_key.sign(\n", 114 | " data,\n", 115 | " ec.ECDSA(hashes.SHA256())\n", 116 | " )\n", 117 | "print(signature.hex())" 118 | ] 119 | }, 120 | { 121 | "cell_type": "markdown", 122 | "metadata": {}, 123 | "source": [ 124 | "The DSA signature is randomized, so if we run the algorithm again we get a different signature. The interface does not ask for a nonce, so we cannot make the mistake of providing repeated nonces." 125 | ] 126 | }, 127 | { 128 | "cell_type": "code", 129 | "execution_count": 7, 130 | "metadata": {}, 131 | "outputs": [ 132 | { 133 | "name": "stdout", 134 | "output_type": "stream", 135 | "text": [ 136 | "30640230151ba8a8dd4ade8486df61b591e823f8420fa24a7d677c164ab41a05abdf68e35b15815448db4b33148976c40d5f3bc902307bbe53fbd31e258a098790ceb23d0bc92fe71b429bfdb6f1608bee652c7429a400fdc6e14cde80829747be7d7ad979b0\n" 137 | ] 138 | } 139 | ], 140 | "source": [ 141 | "signature = private_key.sign(\n", 142 | " data,\n", 143 | " ec.ECDSA(hashes.SHA256())\n", 144 | " )\n", 145 | "print(signature.hex())" 146 | ] 147 | }, 148 | { 149 | "cell_type": "markdown", 150 | "metadata": {}, 151 | "source": [ 152 | "We can decode the signature and retrieve the parameters $r$ and $s$" 153 | ] 154 | }, 155 | { 156 | "cell_type": "code", 157 | "execution_count": 8, 158 | "metadata": {}, 159 | "outputs": [ 160 | { 161 | "name": "stdout", 162 | "output_type": "stream", 163 | "text": [ 164 | "r = 3248825051445311210727916863346667328660788145739950405075486158747012988304000235783552321632121300395825704156105\n", 165 | "s = 19045863015172608162317729194239143865899170147418674052844195550884758407551556669873259350475778252494769103665584\n" 166 | ] 167 | } 168 | ], 169 | "source": [ 170 | "(r,s)=decode_dss_signature(signature)\n", 171 | "print(\"r = %d\\ns = %d\" % (r,s) )" 172 | ] 173 | }, 174 | { 175 | "cell_type": "markdown", 176 | "metadata": {}, 177 | "source": [ 178 | "We can verify the signature using the public key. If the signature is wrong, we get an exception" 179 | ] 180 | }, 181 | { 182 | "cell_type": "code", 183 | "execution_count": 9, 184 | "metadata": {}, 185 | "outputs": [ 186 | { 187 | "name": "stdout", 188 | "output_type": "stream", 189 | "text": [ 190 | "OK\n" 191 | ] 192 | } 193 | ], 194 | "source": [ 195 | "try:\n", 196 | " public_key.verify(signature,data,ec.ECDSA(hashes.SHA256()))\n", 197 | " print(\"OK\")\n", 198 | "except:\n", 199 | " print(\"KO\")" 200 | ] 201 | }, 202 | { 203 | "cell_type": "markdown", 204 | "metadata": {}, 205 | "source": [ 206 | "## Lab" 207 | ] 208 | }, 209 | { 210 | "cell_type": "markdown", 211 | "metadata": {}, 212 | "source": [ 213 | "Find a classmate. One plays as Alice, the other plays as Bob.\n", 214 | "Alice and Bob generate their keys from secret numbers and have published the corresponding public key.\n", 215 | "\n", 216 | "Alice must prove to Bob that she is Alice.\n", 217 | "She derives her private key from her secret number and signs a message.\n", 218 | "Then she sends to Bob the message and the signature.\n", 219 | "Bob verifies the signature using Alice's public key.\n", 220 | "\n", 221 | "Bob proves his identity to Alice in the same way." 222 | ] 223 | }, 224 | { 225 | "cell_type": "code", 226 | "execution_count": 25, 227 | "metadata": {}, 228 | "outputs": [], 229 | "source": [ 230 | "# Alice\n", 231 | "Alice_secret_number = 1234\n", 232 | "Alice_serialized_public_key = b'-----BEGIN PUBLIC KEY-----\\nMHYwEAYHKoZIzj0CAQYFK4EEACIDYgAEe2FShOUqdr8UOfiRMoGoEdS7PY4quCzZ\\nprCsAxK1ft1/zoRprQ4Jsxul09T4aB2hG32SVNtusdPAn8ziDFnNH4rCtVJedReL\\nURy3R3RY1EqiZ/y6ZLLo7NWdnbXWvQ7C\\n-----END PUBLIC KEY-----\\n'\n", 233 | "Alice_message = b\"I'm Alice\"\n", 234 | "\n", 235 | "# You can load the other party's public key with\n", 236 | "# remote_public_key = load_pem_public_key( serialized_public_key )" 237 | ] 238 | }, 239 | { 240 | "cell_type": "code", 241 | "execution_count": 26, 242 | "metadata": {}, 243 | "outputs": [], 244 | "source": [ 245 | "# Bob\n", 246 | "Bob_secret_number = 4321\n", 247 | "Bob_serialized_public_key = b'-----BEGIN PUBLIC KEY-----\\nMHYwEAYHKoZIzj0CAQYFK4EEACIDYgAENyisOoDb9lce4qW9DP+LueJUDXeCm89Y\\n5ZYW0sx50UdwMwgWWk+HMNfct+tBW9AY2jqMRCPJgyIu9BucU38JuwchT6gsZrQi\\nRJTXpndZMbKHD6j31yyaSZ2j9nGxNXkO\\n-----END PUBLIC KEY-----\\n'\n", 248 | "Bob_message = b\"I'm Bob\"\n", 249 | "\n", 250 | "# You can load the other party's public key with\n", 251 | "# remote_public_key = load_pem_public_key( serialized_public_key )" 252 | ] 253 | } 254 | ], 255 | "metadata": { 256 | "kernelspec": { 257 | "display_name": "Python 3 (ipykernel)", 258 | "language": "python", 259 | "name": "python3" 260 | }, 261 | "language_info": { 262 | "codemirror_mode": { 263 | "name": "ipython", 264 | "version": 3 265 | }, 266 | "file_extension": ".py", 267 | "mimetype": "text/x-python", 268 | "name": "python", 269 | "nbconvert_exporter": "python", 270 | "pygments_lexer": "ipython3", 271 | "version": "3.8.8" 272 | } 273 | }, 274 | "nbformat": 4, 275 | "nbformat_minor": 1 276 | } 277 | -------------------------------------------------------------------------------- /Hash Functions.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Some Tests with Hash Functions" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": null, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "from binascii import hexlify\n", 17 | "from os import urandom\n", 18 | "from hashlib import sha256, shake_128\n", 19 | "import numpy as np\n", 20 | "import matplotlib.pyplot as plt" 21 | ] 22 | }, 23 | { 24 | "cell_type": "markdown", 25 | "metadata": {}, 26 | "source": [ 27 | "# Example 1" 28 | ] 29 | }, 30 | { 31 | "cell_type": "markdown", 32 | "metadata": {}, 33 | "source": [ 34 | "Let's hash an ASCII string with SHA256. Print the output will as a hexadecimal string" 35 | ] 36 | }, 37 | { 38 | "cell_type": "code", 39 | "execution_count": null, 40 | "metadata": {}, 41 | "outputs": [], 42 | "source": [ 43 | "x = b\"test message\"\n", 44 | "y = sha256(x).hexdigest()\n", 45 | "y" 46 | ] 47 | }, 48 | { 49 | "cell_type": "markdown", 50 | "metadata": {}, 51 | "source": [ 52 | "# Example 2" 53 | ] 54 | }, 55 | { 56 | "cell_type": "markdown", 57 | "metadata": {}, 58 | "source": [ 59 | "Consider the hash SHAKE128 with output 16 bits. Generate a random digest and find a preimage. Print the number of attempts." 60 | ] 61 | }, 62 | { 63 | "cell_type": "code", 64 | "execution_count": null, 65 | "metadata": {}, 66 | "outputs": [], 67 | "source": [ 68 | "DIGESTSIZE = 2\n", 69 | "\n", 70 | "# Define a hash function h with output size DIGESTSIZE bytes \n", 71 | "h = lambda x: shake_128(x).digest(DIGESTSIZE)\n", 72 | "\n", 73 | "# y is the target\n", 74 | "y = urandom(DIGESTSIZE)\n", 75 | "\n", 76 | "i = 0\n", 77 | "yp = None\n", 78 | "while (yp != y):\n", 79 | " # Generate a random 64 bit input\n", 80 | " xp = urandom(8)\n", 81 | " yp = h(xp)\n", 82 | " i = i + 1\n", 83 | "\n", 84 | "print(\"Attempt #{}\".format(i) )\n", 85 | "print(\"Preimage is {}\".format(hexlify(xp)) ) " 86 | ] 87 | }, 88 | { 89 | "cell_type": "markdown", 90 | "metadata": {}, 91 | "source": [ 92 | "# Example 3" 93 | ] 94 | }, 95 | { 96 | "cell_type": "markdown", 97 | "metadata": {}, 98 | "source": [ 99 | "Find, empirically, the average number of attempts to find a preimage to SHAKE128 with output 8 bits" 100 | ] 101 | }, 102 | { 103 | "cell_type": "code", 104 | "execution_count": null, 105 | "metadata": {}, 106 | "outputs": [], 107 | "source": [ 108 | "DIGESTSIZE = 1\n", 109 | "num_simulations = 1000\n", 110 | "h = lambda x: shake_128(x).digest(DIGESTSIZE)\n", 111 | "\n", 112 | "def simulate(y):\n", 113 | " i = 0\n", 114 | " yp = None\n", 115 | " while (y != yp):\n", 116 | " # Generate a random 64 bit input\n", 117 | " xp = urandom(8)\n", 118 | " yp = h(xp)\n", 119 | " i = i + 1\n", 120 | " return(i) \n", 121 | "\n", 122 | "messages = [ urandom(DIGESTSIZE) for _ in range(num_simulations) ]\n", 123 | "simulations = [simulate(m) for m in messages]\n", 124 | "\n", 125 | "np.mean(simulations)" 126 | ] 127 | }, 128 | { 129 | "cell_type": "code", 130 | "execution_count": null, 131 | "metadata": {}, 132 | "outputs": [], 133 | "source": [ 134 | "plt.hist(simulations,32);\n", 135 | "plt.xlabel('Number of attempts')\n", 136 | "plt.ylabel('Occurrences')" 137 | ] 138 | }, 139 | { 140 | "cell_type": "markdown", 141 | "metadata": {}, 142 | "source": [ 143 | "# Example 4" 144 | ] 145 | }, 146 | { 147 | "cell_type": "markdown", 148 | "metadata": {}, 149 | "source": [ 150 | "Find the empirical probability of finding a second preimage with $q=100$ attempts " 151 | ] 152 | }, 153 | { 154 | "cell_type": "code", 155 | "execution_count": null, 156 | "metadata": {}, 157 | "outputs": [], 158 | "source": [ 159 | "DIGESTSIZE = 2\n", 160 | "# attempts\n", 161 | "q = 100\n", 162 | "# number of simulations\n", 163 | "num_simulations = 10000\n", 164 | "\n", 165 | "h = lambda x: shake_128(x).digest(DIGESTSIZE)\n", 166 | "\n", 167 | "def simulate(message,attempts):\n", 168 | " y = h(message)\n", 169 | " xp = [ urandom(8) for _ in range(attempts) ]\n", 170 | " yp = [ h(x) for x in xp]\n", 171 | " return ( y in yp )\n", 172 | "\n", 173 | "# generate many random messages\n", 174 | "messages = [urandom(8) for _ in range(num_simulations)]\n", 175 | "\n", 176 | "# simulate finding a second preimage\n", 177 | "simulations = [ simulate(m,q) for m in messages]\n", 178 | "\n", 179 | "# result\n", 180 | "p_succ = (sum(simulations)) / num_simulations\n", 181 | "p_theory = q / pow(2,DIGESTSIZE * 8)\n", 182 | "\n", 183 | "print(\"Simulations: p={}\".format(p_succ))\n", 184 | "print(\"Theory: p={}\".format(p_theory))" 185 | ] 186 | }, 187 | { 188 | "cell_type": "markdown", 189 | "metadata": {}, 190 | "source": [ 191 | "# Example 5" 192 | ] 193 | }, 194 | { 195 | "cell_type": "markdown", 196 | "metadata": {}, 197 | "source": [ 198 | "Find the empirical probability of finding a collision with q=30 attempts.\n", 199 | "Use a 16-bit hash function" 200 | ] 201 | }, 202 | { 203 | "cell_type": "code", 204 | "execution_count": null, 205 | "metadata": {}, 206 | "outputs": [], 207 | "source": [ 208 | "DIGESTSIZE = 2\n", 209 | "# attempts\n", 210 | "q = 30\n", 211 | "# number of simulations\n", 212 | "num_simulations = 10000\n", 213 | "\n", 214 | "h = lambda x: shake_128(x).digest(DIGESTSIZE)\n", 215 | "\n", 216 | "def simulate(attempts):\n", 217 | " xp = [ urandom(8) for _ in range(attempts) ]\n", 218 | " yp = [ h(x) for x in xp]\n", 219 | " # remove unique items\n", 220 | " ys = set(yp)\n", 221 | " # if they are the same, there were no duplicates \n", 222 | " return (len(ys) != len(yp))\n", 223 | "\n", 224 | "# simulate finding a collision\n", 225 | "simulations = [ simulate(q) for _ in range(num_simulations)]\n", 226 | "# result\n", 227 | "p_succ = (sum(simulations)) / num_simulations\n", 228 | "p_theory = 1-np.exp(-q**2 / pow(2,DIGESTSIZE*8+1))\n", 229 | "\n", 230 | "print(\"Simulations: p={}\".format(p_succ))\n", 231 | "print(\"Theory: p={}\".format(p_theory))" 232 | ] 233 | }, 234 | { 235 | "cell_type": "markdown", 236 | "metadata": {}, 237 | "source": [ 238 | "# Lab Work\n", 239 | "\n", 240 | "Work in pairs. One is Alice, one is Bob.\n", 241 | "\n", 242 | "Use SHAKE128 with 8 bits output.\n", 243 | "\n", 244 | "Alice generates a random input and sends it to Bob. Bob finds a second preimage and sends it to Alice.\n", 245 | "Alice verifies that the second preimage is correct.\n", 246 | "\n", 247 | "How many attempts did Bob make?\n", 248 | "\n", 249 | "Now swap your roles and use SHAKE128 with 16 bits of output." 250 | ] 251 | }, 252 | { 253 | "cell_type": "markdown", 254 | "metadata": {}, 255 | "source": [ 256 | "# Challenge" 257 | ] 258 | }, 259 | { 260 | "cell_type": "markdown", 261 | "metadata": {}, 262 | "source": [ 263 | "For this challenge use the hash function SHAKE128 with 1 byte of output. Your goal it to find a second preimage for the following input. Measure the time it takes, then increase the output size by one byte and repeat. Plot your measurements in a graph." 264 | ] 265 | }, 266 | { 267 | "cell_type": "code", 268 | "execution_count": null, 269 | "metadata": {}, 270 | "outputs": [], 271 | "source": [ 272 | "input_string = b'The frog jumped into the pond'" 273 | ] 274 | } 275 | ], 276 | "metadata": { 277 | "anaconda-cloud": {}, 278 | "kernelspec": { 279 | "display_name": "Python 3 (ipykernel)", 280 | "language": "python", 281 | "name": "python3" 282 | }, 283 | "language_info": { 284 | "codemirror_mode": { 285 | "name": "ipython", 286 | "version": 3 287 | }, 288 | "file_extension": ".py", 289 | "mimetype": "text/x-python", 290 | "name": "python", 291 | "nbconvert_exporter": "python", 292 | "pygments_lexer": "ipython3", 293 | "version": "3.11.7" 294 | } 295 | }, 296 | "nbformat": 4, 297 | "nbformat_minor": 4 298 | } 299 | -------------------------------------------------------------------------------- /Information-Theoretic Security.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Information-Theoretic Security (examples and exercises)" 8 | ] 9 | }, 10 | { 11 | "cell_type": "markdown", 12 | "metadata": {}, 13 | "source": [ 14 | "## Definitions of Information Entropy" 15 | ] 16 | }, 17 | { 18 | "cell_type": "markdown", 19 | "metadata": {}, 20 | "source": [ 21 | "### Source Entropy\n", 22 | "\n", 23 | "Given a message $m$ taken from a set of messages $\\{m_1, \\dots, m_N\\}$ with probabilities $p(m_i)\\ 1\\leq i \\leq N$, the source entropy is\n", 24 | "\n", 25 | "$H(m) = -\\sum_{i=1}^N p_i \\log_2 p_i$\n", 26 | "\n", 27 | "Source entropy measures the minimum number of bits necessary to encode the source messages." 28 | ] 29 | }, 30 | { 31 | "cell_type": "markdown", 32 | "metadata": {}, 33 | "source": [ 34 | "### Joint Entropy\n", 35 | "\n", 36 | "Given a message $m$ taken from a set of messages $\\{m_i\\}$ with probabilities $p(m_i)$ and a message $c$ taken from a set of messages $\\{c_j\\}$ with probabilities $p(c_j)$ and joint probabilities $p(m_i,c_j)$, the joint entropy is\n", 37 | "\n", 38 | "$H(m,c) = - \\sum_{i,j} p(m_i,c_j) \\log_2 p(m_i,c_j)$" 39 | ] 40 | }, 41 | { 42 | "cell_type": "markdown", 43 | "metadata": {}, 44 | "source": [ 45 | "### Conditional Entropy\n", 46 | "\n", 47 | "Given a message $m$ taken from a set of messages $\\{m_i\\}$ with probabilities $p(m_i)$ and a ciphertext $c$ taken from a set of messages $\\{c_j\\}$ with probabilities $p(c_j)$ and joint probabilities $p(m_i,c_j)$, the conditional entropy is\n", 48 | "\n", 49 | "$H(c|m) = \\sum_i H(c|m_i) \\Pr[m_i]$" 50 | ] 51 | }, 52 | { 53 | "cell_type": "markdown", 54 | "metadata": {}, 55 | "source": [ 56 | "### Chain Rule\n", 57 | "\n", 58 | "$H(m,c) = H(m) + H(c|m) = H(c) + H(m|c)$" 59 | ] 60 | }, 61 | { 62 | "cell_type": "markdown", 63 | "metadata": {}, 64 | "source": [ 65 | "### Bayes' Theorem\n", 66 | "\n", 67 | "$H(m|c) = H(c|m) - H(c) + H(m)$" 68 | ] 69 | }, 70 | { 71 | "cell_type": "markdown", 72 | "metadata": {}, 73 | "source": [ 74 | "### Mutual Information\n", 75 | "\n", 76 | "$I(m;c) = H(m)-H(m|c) = H(c)-H(c|m)$\n", 77 | "\n", 78 | "Mutual information is equal to zero if and only if the message and the ciphertext are independent." 79 | ] 80 | }, 81 | { 82 | "cell_type": "markdown", 83 | "metadata": {}, 84 | "source": [ 85 | "## Exercise 1\n", 86 | "\n", 87 | "Assume that the message space is $M = \\{\\texttt{A}, \\texttt{Z} \\}$.\n", 88 | "Alice chooses $m=\\texttt{A}$ with probability 0.7 and $m=\\texttt{Z}$ with probability 0.3. Alice encrypts the message with some random key.\n", 89 | "\n", 90 | "Eve, an eavesdropper, sees $c=\\texttt{B}$. What can she say about the plaintext?\n", 91 | "What is $I(m;c)$?" 92 | ] 93 | }, 94 | { 95 | "cell_type": "markdown", 96 | "metadata": {}, 97 | "source": [ 98 | "**Solution (analytical)**\n", 99 | "\n", 100 | "The entropy of the message source is:\n", 101 | "\n", 102 | "$H(m) =\n", 103 | "- \\Pr[m=\\texttt{A}] \\log_2 \\Pr[m=\\texttt{A}]\n", 104 | "- \\Pr[m=\\texttt{Z}] \\log_2 \\Pr[m=\\texttt{Z}]\t\n", 105 | "= 0.88\\,\\mathrm{bit}$\n", 106 | "\n", 107 | "Given a plaintext, all ciphertexts are equally likely, so \n", 108 | "\n", 109 | "$H(c|m) = 26 \\times \\frac{1}{26}\\log_2 26 = 4.7\\,\\mathrm{bit}$\n", 110 | "\n", 111 | "We already know that\n", 112 | "\n", 113 | "$\\Pr[c=\\texttt{A}]=\\dots=\\Pr[c=\\texttt{Z}]=1/26$\n", 114 | "\n", 115 | "thus $H(c)=4.7\\,\\mathrm{bit}$.\n", 116 | "\n", 117 | "Therefore:\n", 118 | "\n", 119 | "$I(m;c) = H(c) - H(c|m) = 0$\n", 120 | "\n", 121 | "No information is learned." 122 | ] 123 | }, 124 | { 125 | "cell_type": "markdown", 126 | "metadata": {}, 127 | "source": [ 128 | "**Solution (empirical)**" 129 | ] 130 | }, 131 | { 132 | "cell_type": "code", 133 | "execution_count": 1, 134 | "metadata": {}, 135 | "outputs": [], 136 | "source": [ 137 | "from random import random, randint\n", 138 | "import numpy as np" 139 | ] 140 | }, 141 | { 142 | "cell_type": "code", 143 | "execution_count": 3, 144 | "metadata": {}, 145 | "outputs": [], 146 | "source": [ 147 | "# calculates the entropy of a list of elements\n", 148 | "# builds the empyrical pdf of the list\n", 149 | "# then applies the definition\n", 150 | "def entropy_of_list(l):\n", 151 | " histogram = {x:(l.count(x)+0.0)/len(l) for x in l}\n", 152 | " return sum([-x*log(x,2) for x in histogram.values()])" 153 | ] 154 | }, 155 | { 156 | "cell_type": "code", 157 | "execution_count": null, 158 | "metadata": {}, 159 | "outputs": [], 160 | "source": [ 161 | "# Instantiate a shift cipher over the alphabet\n", 162 | "S = ShiftCryptosystem(AlphabeticStrings());\n", 163 | "\n", 164 | "# Define the random key generator\n", 165 | "def keygen():\n", 166 | " return randint(0,25)" 167 | ] 168 | }, 169 | { 170 | "cell_type": "code", 171 | "execution_count": 2, 172 | "metadata": {}, 173 | "outputs": [], 174 | "source": [ 175 | "# message source\n", 176 | "def message1():\n", 177 | " if random()<=0.7:\n", 178 | " return 'A'\n", 179 | " else:\n", 180 | " return 'Z'" 181 | ] 182 | }, 183 | { 184 | "cell_type": "code", 185 | "execution_count": 11, 186 | "metadata": {}, 187 | "outputs": [ 188 | { 189 | "name": "stdout", 190 | "output_type": "stream", 191 | "text": [ 192 | "Empirical Entropy of source is: 0.893173458377857\n" 193 | ] 194 | } 195 | ], 196 | "source": [ 197 | "# generate several random messages \n", 198 | "m = [S.encoding(message1()) for i in range(1000)]\n", 199 | "\n", 200 | "# print empirical entropy of the message source\n", 201 | "print \"Empirical Entropy of source is:\", entropy_of_list(m)" 202 | ] 203 | }, 204 | { 205 | "cell_type": "code", 206 | "execution_count": 12, 207 | "metadata": {}, 208 | "outputs": [ 209 | { 210 | "name": "stdout", 211 | "output_type": "stream", 212 | "text": [ 213 | "Empirical Entropy of ciphertext is: 4.68926722828748\n" 214 | ] 215 | } 216 | ], 217 | "source": [ 218 | "# Encrypt each message with a different random key\n", 219 | "c = [ S.enciphering( keygen() , x ) for x in m ]\n", 220 | "\n", 221 | "print \"Empirical Entropy of ciphertext is:\", entropy_of_list(c)" 222 | ] 223 | }, 224 | { 225 | "cell_type": "markdown", 226 | "metadata": {}, 227 | "source": [ 228 | "## Exercise 2\n", 229 | "\n", 230 | "Assume that $\\Pr[m=\\texttt{kim}]=0.5$, $\\Pr[m=\\texttt{ann}]=0.2$, and $\\Pr[m=\\texttt{boo}]=0.3$.\n", 231 | "\n", 232 | "What is $I(m;c)$?" 233 | ] 234 | }, 235 | { 236 | "cell_type": "markdown", 237 | "metadata": {}, 238 | "source": [ 239 | "**Solution (analytical)** \n", 240 | "Analogous calculations can be done using information entropy.\n", 241 | "The entropy of the message source is:\n", 242 | "\n", 243 | "$H(m) = 1.49 bit$\n", 244 | "\n", 245 | "Ciphertexts are not equally likely. There are only $26+26=52$ different CTs, half of which obtained from $\\texttt{kim}$ and half from $\\texttt{ann}$ and $\\texttt{boo}$.\n", 246 | "\n", 247 | "$H(c) = - 52 \\frac{1}{52}\\log_2\\frac{1}{52} = 5.7 bit$\n", 248 | "\n", 249 | "Given a plaintex, all ciphertexts are equally likely:\n", 250 | "$H(c|m) = 4.7 bit$.\n", 251 | "\n", 252 | "$I(m;c) = H(c) -H(c|m) = 1 bit$\n", 253 | "meaning that knowledge of $c$ gives us 1 bit of information.\n", 254 | "\n", 255 | "What does it mean that mutual information is 1 bit? It means that there exists an agorithm that can answer a yes/no question about the plaintext by just looking to the ciphertext.\n", 256 | "\n", 257 | "This does not tell us what is the algorithm to obtain this bit. Neither whether this algorithm is efficient. It only tells us that this algorithm exists.\n", 258 | "The following is just an example:\n", 259 | "\n", 260 | "```\n", 261 | "def plaintext_is_kim(ciphertext):\n", 262 | " if last two letters of ciphertext are the same:\n", 263 | " return false # plaintext is not kim\n", 264 | " else\n", 265 | " return true # plaintext is kim\n", 266 | " end if\n", 267 | "```" 268 | ] 269 | }, 270 | { 271 | "cell_type": "markdown", 272 | "metadata": {}, 273 | "source": [ 274 | "**Solution (empirical)**" 275 | ] 276 | }, 277 | { 278 | "cell_type": "code", 279 | "execution_count": 14, 280 | "metadata": {}, 281 | "outputs": [], 282 | "source": [ 283 | "# generate random three letter messages\n", 284 | "def message2():\n", 285 | " r = random() \n", 286 | " if r <= 0.5:\n", 287 | " return 'KIM'\n", 288 | " if r <= 0.7:\n", 289 | " return 'ANN'\n", 290 | " else:\n", 291 | " return 'BOO'" 292 | ] 293 | }, 294 | { 295 | "cell_type": "code", 296 | "execution_count": 15, 297 | "metadata": {}, 298 | "outputs": [ 299 | { 300 | "name": "stdout", 301 | "output_type": "stream", 302 | "text": [ 303 | "Empirical Entropy of source is: 1.47717700758957\n" 304 | ] 305 | } 306 | ], 307 | "source": [ 308 | "# generate random messages \n", 309 | "m = [S.encoding(message2()) for i in range(1000)]\n", 310 | "\n", 311 | "# print empirical entropy of the message source\n", 312 | "print \"Empirical Entropy of source is:\", entropy_of_list(m)" 313 | ] 314 | }, 315 | { 316 | "cell_type": "code", 317 | "execution_count": 16, 318 | "metadata": {}, 319 | "outputs": [ 320 | { 321 | "name": "stdout", 322 | "output_type": "stream", 323 | "text": [ 324 | "Empirical Entropy of ciphertext is: 5.67313984293835\n" 325 | ] 326 | } 327 | ], 328 | "source": [ 329 | "# Encrypt each message with a random key\n", 330 | "c=[S.enciphering( keygen(), x) for x in m]\n", 331 | "\n", 332 | "print \"Empirical Entropy of ciphertext is:\", entropy_of_list(c)" 333 | ] 334 | }, 335 | { 336 | "cell_type": "markdown", 337 | "metadata": {}, 338 | "source": [ 339 | "## Exercise 3\n", 340 | "\n", 341 | "Consider a Vigènere cipher. Messages are 4-character sequences of letters. Passwords are $n$-character long.\n", 342 | "\n", 343 | "Assume that there are three possible messages, with different probabilities.\n", 344 | "\n", 345 | "| Message | Probability |\n", 346 | "|---|---|\n", 347 | "| $m_1$ = BEBA | 0.5 |\n", 348 | "| $m_2$ = DEDA | 0.3 |\n", 349 | "| $m_3$ = CFCB | 0.2 |\n", 350 | "\n", 351 | "1. Calculate the entropy of the source.\n", 352 | "2. Calculate the entropy of the key with $n=1,2,4$.\n", 353 | "3. Calculate the entropy of the ciphertext with $n=1,2,4$.\n", 354 | "4. Calculate the information leak of the ciphertext with $n=1,2,4$\n", 355 | "5. Is the cipher of this exercise perfectly secret?" 356 | ] 357 | }, 358 | { 359 | "cell_type": "markdown", 360 | "metadata": {}, 361 | "source": [ 362 | "### Solution\n", 363 | "\n", 364 | "#### Part 1\n", 365 | "\n", 366 | "$H(M) = -0.5\\log_2 0.5 -0.3\\log_2 0.3 -0.2\\log_2 0.2 = 1.48$\n", 367 | "\n", 368 | "#### Part 2\n", 369 | "\n", 370 | "| $n$ | $H(K)$ |\n", 371 | "| --- | ------ |\n", 372 | "| 1 | 4.7 |\n", 373 | "| 2 | 9.4 |\n", 374 | "| 4 | 18.8 |\n", 375 | "\n", 376 | "\n", 377 | "#### Part 3\n", 378 | "\n", 379 | "Let call $C_i$ the set of possible ciphertexts for message $m_i$.\n", 380 | "\n", 381 | "With $n=1$, we have\n", 382 | "* $C_1 = \\{ \\text{BEBA}, \\text{CFCB}, \\text{DGDC}, \\dots \\}$\n", 383 | "* $C_2 = \\{ \\text{DEDA}, \\text{EFEB}, \\text{FGFB}, \\dots \\}$\n", 384 | "* $C_3 = \\{ \\text{CFCB}, \\text{DGDC}, \\text{EHED}, \\dots \\}$\n", 385 | "\n", 386 | "So $C_1 = C_3$, while $C_2$ is distinct. Thus, we have $2\\times26=52$ ciphertexts. The first set has probability $0.5+0.2=0.7$. The second set has probability $0.3$.\n", 387 | "\n", 388 | "$H(C)=\n", 389 | " -26\\times 0.7/26\\log_2 0.7/26\n", 390 | " -26\\times 0.3/26\\log_2 0.3/26\n", 391 | " = 5.58\n", 392 | "$\n", 393 | "\n", 394 | "With $n=2$, we have again $C_1 = C_2 = C_3$. In fact, the key $(2,0)$ can map BEBA into DEDA, while the key $(1,1)$ can map BEBA into CFCB. Consequently all the messages generate the same $26^2=676$ ciphertexts.\n", 395 | "\n", 396 | "$H(C)= -676 / 676 \\log_2(1/676)=9.4$\n", 397 | "\n", 398 | "With $n=4$, we have $C_1 = C_2 = C_3$. Thus, we have $26^4=456976$ equally likely ciphertexts.\n", 399 | "\n", 400 | "$H(C)= 26^4 / 26^4 \\log_2(1/26^4)=18.8$\n", 401 | "\n", 402 | "#### Part 4\n", 403 | "\n", 404 | "With $n=1$\n", 405 | "\n", 406 | "$H(C|M)=H(C_1)\\Pr(m_1) + H(C_2)\\Pr(m_2) + H(C_3)\\Pr(m_3)= 4.7$\n", 407 | "\n", 408 | "Then\n", 409 | "$\n", 410 | "\tI(M;C) =\n", 411 | "\tH(C)-H(C|M) = 5.58 - 4.7 = 0.88\n", 412 | "$\n", 413 | "\n", 414 | "With $n=2$\n", 415 | "\n", 416 | "$\n", 417 | "\tH(C|M)=H(C_1)\\Pr(m_1) + H(C_2)\\Pr(m_2+m_3)= 9.4\n", 418 | "$\n", 419 | "\n", 420 | "Then\n", 421 | "$\n", 422 | "\tI(M;C) =\n", 423 | "\tH(C)-H(C|M) = 9.4-9.4 = 0\n", 424 | "$\n", 425 | "\n", 426 | "With $n=4$\n", 427 | "\n", 428 | "$\n", 429 | "\tH(C|M)=H(C_1)\\Pr(m_1+m_2+m_3)= 18.8\n", 430 | "$\n", 431 | "\n", 432 | "Then\n", 433 | "$\n", 434 | "\tI(M;C) =\n", 435 | "\tH(C)-H(C|M) = 18.8-18.8=0\n", 436 | "$\n", 437 | "\n", 438 | "#### Part 5\n", 439 | "If these three are the only possible messages, then with $n=2$ and $n=4$ we have perfect secrecy." 440 | ] 441 | }, 442 | { 443 | "cell_type": "code", 444 | "execution_count": null, 445 | "metadata": {}, 446 | "outputs": [], 447 | "source": [] 448 | } 449 | ], 450 | "metadata": { 451 | "kernelspec": { 452 | "display_name": "Python 3 (ipykernel)", 453 | "language": "python", 454 | "name": "python3" 455 | }, 456 | "language_info": { 457 | "codemirror_mode": { 458 | "name": "ipython", 459 | "version": 3 460 | }, 461 | "file_extension": ".py", 462 | "mimetype": "text/x-python", 463 | "name": "python", 464 | "nbconvert_exporter": "python", 465 | "pygments_lexer": "ipython3", 466 | "version": "3.9.7" 467 | } 468 | }, 469 | "nbformat": 4, 470 | "nbformat_minor": 2 471 | } 472 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Network Security and Cryptography course page 2 | 3 | Currently some notebooks require the python3 kernel and some require the sagemath kernel 4 | 5 | Notebooks in this collection: 6 | 1. Classic Ciphers [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Classic%20ciphers.ipynb) 7 | 2. Secret Sharing [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Secret%20Sharing.ipynb) 8 | 3. Secret Sharing (Galois) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Secret%20Sharing%20(gf).ipynb) 9 | 4. Hash Functions [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Hash%20Functions.ipynb) 10 | 5. Stream Ciphers [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/StreamCiphers.ipynb) 11 | 6. Symmetric Ciphers [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Symmetric.ipynb) 12 | 7. RSA [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/RSA.ipynb) 13 | 8. Legacy DSA [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/DSA.ipynb) 14 | 9. Diffie-Hellman [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/DHKE.ipynb) 15 | 8. Elliptic Curves [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master/Elliptic%20Curves.ipynb) 16 | 17 | Open this repository: 18 | [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/gverticale/network-security-and-cryptography/blob/master) 19 | [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/gverticale/network-security-and-cryptography/blob/master) 20 | [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/gverticale/network-security-and-cryptography/HEAD) 21 | -------------------------------------------------------------------------------- /RSA.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Playing with RSA" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": null, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "from math import log2, floor, isqrt\n", 17 | "from timeit import timeit\n", 18 | "from sympy import randprime, nextprime\n", 19 | "\n", 20 | "from cryptography.hazmat.primitives.asymmetric import rsa, padding\n", 21 | "from cryptography.hazmat.primitives import hashes, serialization" 22 | ] 23 | }, 24 | { 25 | "cell_type": "markdown", 26 | "metadata": {}, 27 | "source": [ 28 | "## What happens if p and q are too close?" 29 | ] 30 | }, 31 | { 32 | "cell_type": "code", 33 | "execution_count": null, 34 | "metadata": {}, 35 | "outputs": [], 36 | "source": [ 37 | "# p = next_prime(2^512)\n", 38 | "p = randprime(2**511, 2**512)\n", 39 | "p" 40 | ] 41 | }, 42 | { 43 | "cell_type": "code", 44 | "execution_count": null, 45 | "metadata": {}, 46 | "outputs": [], 47 | "source": [ 48 | "q=nextprime(p)\n", 49 | "q" 50 | ] 51 | }, 52 | { 53 | "cell_type": "code", 54 | "execution_count": null, 55 | "metadata": {}, 56 | "outputs": [], 57 | "source": [ 58 | "n = p * q\n", 59 | "n" 60 | ] 61 | }, 62 | { 63 | "cell_type": "code", 64 | "execution_count": null, 65 | "metadata": {}, 66 | "outputs": [], 67 | "source": [ 68 | "# n has 1024 bit\n", 69 | "log2(n)" 70 | ] 71 | }, 72 | { 73 | "cell_type": "code", 74 | "execution_count": null, 75 | "metadata": {}, 76 | "outputs": [], 77 | "source": [ 78 | "# We start from the square root and look for the next prime\n", 79 | "# isqrt is the integer part of the square root\n", 80 | "t = isqrt(n)\n", 81 | "t" 82 | ] 83 | }, 84 | { 85 | "cell_type": "code", 86 | "execution_count": null, 87 | "metadata": {}, 88 | "outputs": [], 89 | "source": [ 90 | "nextprime(t)" 91 | ] 92 | }, 93 | { 94 | "cell_type": "code", 95 | "execution_count": null, 96 | "metadata": {}, 97 | "outputs": [], 98 | "source": [ 99 | "# Compare with q\n", 100 | "q" 101 | ] 102 | }, 103 | { 104 | "cell_type": "markdown", 105 | "metadata": {}, 106 | "source": [ 107 | "## How long does it take to generate a key?" 108 | ] 109 | }, 110 | { 111 | "cell_type": "code", 112 | "execution_count": null, 113 | "metadata": {}, 114 | "outputs": [], 115 | "source": [ 116 | "def gen():\n", 117 | " return rsa.generate_private_key(\n", 118 | " public_exponent=65537,\n", 119 | " key_size=2048\n", 120 | " )" 121 | ] 122 | }, 123 | { 124 | "cell_type": "code", 125 | "execution_count": null, 126 | "metadata": {}, 127 | "outputs": [], 128 | "source": [ 129 | "timeit(gen, number = 100)" 130 | ] 131 | }, 132 | { 133 | "cell_type": "markdown", 134 | "metadata": {}, 135 | "source": [ 136 | "## What happens if the exponent is too low?" 137 | ] 138 | }, 139 | { 140 | "cell_type": "code", 141 | "execution_count": null, 142 | "metadata": {}, 143 | "outputs": [], 144 | "source": [ 145 | "private_key = rsa.generate_private_key(\n", 146 | " public_exponent=3,\n", 147 | " key_size=1024\n", 148 | ")\n", 149 | "public_key = private_key.public_key()\n", 150 | "\n", 151 | "e = public_key.public_numbers().e\n", 152 | "n = public_key.public_numbers().n\n", 153 | "print(\"e = {}\\nn = {}\".format(e,n))" 154 | ] 155 | }, 156 | { 157 | "cell_type": "code", 158 | "execution_count": null, 159 | "metadata": {}, 160 | "outputs": [], 161 | "source": [ 162 | "message = b\"OK\"\n", 163 | "\n", 164 | "m = int.from_bytes(message,\"big\")\n", 165 | "c_small = pow(m,e,n)\n", 166 | "\n", 167 | "print(\"m = {}\\nc = {}\".format(m,c_small))" 168 | ] 169 | }, 170 | { 171 | "cell_type": "code", 172 | "execution_count": null, 173 | "metadata": {}, 174 | "outputs": [], 175 | "source": [ 176 | "# Can we find the message?\n", 177 | "# If the message is small, the modulo is not effective and we can just take the e-th root\n", 178 | "pow(c_small,1/3)" 179 | ] 180 | }, 181 | { 182 | "cell_type": "code", 183 | "execution_count": null, 184 | "metadata": {}, 185 | "outputs": [], 186 | "source": [ 187 | "# If we use padding the problem is solved\n", 188 | "c = public_key.encrypt(\n", 189 | " message,\n", 190 | " padding.OAEP(\n", 191 | " mgf=padding.MGF1( algorithm=hashes.SHA256() ),\n", 192 | " algorithm=hashes.SHA256(),\n", 193 | " label=None\n", 194 | " )\n", 195 | ")\n", 196 | "print(\"c = {:d}\".format(int.from_bytes(c,\"big\")))" 197 | ] 198 | }, 199 | { 200 | "cell_type": "code", 201 | "execution_count": null, 202 | "metadata": {}, 203 | "outputs": [], 204 | "source": [ 205 | "private_key.decrypt(\n", 206 | " c,\n", 207 | " padding.OAEP(\n", 208 | " mgf=padding.MGF1(algorithm=hashes.SHA256()),\n", 209 | " algorithm=hashes.SHA256(),\n", 210 | " label=None\n", 211 | " )\n", 212 | ")" 213 | ] 214 | }, 215 | { 216 | "cell_type": "markdown", 217 | "metadata": {}, 218 | "source": [ 219 | "## Signature\n", 220 | "\n", 221 | "The key generation is the same as encryption, but we want a fresh key" 222 | ] 223 | }, 224 | { 225 | "cell_type": "code", 226 | "execution_count": null, 227 | "metadata": {}, 228 | "outputs": [], 229 | "source": [ 230 | "private_key = rsa.generate_private_key(\n", 231 | " public_exponent=65537,\n", 232 | " key_size=4096\n", 233 | ")\n", 234 | "public_key = private_key.public_key()" 235 | ] 236 | }, 237 | { 238 | "cell_type": "code", 239 | "execution_count": null, 240 | "metadata": {}, 241 | "outputs": [], 242 | "source": [ 243 | "message = b\"A message I want to sign\"\n", 244 | "signature = private_key.sign(\n", 245 | " message,\n", 246 | " padding.PSS(\n", 247 | " mgf=padding.MGF1( hashes.SHA256() ),\n", 248 | " salt_length=padding.PSS.MAX_LENGTH\n", 249 | " ),\n", 250 | " hashes.SHA256()\n", 251 | ")" 252 | ] 253 | }, 254 | { 255 | "cell_type": "code", 256 | "execution_count": null, 257 | "metadata": {}, 258 | "outputs": [], 259 | "source": [ 260 | "try:\n", 261 | " public_key.verify(\n", 262 | " signature,\n", 263 | " message,\n", 264 | " padding.PSS(\n", 265 | " mgf=padding.MGF1( hashes.SHA256() ),\n", 266 | " salt_length=padding.PSS.MAX_LENGTH\n", 267 | " ),\n", 268 | " hashes.SHA256()\n", 269 | " )\n", 270 | " print(\"OK\")\n", 271 | "except:\n", 272 | " print(\"KO\")" 273 | ] 274 | }, 275 | { 276 | "cell_type": "code", 277 | "execution_count": null, 278 | "metadata": {}, 279 | "outputs": [], 280 | "source": [ 281 | "message = b\"Not the real message\"\n", 282 | "try:\n", 283 | " public_key.verify(\n", 284 | " signature,\n", 285 | " message,\n", 286 | " padding.PSS(\n", 287 | " mgf=padding.MGF1( hashes.SHA256() ),\n", 288 | " salt_length=padding.PSS.MAX_LENGTH\n", 289 | " ),\n", 290 | " hashes.SHA256()\n", 291 | " )\n", 292 | " print(\"OK\")\n", 293 | "except:\n", 294 | " print(\"KO\")" 295 | ] 296 | }, 297 | { 298 | "cell_type": "markdown", 299 | "metadata": {}, 300 | "source": [ 301 | "## How to export and import the public key" 302 | ] 303 | }, 304 | { 305 | "cell_type": "code", 306 | "execution_count": null, 307 | "metadata": {}, 308 | "outputs": [], 309 | "source": [ 310 | "private_key = rsa.generate_private_key(\n", 311 | " public_exponent=65537,\n", 312 | " key_size=4096\n", 313 | ")\n", 314 | "public_key = private_key.public_key()" 315 | ] 316 | }, 317 | { 318 | "cell_type": "code", 319 | "execution_count": null, 320 | "metadata": {}, 321 | "outputs": [], 322 | "source": [ 323 | "#Export public key\n", 324 | "pem = public_key.public_bytes(\n", 325 | " encoding=serialization.Encoding.PEM,\n", 326 | " format=serialization.PublicFormat.SubjectPublicKeyInfo\n", 327 | ")\n", 328 | "pem" 329 | ] 330 | }, 331 | { 332 | "cell_type": "code", 333 | "execution_count": null, 334 | "metadata": {}, 335 | "outputs": [], 336 | "source": [ 337 | "#Import public key\n", 338 | "pem = b'-----BEGIN PUBLIC KEY-----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\\nfcD0yRbK6We+xdN+aYI35jUCAwEAAQ==\\n-----END PUBLIC KEY-----\\n'\n", 339 | "rsa_public_key = serialization.load_pem_public_key(pem)" 340 | ] 341 | }, 342 | { 343 | "cell_type": "code", 344 | "execution_count": null, 345 | "metadata": {}, 346 | "outputs": [], 347 | "source": [] 348 | }, 349 | { 350 | "cell_type": "code", 351 | "execution_count": null, 352 | "metadata": {}, 353 | "outputs": [], 354 | "source": [] 355 | }, 356 | { 357 | "cell_type": "markdown", 358 | "metadata": {}, 359 | "source": [ 360 | "# Laboratory" 361 | ] 362 | }, 363 | { 364 | "cell_type": "markdown", 365 | "metadata": {}, 366 | "source": [ 367 | "1. Measure the key generation time and the signature time for key sizes ranging from 512 to 4096 bits. Plot a graph." 368 | ] 369 | }, 370 | { 371 | "cell_type": "markdown", 372 | "metadata": {}, 373 | "source": [ 374 | "2. The following is my public key. Verify my message \"Hallo class!\"" 375 | ] 376 | }, 377 | { 378 | "cell_type": "code", 379 | "execution_count": null, 380 | "metadata": {}, 381 | "outputs": [], 382 | "source": [ 383 | "# My public key\n", 384 | "b'-----BEGIN PUBLIC KEY-----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\\naq2/Ut7ZungeWwW3TILiW1ECAwEAAQ==\\n-----END PUBLIC KEY-----\\n'" 385 | ] 386 | }, 387 | { 388 | "cell_type": "code", 389 | "execution_count": null, 390 | "metadata": {}, 391 | "outputs": [], 392 | "source": [ 393 | "# The signature of my message \"Hallo class!\"\n", 394 | "b'}\\xf3\\x0b\\xa3\\xda\\xd4a\\xe0@p\\xdfz\\xb0\\xdbY\\xc9\\x10:\\xbe\\xae\\x84!\\xd5e\\xf4\\xe2\\xebD\\xb4\\xd4H\\xcd\\x1b\\xf4r\\x7f\\xec\\x0e\\x83\\xa7\\x9a\\xdbY\\x8d\\xdd\\xc9\\xect\\xea\\x1b\\x01\\x02\\xbd\\xb5\\x9dYz\\xd9\\x0fv\\xf7\\x8d\\xbc\\xa1\\x1d\\x80\\x91}\\xd6\\xbc:\\xbb\\xce\\x00o\\xa8\\'\\x08@\\x99\\xfa\\xec;a\\xb2lEy]\\x9a\\xccV\\x00\\xb0\\x8d\\x9a\\xf0F\\x0f\\x81\\x8f\\xf7\\xb8\\xf8\\xc7i,\\x0bCn93\\xf2\\x7f\\x08\\xbb\\x00\\xe9]\\xb8\\xcc\\x04i\\x93\\xd6y\\xbc\\xa9N\\xe7\\xb5tI\\xe2\\x9a\\xc4{4\\x8c\\xa6&\\xd4b\\xa6\\xdf\\x10I\\xe7ZI3\\xee)\\xba_\\x0c)\\xa3\\x04\\xd7\\xe1\\'\\xcaA]J\\x1d\\xf0?M\\xf2\\xb1Ft\\x9a\\xad\\xf6y\\x8a3\\x87\\xc02\\xbe%\\xfaOk\\xfa\\x99\\xe6\\x14\\x82\\x07wN\\xe9%\\xd5\\xbd\\xd4\\x84\\x0b]\\xd3\\xfa\\xab#\\xf9\\x84\\xf9U\\xbd\\x0f\\xa0\\x85NL!\\x9f\\xda\\xf18\\xad\\x81\\x0fg(\\x94\\xd8mt\\x98Z\\xed`t\\tm\\xf1O\\x85\\x1f\\xa8\\x7f\\xb0\\x1d\\x87\\xf9c\\x89\\x1f\\xb6$\\x05\\xd3Z\\xab\\xc0a\\x85\\xe9\\x8e\\x91\\xbb\\xf9\\xe8\\x99\\xec(\\xc0\\xd7\\xdcf\\xbf|\\x8e.\\xf8\\xd7H\\x04\\xec\\n\\xffDKq\\x94\\xb9\\xe4\\r\\xd6r\\xa1Gf=V\\x16%\\x1e\\x9e\\xbc\\xb3\\xabQ \\xd0\\x19\\x92~\\x19?]\\x15\\xf3\"\\xdc\\x9876\\x8f\\x8a\\xca\\\\\\x153d\\xb6\\xab:\\xc8\\xec)Q\\xcb*\\xac\\x92\\xc3\\x06\\xe9A\\x92\\x95\\x1a\\xae\\xcf\\xf5\\x95<6\\x94n\\xf9\\xd8_\\xbf\\x05\\xc4T\\xc3\\xed\\xaf\\xf9IK\\x8e\\x16B\\x95\\x15\\xdd\\x8a\\xb0V\\xa8^\\xfd\\x06:\\x08)\\xcff\\xe9uY\\xee\\xad)\\x07S}\\x9b\\xb0\\xf4\\xd9\\xa7\\x92m\\x84\\xe5\\xab\\x9c\\x84\\x96k\\xc2+Y\\x0f\\x9ci#p\\xd6z\\xd6\\x81\\xe9\\xb2D\\x99h\\'z\\x0b\\xd0\\xe4W\\x8a|C\\xea\\xf3\\x94\\xb8}\\x87t\\x13\\xf5u\\xbd\\xcb\\x89\\t\\xb6\\xa1a\\x8a\\x9cC\\r\\xab\\xd3\\xa1\\xdf\\xffW\\xe0LY\\x18\\x83\\x05*\\x85\\xa9u\\xf4\\xb4\\xe4\\xfc\\xda\\xb4JB~\\xdeTd\\x07\\xcc\\xed\\x1a\\x80\\x08`\\xb1\\xefB\\xf9j:\\x85\\xbc\\xb0\\x94\\x8dbR\\xe8\\n>w8\\xf6\\x11F'" 395 | ] 396 | }, 397 | { 398 | "cell_type": "markdown", 399 | "metadata": {}, 400 | "source": [ 401 | "3. Work in pairs. One plays as Alice, one as Bob.\n", 402 | "Alice generates a keypair and sends her public key to Bob. Bob encrypts a message and sends it to Alice.\n", 403 | "Alice decrypts the message." 404 | ] 405 | }, 406 | { 407 | "cell_type": "markdown", 408 | "metadata": {}, 409 | "source": [ 410 | "4. Work in pairs. One plays as Alice, one as Bob. Bob generates a keypair and sends his public key to Alice.\n", 411 | "Alice generates a random key and sends it to Bob using RSA. Then Alice encrypts a message using AES-CTR and sends it to Bob.\n", 412 | "Bob decrypts the message." 413 | ] 414 | }, 415 | { 416 | "cell_type": "code", 417 | "execution_count": null, 418 | "metadata": {}, 419 | "outputs": [], 420 | "source": [] 421 | } 422 | ], 423 | "metadata": { 424 | "kernelspec": { 425 | "display_name": "Python 3 (ipykernel)", 426 | "language": "python", 427 | "name": "python3" 428 | }, 429 | "language_info": { 430 | "codemirror_mode": { 431 | "name": "ipython", 432 | "version": 3 433 | }, 434 | "file_extension": ".py", 435 | "mimetype": "text/x-python", 436 | "name": "python", 437 | "nbconvert_exporter": "python", 438 | "pygments_lexer": "ipython3", 439 | "version": "3.9.12" 440 | } 441 | }, 442 | "nbformat": 4, 443 | "nbformat_minor": 2 444 | } 445 | -------------------------------------------------------------------------------- /Secret Sharing (gf).ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Secret Sharing" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": null, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "# Uncomment and execute the last line\n", 17 | "# if the pycryptodomex package is missing \n", 18 | "# !pip install pycryptodomex" 19 | ] 20 | }, 21 | { 22 | "cell_type": "code", 23 | "execution_count": null, 24 | "metadata": {}, 25 | "outputs": [], 26 | "source": [ 27 | "from binascii import hexlify, unhexlify\n", 28 | "from Cryptodome.Random import get_random_bytes\n", 29 | "from Cryptodome.Protocol.SecretSharing import Shamir" 30 | ] 31 | }, 32 | { 33 | "cell_type": "markdown", 34 | "metadata": {}, 35 | "source": [ 36 | "The Shamir scheme can be implemented over the Galois Field $GF(2^{128})$" 37 | ] 38 | }, 39 | { 40 | "cell_type": "code", 41 | "execution_count": null, 42 | "metadata": {}, 43 | "outputs": [], 44 | "source": [ 45 | "# Generate a random secret\n", 46 | "secret = get_random_bytes(16)\n", 47 | "hexlify(secret)" 48 | ] 49 | }, 50 | { 51 | "cell_type": "code", 52 | "execution_count": null, 53 | "metadata": {}, 54 | "outputs": [], 55 | "source": [ 56 | "# Now split the shares\n", 57 | "t = 2\n", 58 | "w = 5\n", 59 | "shares = Shamir.split(t, w, secret)" 60 | ] 61 | }, 62 | { 63 | "cell_type": "code", 64 | "execution_count": null, 65 | "metadata": { 66 | "scrolled": true 67 | }, 68 | "outputs": [], 69 | "source": [ 70 | "# Now print the shares\n", 71 | "for idx, share in shares:\n", 72 | " print(\"x = %d, y = %s\" % (idx, hexlify(share)) )" 73 | ] 74 | }, 75 | { 76 | "cell_type": "code", 77 | "execution_count": null, 78 | "metadata": {}, 79 | "outputs": [], 80 | "source": [ 81 | "# Pick two shares (e.g. 2 and 4)\n", 82 | "available = [ shares[1], shares[3] ]\n", 83 | "hexlify( Shamir.combine(available) )" 84 | ] 85 | }, 86 | { 87 | "cell_type": "code", 88 | "execution_count": null, 89 | "metadata": {}, 90 | "outputs": [], 91 | "source": [ 92 | "# if we use only one share\n", 93 | "available = [ shares[1] ]\n", 94 | "hexlify( Shamir.combine(available) )" 95 | ] 96 | }, 97 | { 98 | "cell_type": "markdown", 99 | "metadata": {}, 100 | "source": [ 101 | "# Lab work\n", 102 | "\n", 103 | "We have the following shares with $t = 2$.\n", 104 | "Are there any wrong shares? Which ones?" 105 | ] 106 | }, 107 | { 108 | "cell_type": "code", 109 | "execution_count": null, 110 | "metadata": {}, 111 | "outputs": [], 112 | "source": [ 113 | "hexshares = [(1, b'ce9e5ad55b4c7d13389238afee6d1911'),\n", 114 | " (2, b'0563b1a58880f544769690f3d6ab1757'),\n", 115 | " (3, b'43c8e88a39c48d76b36af7383ee91295'),\n", 116 | " (4, b'929867442f19efeaea9fc14ba7270b5c'),\n", 117 | " (5, b'd4333e6b9e5d9dd82f63a7804f650e9e')]" 118 | ] 119 | }, 120 | { 121 | "cell_type": "code", 122 | "execution_count": null, 123 | "metadata": {}, 124 | "outputs": [], 125 | "source": [ 126 | "shares = [(x,unhexlify(y)) for x,y in hexshares]" 127 | ] 128 | }, 129 | { 130 | "cell_type": "code", 131 | "execution_count": null, 132 | "metadata": {}, 133 | "outputs": [], 134 | "source": [ 135 | "hexlify( Shamir.combine( (shares[0],shares[1]) ) )" 136 | ] 137 | }, 138 | { 139 | "cell_type": "code", 140 | "execution_count": null, 141 | "metadata": {}, 142 | "outputs": [], 143 | "source": [ 144 | "hexlify( Shamir.combine( (shares[0],shares[2]) ) )" 145 | ] 146 | }, 147 | { 148 | "cell_type": "code", 149 | "execution_count": null, 150 | "metadata": {}, 151 | "outputs": [], 152 | "source": [ 153 | "hexlify( Shamir.combine( (shares[0],shares[3]) ) )" 154 | ] 155 | }, 156 | { 157 | "cell_type": "code", 158 | "execution_count": null, 159 | "metadata": {}, 160 | "outputs": [], 161 | "source": [ 162 | "hexlify( Shamir.combine( (shares[0],shares[4]) ) )" 163 | ] 164 | }, 165 | { 166 | "cell_type": "code", 167 | "execution_count": null, 168 | "metadata": {}, 169 | "outputs": [], 170 | "source": [] 171 | } 172 | ], 173 | "metadata": { 174 | "kernelspec": { 175 | "display_name": "Python 3 (ipykernel)", 176 | "language": "python", 177 | "name": "python3" 178 | }, 179 | "language_info": { 180 | "codemirror_mode": { 181 | "name": "ipython", 182 | "version": 3 183 | }, 184 | "file_extension": ".py", 185 | "mimetype": "text/x-python", 186 | "name": "python", 187 | "nbconvert_exporter": "python", 188 | "pygments_lexer": "ipython3", 189 | "version": "3.9.7" 190 | } 191 | }, 192 | "nbformat": 4, 193 | "nbformat_minor": 2 194 | } 195 | -------------------------------------------------------------------------------- /Secret Sharing.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Secret Sharing" 8 | ] 9 | }, 10 | { 11 | "cell_type": "code", 12 | "execution_count": null, 13 | "metadata": {}, 14 | "outputs": [], 15 | "source": [ 16 | "from sympy.matrices import Matrix" 17 | ] 18 | }, 19 | { 20 | "cell_type": "markdown", 21 | "metadata": {}, 22 | "source": [ 23 | "Suppose we want to share the secret $m=5$ to five players so that three of them must cooperate.\n", 24 | "We create the random polynomial\n", 25 | "\n", 26 | "$ s(x) = 5 + 7x +3x^2 \\mod 13 $\n" 27 | ] 28 | }, 29 | { 30 | "cell_type": "code", 31 | "execution_count": null, 32 | "metadata": {}, 33 | "outputs": [], 34 | "source": [ 35 | "p = 13\n", 36 | "s = lambda x: (5 + 7*x + 3*x*x) % p" 37 | ] 38 | }, 39 | { 40 | "cell_type": "markdown", 41 | "metadata": {}, 42 | "source": [ 43 | "We distribute the shares to players 1, 2, 3, 4, and 5" 44 | ] 45 | }, 46 | { 47 | "cell_type": "code", 48 | "execution_count": null, 49 | "metadata": {}, 50 | "outputs": [], 51 | "source": [ 52 | "# the share with index 0 is the secret\n", 53 | "# Do not share it!\n", 54 | "s(0)" 55 | ] 56 | }, 57 | { 58 | "cell_type": "code", 59 | "execution_count": null, 60 | "metadata": {}, 61 | "outputs": [], 62 | "source": [ 63 | "share = [(i,s(i)) for i in range(1,6)]\n", 64 | "share" 65 | ] 66 | }, 67 | { 68 | "cell_type": "markdown", 69 | "metadata": {}, 70 | "source": [ 71 | "We pick three shares and rebuild the secret.\n", 72 | "We use (2,5), (3,1), and (5,11)" 73 | ] 74 | }, 75 | { 76 | "cell_type": "code", 77 | "execution_count": null, 78 | "metadata": {}, 79 | "outputs": [], 80 | "source": [ 81 | "M = Matrix( \n", 82 | " [ \n", 83 | " [ 1, 2, pow(2,2,p) ], \n", 84 | " [ 1, 3, pow(3,2,p) ], \n", 85 | " [ 1, 5, pow(5,2,p) ] \n", 86 | " ] \n", 87 | ")\n", 88 | "M" 89 | ] 90 | }, 91 | { 92 | "cell_type": "code", 93 | "execution_count": null, 94 | "metadata": {}, 95 | "outputs": [], 96 | "source": [ 97 | "V = Matrix([5, 1, 11])\n", 98 | "V" 99 | ] 100 | }, 101 | { 102 | "cell_type": "code", 103 | "execution_count": null, 104 | "metadata": {}, 105 | "outputs": [], 106 | "source": [ 107 | "# We need the inverse of M modulo p\n", 108 | "Mi = M.inv_mod(p)\n", 109 | "Mi" 110 | ] 111 | }, 112 | { 113 | "cell_type": "code", 114 | "execution_count": null, 115 | "metadata": {}, 116 | "outputs": [], 117 | "source": [ 118 | "# The result is the vector of coefficients of s\n", 119 | "# The first value is the secret\n", 120 | "Mi * V %p" 121 | ] 122 | }, 123 | { 124 | "cell_type": "markdown", 125 | "metadata": {}, 126 | "source": [ 127 | "## Lab Work\n", 128 | "Let's test the homomorphic properties of Shamir's scheme. Work in groups of four: a Dealer, two Players (Alice and Bob) and an Observer (Charlie).\n", 129 | "\n", 130 | "1. Use the parameter $p=97$.\n", 131 | "2. The Dealer chooses two secrets: $m$ and $n$.\n", 132 | "3. The Dealer generates a first-degree random polynomial to share secret $m$.\n", 133 | "4. The Dealer gives share (1,ym1) to Alice and share (2,ym2) to Bob.\n", 134 | "5. The Dealer generates a new first-degree random polynomial to share secret $n$. Remember to use fresh random coefficients.\n", 135 | "6. The Dealer gives share (1,yn1) to Alice and share (2,yn2) to Bob.\n", 136 | "7. Alice computes the new share (1,ym1 + yn1 mod p) and gives it to Charlie\n", 137 | "8. Bob computes the new share (2,ym2 + yn2 mod p) and gives it to Charlie\n", 138 | "9. Charlie recovers the secret $z$ from the two shares. The correct result is $z=m+n \\mod p$." 139 | ] 140 | }, 141 | { 142 | "cell_type": "markdown", 143 | "metadata": {}, 144 | "source": [ 145 | "# Challenge" 146 | ] 147 | }, 148 | { 149 | "cell_type": "markdown", 150 | "metadata": {}, 151 | "source": [ 152 | "We provide 5 shares of a secret with threshold $t=3$. One of them is wrong.\n", 153 | "Your goal is to find the secret." 154 | ] 155 | }, 156 | { 157 | "cell_type": "code", 158 | "execution_count": 1, 159 | "metadata": {}, 160 | "outputs": [], 161 | "source": [ 162 | "#Public prime number\n", 163 | "p = 23" 164 | ] 165 | }, 166 | { 167 | "cell_type": "code", 168 | "execution_count": 2, 169 | "metadata": {}, 170 | "outputs": [], 171 | "source": [ 172 | "shares = [(1, 19), (2, 17), (3, 4), (4, 9), (5, 14)]" 173 | ] 174 | }, 175 | { 176 | "cell_type": "code", 177 | "execution_count": null, 178 | "metadata": {}, 179 | "outputs": [], 180 | "source": [] 181 | } 182 | ], 183 | "metadata": { 184 | "kernelspec": { 185 | "display_name": "Python 3 (ipykernel)", 186 | "language": "python", 187 | "name": "python3" 188 | }, 189 | "language_info": { 190 | "codemirror_mode": { 191 | "name": "ipython", 192 | "version": 3 193 | }, 194 | "file_extension": ".py", 195 | "mimetype": "text/x-python", 196 | "name": "python", 197 | "nbconvert_exporter": "python", 198 | "pygments_lexer": "ipython3", 199 | "version": "3.11.7" 200 | } 201 | }, 202 | "nbformat": 4, 203 | "nbformat_minor": 4 204 | } 205 | -------------------------------------------------------------------------------- /StreamCiphers.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "code", 5 | "execution_count": null, 6 | "id": "eb51cc5f-e9f7-40c9-9151-684b416713af", 7 | "metadata": {}, 8 | "outputs": [], 9 | "source": [ 10 | "%pip install -q cryptography mediapy\n", 11 | "\n", 12 | "# Some imports\n", 13 | "from os import urandom\n", 14 | "import struct\n", 15 | "from binascii import hexlify\n", 16 | "import numpy as np\n", 17 | "import mediapy as media\n", 18 | "\n", 19 | "from cryptography.hazmat.primitives.ciphers import Cipher, algorithms, modes\n", 20 | "# from cryptography.hazmat.backends import default_backend" 21 | ] 22 | }, 23 | { 24 | "cell_type": "markdown", 25 | "id": "7c9079ed-6e89-4bc7-9a87-87d3224d19c7", 26 | "metadata": {}, 27 | "source": [ 28 | "# Using ChaCha20" 29 | ] 30 | }, 31 | { 32 | "cell_type": "code", 33 | "execution_count": null, 34 | "id": "266ae1b8-6fca-4f55-a738-1964e7404997", 35 | "metadata": {}, 36 | "outputs": [], 37 | "source": [ 38 | "# key is 256 bits (32 bytes)\n", 39 | "key = urandom(32)\n", 40 | "\n", 41 | "# nonce is a random value (64 bits) concatenated to a counter (64 bits)\n", 42 | "# note that IETF standard has a different split (96/32) \n", 43 | "nonce = urandom(8)\n", 44 | "counter = 0\n", 45 | "full_nonce = struct.pack(\" \"ECB cannot be used because you can see the penguin\"" 200 | ] 201 | }, 202 | { 203 | "cell_type": "code", 204 | "execution_count": null, 205 | "metadata": {}, 206 | "outputs": [], 207 | "source": [ 208 | "# Load the penguin image\n", 209 | "# Uncomment the following line if you get an error about a missing file\n", 210 | "#!wget -q https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/master/tux_gray.png\n", 211 | "tux_png = media.read_image('https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/master/tux_gray.png')\n", 212 | "\n", 213 | "media.show_image(tux_png,height=480, title='Tux')" 214 | ] 215 | }, 216 | { 217 | "cell_type": "code", 218 | "execution_count": null, 219 | "metadata": {}, 220 | "outputs": [], 221 | "source": [ 222 | "# Keep only luminance and serialize the image\n", 223 | "# The plaintext is a string of bytes\n", 224 | "tux = tux_png[:,:,1] * 255\n", 225 | "tux = tux.astype(np.uint8)\n", 226 | "p = tux.tobytes()" 227 | ] 228 | }, 229 | { 230 | "cell_type": "code", 231 | "execution_count": null, 232 | "metadata": {}, 233 | "outputs": [], 234 | "source": [ 235 | "# Generate a key and an instance of AES\n", 236 | "k = urandom(KEYLEN)\n", 237 | "cipher = Cipher(algorithms.AES(k), modes.ECB())" 238 | ] 239 | }, 240 | { 241 | "cell_type": "code", 242 | "execution_count": null, 243 | "metadata": {}, 244 | "outputs": [], 245 | "source": [ 246 | "# Encrypt the image\n", 247 | "aes_encrypt = cipher.encryptor()\n", 248 | "c = aes_encrypt.update(p) + aes_encrypt.finalize()" 249 | ] 250 | }, 251 | { 252 | "cell_type": "code", 253 | "execution_count": null, 254 | "metadata": {}, 255 | "outputs": [], 256 | "source": [ 257 | "# Reshape the ciphertext into a matrix of bytes \n", 258 | "# Show the ciphertext\n", 259 | "encrypted_tux = np.frombuffer(c, dtype = np.uint8).reshape(tux.shape)\n", 260 | "\n", 261 | "media.show_image(encrypted_tux,height=480, title='Encrypted Tux')" 262 | ] 263 | }, 264 | { 265 | "cell_type": "markdown", 266 | "metadata": {}, 267 | "source": [ 268 | "## Example 4\n", 269 | "We encrypt the Linux mascot with deterministic CTR, and see that patterns disappear." 270 | ] 271 | }, 272 | { 273 | "cell_type": "code", 274 | "execution_count": null, 275 | "metadata": {}, 276 | "outputs": [], 277 | "source": [ 278 | "# Generate a key and an instance of AES\n", 279 | "k = urandom(KEYLEN)" 280 | ] 281 | }, 282 | { 283 | "cell_type": "code", 284 | "execution_count": null, 285 | "metadata": {}, 286 | "outputs": [], 287 | "source": [ 288 | "# Generate the starting counter (in deterministic CTR, we start from 0)\n", 289 | "iv = b'\\x00'*BLOCKLEN" 290 | ] 291 | }, 292 | { 293 | "cell_type": "code", 294 | "execution_count": null, 295 | "metadata": {}, 296 | "outputs": [], 297 | "source": [ 298 | "# Generate an instance of AES with the given key and nonce\n", 299 | "cipher = Cipher(algorithms.AES(k), modes.CTR(iv))" 300 | ] 301 | }, 302 | { 303 | "cell_type": "code", 304 | "execution_count": null, 305 | "metadata": {}, 306 | "outputs": [], 307 | "source": [ 308 | "# Encrypt the image\n", 309 | "aes_encrypt = cipher.encryptor()\n", 310 | "c = aes_encrypt.update(p) + aes_encrypt.finalize()" 311 | ] 312 | }, 313 | { 314 | "cell_type": "code", 315 | "execution_count": null, 316 | "metadata": {}, 317 | "outputs": [], 318 | "source": [ 319 | "# Reshape the ciphertext into a matrix of bytes \n", 320 | "# Show the ciphertext\n", 321 | "encrypted_tux = np.frombuffer(c, dtype = np.uint8).reshape(tux.shape)\n", 322 | "\n", 323 | "media.show_image(encrypted_tux,height=480, title='Encrypted Tux')" 324 | ] 325 | }, 326 | { 327 | "cell_type": "markdown", 328 | "metadata": {}, 329 | "source": [ 330 | "## Example 5\n", 331 | "We generate a new ciphertext with the same key and still use deterministic CTR.\n", 332 | "We then see that we can then learn about the plaintext " 333 | ] 334 | }, 335 | { 336 | "cell_type": "code", 337 | "execution_count": null, 338 | "metadata": {}, 339 | "outputs": [], 340 | "source": [ 341 | "# Load \"The Black Cat\" by Edgar Allan Poe \n", 342 | "# Uncomment the following line if you get an error about a missing file\n", 343 | "# !wget -q https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/master/cat.txt\n", 344 | "\n", 345 | "catfile = requests.get('https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/master/cat.txt')\n", 346 | "cat = catfile.content\n", 347 | "\n", 348 | "#with open('cat.txt', 'rb') as myfile:\n", 349 | "# cat=myfile.read()\n", 350 | "\n", 351 | "# The file is shorter than the image, so we make it the same size\n", 352 | "plaintext_cat = 50 * cat\n", 353 | "plaintext_cat = plaintext_cat[0:len(p)]" 354 | ] 355 | }, 356 | { 357 | "cell_type": "code", 358 | "execution_count": null, 359 | "metadata": {}, 360 | "outputs": [], 361 | "source": [ 362 | "# Encrypt the text with the same key and IV\n", 363 | "cipher = Cipher(algorithms.AES(k), modes.CTR(iv))\n", 364 | "aes_encrypt = cipher.encryptor()\n", 365 | "encrypted_cat = aes_encrypt.update(plaintext_cat) + aes_encrypt.finalize()" 366 | ] 367 | }, 368 | { 369 | "cell_type": "code", 370 | "execution_count": null, 371 | "metadata": {}, 372 | "outputs": [], 373 | "source": [ 374 | "# Reshape the encrypted poem into a matrix\n", 375 | "encrypted_cat_rect = np.frombuffer(encrypted_cat, dtype = np.uint8).reshape(tux.shape)" 376 | ] 377 | }, 378 | { 379 | "cell_type": "code", 380 | "execution_count": null, 381 | "metadata": {}, 382 | "outputs": [], 383 | "source": [ 384 | "# Calculate the XOR between the ciphertexts\n", 385 | "ciphertext_xor = np.bitwise_xor(encrypted_cat_rect, encrypted_tux)\n", 386 | "\n", 387 | "media.show_image(ciphertext_xor,height=480, title='Xored Tux')" 388 | ] 389 | }, 390 | { 391 | "cell_type": "markdown", 392 | "metadata": {}, 393 | "source": [ 394 | "# Example 6" 395 | ] 396 | }, 397 | { 398 | "cell_type": "markdown", 399 | "metadata": {}, 400 | "source": [ 401 | "Let's use an authenticated cipher: AES-GCM" 402 | ] 403 | }, 404 | { 405 | "cell_type": "code", 406 | "execution_count": null, 407 | "metadata": {}, 408 | "outputs": [], 409 | "source": [ 410 | "# Generate a key and an instance of AES-GCM\n", 411 | "# The standard requires an 96-bit random IV\n", 412 | "k = urandom(KEYLEN)\n", 413 | "iv = urandom(12)\n", 414 | "cipher = Cipher(algorithms.AES(k), modes.GCM(iv))" 415 | ] 416 | }, 417 | { 418 | "cell_type": "code", 419 | "execution_count": null, 420 | "metadata": {}, 421 | "outputs": [], 422 | "source": [ 423 | "# Encrypt \"The Black Cat\" and authenticate the string \"A poem\"\n", 424 | "gcm_encrypt = cipher.encryptor()\n", 425 | "gcm_encrypt.authenticate_additional_data(b\"A poem\")\n", 426 | "encrypted_cat = gcm_encrypt.update(plaintext_cat) + gcm_encrypt.finalize()\n", 427 | "tag = gcm_encrypt.tag" 428 | ] 429 | }, 430 | { 431 | "cell_type": "code", 432 | "execution_count": null, 433 | "metadata": {}, 434 | "outputs": [], 435 | "source": [ 436 | "# Now decrypt\n", 437 | "cipher = Cipher(algorithms.AES(k), modes.GCM(iv,tag))" 438 | ] 439 | }, 440 | { 441 | "cell_type": "code", 442 | "execution_count": null, 443 | "metadata": {}, 444 | "outputs": [], 445 | "source": [ 446 | "gcm_decrypt = cipher.decryptor()\n", 447 | "gcm_decrypt.authenticate_additional_data(b\"A new poem\")\n", 448 | "try:\n", 449 | " decrypted_cat = gcm_decrypt.update(encrypted_cat) + gcm_decrypt.finalize()\n", 450 | " print(decrypted_cat)\n", 451 | "except InvalidTag:\n", 452 | " print(\"Not authentic\")" 453 | ] 454 | }, 455 | { 456 | "cell_type": "markdown", 457 | "metadata": {}, 458 | "source": [ 459 | "# Lab Activity: randomized CTR" 460 | ] 461 | }, 462 | { 463 | "cell_type": "markdown", 464 | "metadata": {}, 465 | "source": [ 466 | "1. Encrypt tux with randomized CTR\n", 467 | "2. Encrypt cat with randomized CTR\n", 468 | "3. Compute the bitwise xor of the ciphertexts\n", 469 | "4. Show the result" 470 | ] 471 | }, 472 | { 473 | "cell_type": "markdown", 474 | "metadata": {}, 475 | "source": [ 476 | "# Lab Activity: Malleability of AES-CTR" 477 | ] 478 | }, 479 | { 480 | "cell_type": "markdown", 481 | "metadata": {}, 482 | "source": [ 483 | "1. Start from \"THE BLACK CAT\"\n", 484 | "2. Encrypt it using AES-CTR (use a random IV)\n", 485 | "3. Using ciphertext malleability change the ciphertext so that the title of the poem becomes \"THE GREEN CAT\"\n", 486 | "4. Decrypt it showing the modified text\n", 487 | "5. Repeat using AES-GCM and verify that it fails" 488 | ] 489 | }, 490 | { 491 | "cell_type": "markdown", 492 | "metadata": {}, 493 | "source": [ 494 | "_Quick note on the python library: the output of the encryption is immutable, you need to convert it to a list to modify it (command `list`) , then you need to convert it back to a bytes object (command `bytes`)_" 495 | ] 496 | }, 497 | { 498 | "cell_type": "code", 499 | "execution_count": null, 500 | "metadata": {}, 501 | "outputs": [], 502 | "source": [] 503 | } 504 | ], 505 | "metadata": { 506 | "kernelspec": { 507 | "display_name": "Python 3 (ipykernel)", 508 | "language": "python", 509 | "name": "python3" 510 | }, 511 | "language_info": { 512 | "codemirror_mode": { 513 | "name": "ipython", 514 | "version": 3 515 | }, 516 | "file_extension": ".py", 517 | "mimetype": "text/x-python", 518 | "name": "python", 519 | "nbconvert_exporter": "python", 520 | "pygments_lexer": "ipython3", 521 | "version": "3.11.7" 522 | } 523 | }, 524 | "nbformat": 4, 525 | "nbformat_minor": 4 526 | } 527 | -------------------------------------------------------------------------------- /cat.txt: -------------------------------------------------------------------------------- 1 | THE BLACK CAT 2 | 3 | by Edgar Allan Poe 4 | (1843) 5 | 6 | FOR the most wild, yet most homely narrative which I am about to pen, I 7 | neither expect nor solicit belief. Mad indeed would I be to expect it, in a 8 | case where my very senses reject their own evidence. Yet, mad am I not --and 9 | very surely do I not dream. But to-morrow I die, and to-day I would 10 | unburthen my soul. My immediate purpose is to place before the world, 11 | plainly, succinctly, and without comment, a series of mere household events. 12 | In their consequences, these events have terrified --have tortured --have 13 | destroyed me. Yet I will not attempt to expound them. To me, they have 14 | presented little but Horror --to many they will seem less terrible than 15 | baroques. Hereafter, perhaps, some intellect may be found which will reduce 16 | my phantasm to the common-place --some intellect more calm, more logical, 17 | and far less excitable than my own, which will perceive, in the 18 | circumstances I detail with awe, nothing more than an ordinary succession of 19 | very natural causes and effects. 20 | From my infancy I was noted for the docility and humanity of my 21 | disposition. My tenderness of heart was even so conspicuous as to make me 22 | the jest of my companions. I was especially fond of animals, and was 23 | indulged by my parents with a great variety of pets. With these I spent most 24 | of my time, and never was so happy as when feeding and caressing them. This 25 | peculiar of character grew with my growth, and in my manhood, I derived from 26 | it one of my principal sources of pleasure. To those who have cherished an 27 | affection for a faithful and sagacious dog, I need hardly be at the trouble 28 | of explaining the nature or the intensity of the gratification thus 29 | derivable. There is something in the unselfish and self-sacrificing love of 30 | a brute, which goes directly to the heart of him who has had frequent 31 | occasion to test the paltry friendship and gossamer fidelity of mere Man. 32 | I married early, and was happy to find in my wife a disposition not 33 | uncongenial with my own. Observing my partiality for domestic pets, she lost 34 | no opportunity of procuring those of the most agreeable kind. We had birds, 35 | gold fish, a fine dog, rabbits, a small monkey, and a cat. 36 | This latter was a remarkably large and beautiful animal, entirely black, 37 | and sagacious to an astonishing degree. In speaking of his intelligence, my 38 | wife, who at heart was not a little tinctured with superstition, made 39 | frequent allusion to the ancient popular notion, which regarded all black 40 | cats as witches in disguise. Not that she was ever serious upon this point 41 | --and I mention the matter at all for no better reason than that it happens, 42 | just now, to be remembered. 43 | Pluto --this was the cat's name --was my favorite pet and playmate. I 44 | alone fed him, and he attended me wherever I went about the house. It was 45 | even with difficulty that I could prevent him from following me through the 46 | streets. 47 | Our friendship lasted, in this manner, for several years, during which my 48 | general temperament and character --through the instrumentality of the Fiend 49 | Intemperance --had (I blush to confess it) experienced a radical alteration 50 | for the worse. I grew, day by day, more moody, more irritable, more 51 | regardless of the feelings of others. I suffered myself to use intemperate 52 | language to my At length, I even offered her personal violence. My pets, of 53 | course, were made to feel the change in my disposition. I not only 54 | neglected, but ill-used them. For Pluto, however, I still retained 55 | sufficient regard to restrain me from maltreating him, as I made no scruple 56 | of maltreating the rabbits, the monkey, or even the dog, when by accident, 57 | or through affection, they came in my way. But my disease grew upon me --for 58 | what disease is like Alcohol! --and at length even Pluto, who was now 59 | becoming old, and consequently somewhat peevish --even Pluto began to 60 | experience the effects of my ill temper. 61 | One night, returning home, much intoxicated, from one of my haunts about 62 | town, I fancied that the cat avoided my presence. I seized him; when, in his 63 | fright at my violence, he inflicted a slight wound upon my hand with his 64 | teeth. The fury of a demon instantly possessed me. I knew myself no longer. 65 | My original soul seemed, at once, to take its flight from my body; and a 66 | more than fiendish malevolence, gin-nurtured, thrilled every fibre of my 67 | frame. I took from my waistcoat-pocket a pen-knife, opened it, grasped the 68 | poor beast by the throat, and deliberately cut one of its eyes from the 69 | socket! I blush, I burn, I shudder, while I pen the damnable atrocity. 70 | When reason returned with the morning --when I had slept off the fumes of 71 | the night's debauch --I experienced a sentiment half of horror, half of 72 | remorse, for the crime of which I had been guilty; but it was, at best, a 73 | feeble and equivocal feeling, and the soul remained untouched. I again 74 | plunged into excess, and soon drowned in wine all memory of the deed. 75 | In the meantime the cat slowly recovered. The socket of the lost eye 76 | presented, it is true, a frightful appearance, but he no longer appeared to 77 | suffer any pain. He went about the house as usual, but, as might be 78 | expected, fled in extreme terror at my approach. I had so much of my old 79 | heart left, as to be at first grieved by this evident dislike on the part of 80 | a creature which had once so loved me. But this feeling soon gave place to 81 | irritation. And then came, as if to my final and irrevocable overthrow, the 82 | spirit of PERVERSENESS. Of this spirit philosophy takes no account. Yet I am 83 | not more sure that my soul lives, than I am that perverseness is one of the 84 | primitive impulses of the human heart --one of the indivisible primary 85 | faculties, or sentiments, which give direction to the character of Man. Who 86 | has not, a hundred times, found himself committing a vile or a silly action, 87 | for no other reason than because he knows he should not? Have we not a 88 | perpetual inclination, in the teeth of our best judgment, to violate that 89 | which is Law, merely because we understand it to be such? This spirit of 90 | perverseness, I say, came to my final overthrow. It was this unfathomable 91 | longing of the soul to vex itself --to offer violence to its own nature --to 92 | do wrong for the wrong's sake only --that urged me to continue and finally 93 | to consummate the injury I had inflicted upon the unoffending brute. One 94 | morning, in cool blood, I slipped a noose about its neck and hung it to the 95 | limb of a tree; --hung it with the tears streaming from my eyes, and with 96 | the bitterest remorse at my heart; --hung it because I knew that it had 97 | loved me, and because I felt it had given me no reason of offence; --hung it 98 | because I knew that in so doing I was committing a sin --a deadly sin that 99 | would so jeopardize my immortal soul as to place it --if such a thing were 100 | possible --even beyond the reach of the infinite mercy of the Most Merciful 101 | and Most Terrible God. 102 | On the night of the day on which this cruel deed was done, I was aroused 103 | from sleep by the cry of fire. The curtains of my bed were in flames. The 104 | whole house was blazing. It was with great difficulty that my wife, a 105 | servant, and myself, made our escape from the conflagration. The destruction 106 | was complete. My entire worldly wealth was swallowed up, and I resigned 107 | myself thenceforward to despair. 108 | I am above the weakness of seeking to establish a sequence of cause and 109 | effect, between the disaster and the atrocity. But I am detailing a chain of 110 | facts --and wish not to leave even a possible link imperfect. On the day 111 | succeeding the fire, I visited the ruins. The walls, with one exception, had 112 | fallen in. This exception was found in a compartment wall, not very thick, 113 | which stood about the middle of the house, and against which had rested the 114 | head of my bed. The plastering had here, in great measure, resisted the 115 | action of the fire --a fact which I attributed to its having been recently 116 | spread. About this wall a dense crowd were collected, and many persons 117 | seemed to be examining a particular portion of it with every minute and 118 | eager attention. The words "strange!" "singular!" and other similar 119 | expressions, excited my curiosity. I approached and saw, as if graven in bas 120 | relief upon the white surface, the figure of a gigantic cat. The impression 121 | was given with an accuracy truly marvellous. There was a rope about the 122 | animal's neck. 123 | When I first beheld this apparition --for I could scarcely regard it as 124 | less --my wonder and my terror were extreme. But at length reflection came 125 | to my aid. The cat, I remembered, had been hung in a garden adjacent to the 126 | house. Upon the alarm of fire, this garden had been immediately filled by 127 | the crowd --by some one of whom the animal must have been cut from the tree 128 | and thrown, through an open window, into my chamber. This had probably been 129 | done with the view of arousing me from sleep. The falling of other walls had 130 | compressed the victim of my cruelty into the substance of the freshly-spread 131 | plaster; the lime of which, had then with the flames, and the ammonia from 132 | the carcass, accomplished the portraiture as I saw it. 133 | Although I thus readily accounted to my reason, if not altogether to my 134 | conscience, for the startling fact 'just detailed, it did not the less fall 135 | to make a deep impression upon my fancy. For months I could not rid myself 136 | of the phantasm of the cat; and, during this period, there came back into my 137 | spirit a half-sentiment that seemed, but was not, remorse. I went so far as 138 | to regret the loss of the animal, and to look about me, among the vile 139 | haunts which I now habitually frequented, for another pet of the same 140 | species, and of somewhat similar appearance, with which to supply its place. 141 | 142 | One night as I sat, half stupefied, in a den of more than infamy, my 143 | attention was suddenly drawn to some black object, reposing upon the head of 144 | one of the immense hogsheads of Gin, or of Rum, which constituted the chief 145 | furniture of the apartment. I had been looking steadily at the top of this 146 | hogshead for some minutes, and what now caused me surprise was the fact that 147 | I had not sooner perceived the object thereupon. I approached it, and 148 | touched it with my hand. It was a black cat --a very large one --fully as 149 | large as Pluto, and closely resembling him in every respect but one. Pluto 150 | had not a white hair upon any portion of his body; but this cat had a large, 151 | although indefinite splotch of white, covering nearly the whole region of 152 | the breast. 153 | Upon my touching him, he immediately arose, purred loudly, rubbed against 154 | my hand, and appeared delighted with my notice. This, then, was the very 155 | creature of which I was in search. I at once offered to purchase it of the 156 | landlord; but this person made no claim to it --knew nothing of it --had 157 | never seen it before. 158 | I continued my caresses, and, when I prepared to go home, the animal 159 | evinced a disposition to accompany me. I permitted it to do so; occasionally 160 | stooping and patting it as I proceeded. When it reached the house it 161 | domesticated itself at once, and became immediately a great favorite with my 162 | wife. 163 | For my own part, I soon found a dislike to it arising within me. This was 164 | just the reverse of what I had anticipated; but I know not how or why it was 165 | --its evident fondness for myself rather disgusted and annoyed. By slow 166 | degrees, these feelings of disgust and annoyance rose into the bitterness of 167 | hatred. I avoided the creature; a certain sense of shame, and the 168 | remembrance of my former deed of cruelty, preventing me from physically 169 | abusing it. I did not, for some weeks, strike, or otherwise violently ill 170 | use it; but gradually --very gradually --I came to look upon it with 171 | unutterable loathing, and to flee silently from its odious presence, as from 172 | the breath of a pestilence. 173 | What added, no doubt, to my hatred of the beast, was the discovery, on 174 | the morning after I brought it home, that, like Pluto, it also had been 175 | deprived of one of its eyes. This circumstance, however, only endeared it to 176 | my wife, who, as I have already said, possessed, in a high degree, that 177 | humanity of feeling which had once been my distinguishing trait, and the 178 | source of many of my simplest and purest pleasures. 179 | With my aversion to this cat, however, its partiality for myself seemed 180 | to increase. It followed my footsteps with a pertinacity which it would be 181 | difficult to make the reader comprehend. Whenever I sat, it would crouch 182 | beneath my chair, or spring upon my knees, covering me with its loathsome 183 | caresses. If I arose to walk it would get between my feet and thus nearly 184 | throw me down, or, fastening its long and sharp claws in my dress, clamber, 185 | in this manner, to my breast. At such times, although I longed to destroy it 186 | with a blow, I was yet withheld from so doing, partly it at by a memory of 187 | my former crime, but chiefly --let me confess it at once --by absolute dread 188 | of the beast. 189 | This dread was not exactly a dread of physical evil-and yet I should be 190 | at a loss how otherwise to define it. I am almost ashamed to own --yes, even 191 | in this felon's cell, I am almost ashamed to own --that the terror and 192 | horror with which the animal inspired me, had been heightened by one of the 193 | merest chimaeras it would be possible to conceive. My wife had called my 194 | attention, more than once, to the character of the mark of white hair, of 195 | which I have spoken, and which constituted the sole visible difference 196 | between the strange beast and the one I had y si destroyed. The reader will 197 | remember that this mark, although large, had been originally very 198 | indefinite; but, by slow degrees --degrees nearly imperceptible, and which 199 | for a long time my Reason struggled to reject as fanciful --it had, at 200 | length, assumed a rigorous distinctness of outline. It was now the 201 | representation of an object that I shudder to name --and for this, above 202 | all, I loathed, and dreaded, and would have rid myself of the monster had I 203 | dared --it was now, I say, the image of a hideous --of a ghastly thing --of 204 | the GALLOWS! --oh, mournful and terrible engine of Horror and of Crime --of 205 | Agony and of Death! 206 | And now was I indeed wretched beyond the wretchedness of mere Humanity. 207 | And a brute beast --whose fellow I had contemptuously destroyed --a brute 208 | beast to work out for me --for me a man, fashioned in the image of the High 209 | God --so much of insufferable wo! Alas! neither by day nor by night knew I 210 | the blessing of Rest any more! During the former the creature left me no 211 | moment alone; and, in the latter, I started, hourly, from dreams of 212 | unutterable fear, to find the hot breath of the thing upon my face, and its 213 | vast weight --an incarnate Night-Mare that I had no power to shake off 214 | --incumbent eternally upon my heart! 215 | Beneath the pressure of torments such as these, the feeble remnant of the 216 | good within me succumbed. Evil thoughts became my sole intimates --the 217 | darkest and most evil of thoughts. The moodiness of my usual temper 218 | increased to hatred of all things and of all mankind; while, from the 219 | sudden, frequent, and ungovernable outbursts of a fury to which I now 220 | blindly abandoned myself, my uncomplaining wife, alas! was the most usual 221 | and the most patient of sufferers. 222 | One day she accompanied me, upon some household errand, into the cellar 223 | of the old building which our poverty compelled us to inhabit. The cat 224 | followed me down the steep stairs, and, nearly throwing me headlong, 225 | exasperated me to madness. Uplifting an axe, and forgetting, in my wrath, 226 | the childish dread which had hitherto stayed my hand, I aimed a blow at the 227 | animal which, of course, would have proved instantly fatal had it descended 228 | as I wished. But this blow was arrested by the hand of my wife. Goaded, by 229 | the interference, into a rage more than demoniacal, I withdrew my arm from 230 | her grasp and buried the axe in her brain. She fell dead upon the spot, 231 | without a groan. 232 | This hideous murder accomplished, I set myself forthwith, and with entire 233 | deliberation, to the task of concealing the body. I knew that I could not 234 | remove it from the house, either by day or by night, without the risk of 235 | being observed by the neighbors. Many projects entered my mind. At one 236 | period I thought of cutting the corpse into minute fragments, and destroying 237 | them by fire. At another, I resolved to dig a grave for it in the floor of 238 | the cellar. Again, I deliberated about casting it in the well in the yard 239 | --about packing it in a box, as if merchandize, with the usual arrangements, 240 | and so getting a porter to take it from the house. Finally I hit upon what I 241 | considered a far better expedient than either of these. I determined to wall 242 | it up in the cellar --as the monks of the middle ages are recorded to have 243 | walled up their victims. 244 | For a purpose such as this the cellar was well adapted. Its walls were 245 | loosely constructed, and had lately been plastered throughout with a rough 246 | plaster, which the dampness of the atmosphere had prevented from hardening. 247 | Moreover, in one of the walls was a projection, caused by a false chimney, 248 | or fireplace, that had been filled up, and made to resemble the rest of the 249 | cellar. I made no doubt that I could readily displace the at this point, 250 | insert the corpse, and wall the whole up as before, so that no eye could 251 | detect anything suspicious. 252 | And in this calculation I was not deceived. By means of a crow-bar I 253 | easily dislodged the bricks, and, having carefully deposited the body 254 | against the inner wall, I propped it in that position, while, with little 255 | trouble, I re-laid the whole structure as it originally stood. Having 256 | procured mortar, sand, and hair, with every possible precaution, I prepared 257 | a plaster could not every poss be distinguished from the old, and with this 258 | I very carefully went over the new brick-work. When I had finished, I felt 259 | satisfied that all was right. The wall did not present the slightest 260 | appearance of having been disturbed. The rubbish on the floor was picked up 261 | with the minutest care. I looked around triumphantly, and said to myself 262 | --"Here at least, then, my labor has not been in vain." 263 | My next step was to look for the beast which had been the cause of so 264 | much wretchedness; for I had, at length, firmly resolved to put it to death. 265 | Had I been able to meet with it, at the moment, there could have been no 266 | doubt of its fate; but it appeared that the crafty animal had been alarmed 267 | at the violence of my previous anger, and forebore to present itself in my 268 | present mood. It is impossible to describe, or to imagine, the deep, the 269 | blissful sense of relief which the absence of the detested creature 270 | occasioned in my bosom. It did not make its appearance during the night 271 | --and thus for one night at least, since its introduction into the house, I 272 | soundly and tranquilly slept; aye, slept even with the burden of murder upon 273 | my soul! 274 | The second and the third day passed, and still my tormentor came not. 275 | Once again I breathed as a free-man. The monster, in terror, had fled the 276 | premises forever! I should behold it no more! My happiness was supreme! The 277 | guilt of my dark deed disturbed me but little. Some few inquiries had been 278 | made, but these had been readily answered. Even a search had been instituted 279 | --but of course nothing was to be discovered. I looked upon my future 280 | felicity as secured. 281 | Upon the fourth day of the assassination, a party of the police came, 282 | very unexpectedly, into the house, and proceeded again to make rigorous 283 | investigation of the premises. Secure, however, in the inscrutability of my 284 | place of concealment, I felt no embarrassment whatever. The officers bade me 285 | accompany them in their search. They left no nook or corner unexplored. At 286 | length, for the third or fourth time, they descended into the cellar. I 287 | quivered not in a muscle. My heart beat calmly as that of one who slumbers 288 | in innocence. I walked the cellar from end to end. I folded my arms upon my 289 | bosom, and roamed easily to and fro. The police were thoroughly satisfied 290 | and prepared to depart. The glee at my heart was too strong to be 291 | restrained. I burned to say if but one word, by way of triumph, and to 292 | render doubly sure their assurance of my guiltlessness. 293 | "Gentlemen," I said at last, as the party ascended the steps, "I delight 294 | to have allayed your suspicions. I wish you all health, and a little more 295 | courtesy. By the bye, gentlemen, this --this is a very well constructed 296 | house." (In the rabid desire to say something easily, I scarcely knew what I 297 | uttered at all.) --"I may say an excellently well constructed house. These 298 | walls --are you going, gentlemen? --these walls are solidly put together"; 299 | and here, through the mere phrenzy of bravado, I rapped heavily, with a cane 300 | which I held in my hand, upon that very portion of the brick-work behind 301 | which stood the corpse of the wife of my bosom. 302 | But may God shield and deliver me from the fangs of the Arch-Fiend! No 303 | sooner had the reverberation of my blows sunk into silence than I was 304 | answered by a voice from within the tomb! --by a cry, at first muffled and 305 | broken, like the sobbing of a child, and then quickly swelling into one 306 | long, loud, and continuous scream, utterly anomalous and inhuman --a howl 307 | --a wailing shriek, half of horror and half of triumph, such as might have 308 | arisen only out of hell, conjointly from the throats of the damned in their 309 | agony and of the demons that exult in the damnation. 310 | Of my own thoughts it is folly to speak. Swooning, I staggered to the 311 | opposite wall. For one instant the party upon the stairs remained 312 | motionless, through extremity of terror and of awe. In the next, a dozen 313 | stout arms were tolling at the wall. It fell bodily. The corpse, already 314 | greatly decayed and clotted with gore, stood erect before the eyes of the 315 | spectators. Upon its head, with red extended mouth and solitary eye of fire, 316 | sat the hideous beast whose craft had seduced me into murder, and whose 317 | informing voice had consigned me to the hangman. I had walled the monster up 318 | within the tomb! -------------------------------------------------------------------------------- /encrypted_cat.txt: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/abe1e414a0f587528aa51059f1f61d56aae3555b/encrypted_cat.txt -------------------------------------------------------------------------------- /requirements.txt: -------------------------------------------------------------------------------- 1 | matplotlib 2 | sympy 3 | pycryptodomex -------------------------------------------------------------------------------- /tux.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/abe1e414a0f587528aa51059f1f61d56aae3555b/tux.png -------------------------------------------------------------------------------- /tux_gray.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gverticale/network-security-and-cryptography/abe1e414a0f587528aa51059f1f61d56aae3555b/tux_gray.png --------------------------------------------------------------------------------