├── requirements.txt ├── ZK Sudoku.pdf ├── hashUtils.py ├── README.md ├── .gitignore ├── interactiveSudoku.py ├── zkSudoku.py ├── subsetProverRsa.py ├── rsaCommitment.py └── LICENSE /requirements.txt: -------------------------------------------------------------------------------- 1 | gmpy2==2.0.8 2 | -------------------------------------------------------------------------------- /ZK Sudoku.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/naure/zk/HEAD/ZK Sudoku.pdf -------------------------------------------------------------------------------- /hashUtils.py: -------------------------------------------------------------------------------- 1 | 2 | import sys 3 | from hashlib import sha3_256 4 | 5 | if sys.version[0] == '2': 6 | safe_ord = ord 7 | else: 8 | safe_ord = lambda x: x 9 | 10 | def to_bytes(obj): 11 | return bytes(str(obj), "utf8") 12 | 13 | def bytes_to_int(x): 14 | o = 0 15 | for b in x: 16 | o = (o << 8) + safe_ord(b) 17 | return o 18 | 19 | def str_to_int(x): 20 | return bytes_to_int(to_bytes(x)) 21 | 22 | def hashBytes(data): 23 | return sha3_256(data).hexdigest() 24 | 25 | def hashObject(obj): 26 | return sha3_256(to_bytes(obj)).hexdigest() 27 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Zero-knowledge non-interactive proofs 2 | 3 | This is an experimental framework to build Zero-knowledge non-interactive proofs, 4 | based on the Fiat-Shamir heuristic, a proof-of-work, and a constant-size commitment scheme. 5 | 6 | It turns an interactive system with many challenges into a compact static proof. 7 | 8 | The proof-of-work sets the minimum effort required from an attacker to try a 9 | commitment, if looking for favorable challenges. 10 | 11 | ## Concise commitment scheme 12 | 13 | The commitment scheme turns the list of hidden responses into a single number. 14 | After the responses to reveal are chosen, it produces a proof that those were 15 | indeed parts of the commitment. 16 | 17 | See https://medium.com/@aurelcode/cryptographic-accumulators-da3aa4561d77. 18 | 19 | ## Demo with Sudokus 20 | 21 | A demonstration with the obligatory Sudoku interactive proof. 22 | 23 | See the file `zkSudoku.py`. 24 | 25 | ### The underlying interactive protocol 26 | 27 | 1. Find a secret Sudoku grid. 28 | 29 | 2. Prover generates many encrypted versions of the grid, and keeps them hidden. 30 | 31 | 3. Verifier picks a row, file or block to reveal from each grid, and 32 | checks that they do contain the numbers from 1 to 9. 33 | 34 | See `ZK Sudoku.pdf` and print it on paper to try it out by hand. 35 | 36 | 37 | ### Make it non-interactive 38 | 39 | 3. Commit to the encrypted values. 40 | 41 | 4. Execute a proof-of-work. 42 | 43 | 5. Pick pseudo-random challenges from the commitment and p-o-w. 44 | 45 | 6. Collect responses and prove that they were committed to. 46 | 47 | 7. Serialize / deserialize and measure the proof size. 48 | 49 | 8. Verify the proof. 50 | -------------------------------------------------------------------------------- /.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 | env/ 12 | build/ 13 | develop-eggs/ 14 | dist/ 15 | downloads/ 16 | eggs/ 17 | .eggs/ 18 | lib/ 19 | lib64/ 20 | parts/ 21 | sdist/ 22 | var/ 23 | wheels/ 24 | *.egg-info/ 25 | .installed.cfg 26 | *.egg 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 | 49 | # Translations 50 | *.mo 51 | *.pot 52 | 53 | # Django stuff: 54 | *.log 55 | local_settings.py 56 | 57 | # Flask stuff: 58 | instance/ 59 | .webassets-cache 60 | 61 | # Scrapy stuff: 62 | .scrapy 63 | 64 | # Sphinx documentation 65 | docs/_build/ 66 | 67 | # PyBuilder 68 | target/ 69 | 70 | # Jupyter Notebook 71 | .ipynb_checkpoints 72 | 73 | # pyenv 74 | .python-version 75 | 76 | # celery beat schedule file 77 | celerybeat-schedule 78 | 79 | # SageMath parsed files 80 | *.sage.py 81 | 82 | # dotenv 83 | .env 84 | 85 | # virtualenv 86 | .venv 87 | venv/ 88 | ENV/ 89 | 90 | # Spyder project settings 91 | .spyderproject 92 | .spyproject 93 | 94 | # Rope project settings 95 | .ropeproject 96 | 97 | # mkdocs documentation 98 | /site 99 | 100 | # mypy 101 | .mypy_cache/ 102 | -------------------------------------------------------------------------------- /interactiveSudoku.py: -------------------------------------------------------------------------------- 1 | # ZK Sudoku intro: http://blog.computationalcomplexity.org/2006/08/zero-knowledge-sudoku.html 2 | 3 | #%% Generate a secret Sudoku secretGrid 4 | 5 | import numpy as np 6 | 7 | secretGrid = np.zeros((9,9), dtype=int) 8 | r = np.arange(9) 9 | 10 | # Puzzle constraints. Simply 9 squares that contain 1 to 9, to be easy to check. 11 | puzzleIndices = np.array([0, 7, 5, 2, 0, 7, 4, 2, 0]) + np.arange(9) * 9 12 | 13 | # First group of 3 rows 14 | secretGrid[0] = np.roll(r, 0) 15 | secretGrid[1] = np.roll(r, -3) 16 | secretGrid[2] = np.roll(r, -6) 17 | # Second group of 3 rows 18 | secretGrid[3:6] = np.roll(secretGrid[0:3], -1, axis=1) 19 | # Third group of 3 rows 20 | secretGrid[6:9] = np.roll(secretGrid[0:3], -2, axis=1) 21 | 22 | 23 | def checkDigits(block): 24 | return np.all(np.sort(block.flatten()) == r) 25 | 26 | def assertIsSudoku(grid): 27 | for i in range(9): 28 | assert checkDigits(grid[i,:]) 29 | assert checkDigits(grid[:,i]) 30 | for i in range(3): 31 | for j in range(3): 32 | assert checkDigits(grid[3*i:3*i+3, 3*j:3*j+3]) 33 | 34 | assertIsSudoku(secretGrid) 35 | checkDigits(secretGrid.flat[puzzleIndices]) 36 | print(secretGrid+1) 37 | 38 | 39 | #%% Transform into some encrypted grids 40 | 41 | import os 42 | import struct 43 | import numpy as np 44 | 45 | # Reseed at each round to mitigate the weakness of numpy random. 46 | # TODO: Switch to crypto-grade PRNG. 47 | def reseed(): 48 | np.random.seed(struct.unpack("I", os.urandom(4))[0]) 49 | 50 | 51 | def makeHiddenSudoku(grid): 52 | reseed() 53 | # Pick a random mapping of digits 54 | key = np.random.permutation(r) 55 | # Encrypt the grid 56 | encrypted = key[grid] 57 | assertIsSudoku(encrypted) 58 | return key, encrypted 59 | 60 | 61 | def makeManyHiddenSudokus(grid, nChallenges): 62 | keys = np.zeros((nChallenges, 9), dtype=int) 63 | grids = np.zeros((nChallenges, 9,9), dtype=int) 64 | 65 | for i in range(nChallenges): 66 | key, encrypted = makeHiddenSudoku(grid) 67 | # Store with digits between 1-9 68 | keys[i] = key + 1 69 | grids[i] = encrypted + 1 70 | 71 | return keys, grids 72 | 73 | 74 | #%% Format for paper 75 | 76 | def printPaperSudoku(): 77 | keys, grids = makeManyHiddenSudokus(secretGrid, 9) 78 | 79 | # Keep trivial original as demo 80 | keys[0] = r + 1 81 | grids[0] = secretGrid + 1 82 | 83 | put = lambda *args: print(*args, sep="", end="") 84 | 85 | put(""" 86 | Zero-knowledge proof of Sudoku 87 | —————————————————————————————— 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | """) 97 | 98 | # Display 3 groups of 3 grids, line-by-line 99 | for gridI in range(3): 100 | for gridJ in range(3): 101 | put("Key: ", keys[gridI*3 + gridJ], " "*10) 102 | put("\n\n") 103 | for blockI in range(0,9,3): 104 | for line in range(blockI, blockI+3): 105 | for gridJ in range(3): 106 | grid = grids[gridI*3 + gridJ] 107 | for triple in range(0,9,3): 108 | put(grid[line, triple:triple+3], " ") 109 | if gridJ<2: put(" "*7) 110 | put("\n") 111 | put("\n") 112 | put("\n\n") 113 | 114 | # Print mask cutouts 115 | put("\n"*25) # New page 116 | put("\n"*20) # Middle of the page 117 | # Shape of the mask 118 | put("""Masks to cut out: 119 | XXXXXXXXXXXXXXXXXXXXXXXXX 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | XXXXXXX 141 | XXXXXXX 142 | XXXXXXX 143 | 144 | """) 145 | 146 | 147 | if __name__ == "__main__": 148 | printPaperSudoku() 149 | -------------------------------------------------------------------------------- /zkSudoku.py: -------------------------------------------------------------------------------- 1 | """ Crypto-based non-interactive proof 2 | """ 3 | 4 | #%% Settings 5 | challengeMax = 9 * 3 + 1 # Possible choices: 9 per "type", plus 1 for "constraints" 6 | nChallenges = 256 # Security factor of the interactive protocol 7 | difficulty4bits = 16//4 # Extra security factor with proof-of-work (in 4 bits increments) 8 | print("False-positive rate", ((challengeMax - 1) / challengeMax)**nChallenges) 9 | 10 | # Reasonnable values: 512 challenges and 28 bits proof-of-work. 11 | 12 | 13 | #%% Phase 1: proof commitment, ultra compact. 14 | 15 | import numpy as np 16 | from hashUtils import hashObject, str_to_int 17 | from rsaCommitment import RSACommitmentValues 18 | from interactiveSudoku import secretGrid, puzzleIndices, makeManyHiddenSudokus, checkDigits 19 | 20 | keys, grids = makeManyHiddenSudokus(secretGrid, nChallenges) 21 | 22 | committer = RSACommitmentValues(nbits=4) 23 | commitsRoot = committer.commitValues(grids.flatten()) 24 | 25 | 26 | #%% Phase 2: proof-of-work 27 | # Increase the security factor by requiring some work per proof attempt. 28 | # TODO: Support extra fields like blockchain hashes to prove recency (harder to brute-force). 29 | # Alternative: retrieve some public randomness like blockchain hashes. 30 | 31 | difficulty = "f" * difficulty4bits 32 | print("Proof-of-work difficulty:", len(difficulty) * 4, "bits") 33 | 34 | def makeProofOfWork(commitsRoot, nonce): 35 | return hashObject(str(commitsRoot) + str(nonce)) 36 | 37 | def searchProofOfWork(commitsRoot): 38 | nonce = 0 39 | while makeProofOfWork(commitsRoot, nonce) < difficulty: 40 | nonce += 1 41 | return nonce 42 | 43 | nonce = searchProofOfWork(commitsRoot) 44 | 45 | 46 | #%% Phase 3: Fiat–Shamir transformation. 47 | # Derive pseudo-random challenges from the commitment and proof-of-work. 48 | 49 | # Sudoku: encoding a test as a number: n = type*9 + choice_within_type 50 | # Test types: 0=Row, 1=column, 2=block, 3=constraints (then choice=0) 51 | 52 | PoW = makeProofOfWork(commitsRoot, nonce) 53 | assert PoW >= difficulty 54 | 55 | def makeChallenges(commitsRoot, PoW): 56 | seed = str_to_int(hashObject(str(commitsRoot) + PoW)) % 2**32 57 | rd = np.random.RandomState(seed) 58 | 59 | # For each grid, pick a line, a column, and a block to challenge 60 | return rd.randint(challengeMax, size=nChallenges) 61 | 62 | # Derive challenges from random data 63 | challenges = makeChallenges(commitsRoot, PoW) 64 | 65 | 66 | #%% Phase 4: reveal 67 | 68 | responses = np.zeros(( 69 | nChallenges, # Grids 70 | 9), # Values per set 71 | dtype=int) 72 | 73 | 74 | def getResponse(grid, challenge): 75 | " Interpret an integer challenge as a set of digits (rows, ...). " 76 | cType = challenge // 9 77 | cChoice = challenge % 9 78 | if cType == 0: return grid[cChoice, :] # Row 79 | if cType == 1: return grid[:, cChoice] # Column 80 | if cType == 2: # Block 81 | y = (cChoice // 3) * 3 82 | x = (cChoice % 3) * 3 83 | return grid[y:y+3, x:x+3].flatten() 84 | # Otherwise return the puzzle constraints 85 | return grid.flat[puzzleIndices] 86 | 87 | 88 | # Map of grid positions to flat indices 89 | idGrid = np.arange(9 * 9).reshape(9, 9) 90 | 91 | def getSquareIds(gridI, challenge): 92 | " Convert the challenge into flat indices of the Merkle tree. " 93 | gridOffset = gridI * idGrid.size 94 | idsInGrid = getResponse(idGrid, challenges[gridI]) 95 | return idsInGrid + gridOffset 96 | 97 | responseIds = [] 98 | 99 | # Collect the responses and the Merkle paths for all challenges 100 | for gridI in range(len(challenges)): 101 | challenge = challenges[gridI] 102 | response = getResponse(grids[gridI], challenge) 103 | responses[gridI] = response 104 | responseIds.extend(getSquareIds(gridI, challenge)) 105 | 106 | assert len(responseIds) == len(set(responseIds)) == responses.size 107 | 108 | proofOfCommitment = committer.proveValues(responseIds) 109 | 110 | 111 | #%% Phase 5: Pack the proof into a single message 112 | import json 113 | import gzip 114 | 115 | # Simple format with JSON 116 | proof = { 117 | "commitment to set": commitsRoot, 118 | "proof-of-work nonce": nonce, 119 | "responses to challenges": responses.tolist(), 120 | "proof that responses were committed": proofOfCommitment, 121 | "nBits": committer.nbits, 122 | } 123 | 124 | # GZIP will remove most inefficiencies of encodings, duplicate values, etc. 125 | serializedProof = gzip.compress(json.dumps(proof).encode("utf8")) 126 | print("Proof size: %.0fK for %i challenges." % (len(serializedProof) / 1024, nChallenges)) 127 | 128 | v_proof = json.loads(gzip.decompress(serializedProof).decode("utf8")) 129 | assert v_proof == proof 130 | 131 | 132 | #%% Phase 6: Verify 133 | # The v_ prefix indicates verifier's variables. 134 | 135 | # commitsRoot and noonce must be a proof-of-work. 136 | v_commitsRoot = v_proof["commitment to set"] 137 | v_PoW = makeProofOfWork(v_commitsRoot, v_proof["proof-of-work nonce"]) 138 | assert v_PoW >= difficulty, "Too little difficulty." 139 | 140 | # Recompute challenges from PoW data 141 | v_challenges = makeChallenges(v_commitsRoot, v_PoW) 142 | assert len(v_challenges) >= nChallenges, "Too few challenges." 143 | 144 | v_responses = v_proof["responses to challenges"] 145 | assert len(v_challenges) == len(v_responses) 146 | 147 | v_responseValues = np.array(v_responses).flatten() 148 | v_responseIds = [] 149 | 150 | for gridI in range(len(v_challenges)): 151 | challenge = v_challenges[gridI] 152 | response = v_responses[gridI] 153 | v_responseIds.extend(getSquareIds(gridI, challenge)) 154 | 155 | # Verify that the solution is from a valid Sudoku: 156 | # * Each set must be all 1-9 digits. 157 | # * Or, check the puzzle constraints (in that case, it's also 1-9 digits). 158 | assert checkDigits(np.array(response) - 1), "The response is not a valid solution." 159 | 160 | 161 | v_committer = RSACommitmentValues(nbits=v_proof["nBits"]) 162 | v_proofOfResponse = v_proof["proof that responses were committed"] 163 | v_wasCommitted = v_committer.verifyValues(v_responseIds, v_responseValues, v_proofOfResponse, v_commitsRoot) 164 | assert v_wasCommitted, "The responses are not all included in the commitment." 165 | 166 | print("Proof verified!") 167 | 168 | # This is Zero-knowledge where the verifier is anyone who trusts the proof-of-work randomness. 169 | -------------------------------------------------------------------------------- /subsetProverRsa.py: -------------------------------------------------------------------------------- 1 | """ 2 | Cryptographic accumulator. 3 | Prove that some items are a subset of a committed set. 4 | 5 | Next steps: 6 | 7 | Commit to an array using successive (small) primes instead of prime-hashing. 8 | 9 | Treat each prime p as a slot holding a binary number: 10 | 0 = Do not include p it and prove non-membership. 11 | 1 = Include p and prove membership. 12 | 13 | To encode an array of values, use several consecutive slots to 14 | include/exclude the bits of each value. 15 | 16 | Alternative for small values v, or to prove range (a <= v < b): 17 | Let each slot p store a value v. Include p^v in the set and: 18 | Prove that p^a is a member, so v >= a. 19 | Prove that p^(a+1) is not a member, so v < a+1 and v = a. 20 | 21 | """ 22 | 23 | #%% Hash and bytes utilities 24 | import sys 25 | import math 26 | from hashlib import sha3_256 27 | import numpy as np 28 | import gmpy2 29 | 30 | HASH_BYTES = 16 31 | 32 | safe_ord = ord if sys.version[0] == '2' else lambda x: x 33 | 34 | def to_bytes(obj): 35 | return bytes(str(obj), "utf8") 36 | 37 | def bytes_to_int(x): 38 | o = 0 39 | for b in x: 40 | o = (o << 8) + safe_ord(b) 41 | return o 42 | 43 | 44 | def intHash(data): 45 | h = sha3_256(data).digest() 46 | i = bytes_to_int(h[:HASH_BYTES]) 47 | return i 48 | 49 | def primeHash(data): 50 | " Derive a prime from the data. " 51 | i = intHash(data) 52 | p = int(gmpy2.next_prime(i)) 53 | return p 54 | 55 | 56 | def bits(n): 57 | return math.ceil(math.log2(n)) 58 | 59 | 60 | def prod(xs): 61 | y = 1 62 | for x in xs: 63 | y *= x 64 | return y 65 | 66 | def pows(g, exponents, mod): 67 | " Successive exponentiations in a group of unknown order. " 68 | y = g 69 | for e in exponents: 70 | y = pow(y, int(e), mod) 71 | return y 72 | 73 | def extended_euclidean_algorithm(a, b): 74 | """ 75 | Returns a three-tuple (gcd, x, y) such that 76 | a * x + b * y == gcd, where gcd is the greatest 77 | common divisor of a and b. 78 | 79 | This function implements the extended Euclidean 80 | algorithm and runs in O(log b) in the worst case. 81 | """ 82 | s, old_s = 0, 1 83 | t, old_t = 1, 0 84 | r, old_r = b, a 85 | 86 | while r != 0: 87 | quotient = old_r // r 88 | old_r, r = r, old_r - quotient * r 89 | old_s, s = s, old_s - quotient * s 90 | old_t, t = t, old_t - quotient * t 91 | 92 | return old_r, old_s, old_t 93 | 94 | 95 | class MaxHash(object): 96 | # 5 buckets of 6 bytes / 262144 values. 97 | def __init__(self): 98 | self.maxs = [0] * 5 99 | 100 | def add(self, h): 101 | for i in range(5): 102 | v = bytes_to_int(h[6*i : 6*(i+1)]) 103 | if v > self.maxs[i]: 104 | self.maxs[i] = v 105 | return h 106 | 107 | 108 | class SubsetProverRsa(object): 109 | " Same as plain, but pass the commit through the group. Verifier needs an extra subset-dependent proof. " 110 | 111 | # Using the RSA-2048 challenge modulus. 112 | # The factors and group order, equivalent to the private key, are believed to be unknown! 113 | # https://en.wikipedia.org/wiki/RSA_numbers#RSA-2048 114 | MOD = RSA2048 = 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357; 115 | # Any prime or coprime is a generator. 116 | G = 2**256 - 2**32 - 977 117 | assert gmpy2.is_prime(G) 118 | assert MOD % G != 0 119 | 120 | def __init__(self, items): 121 | self.intHashes, self.maxs = self.hashItems(items) 122 | assert len(items) == len(set(self.intHashes)), "Duplicates are not supported yet." 123 | 124 | def hashItems(self, items): 125 | primes = [] 126 | maxHash = MaxHash() 127 | 128 | for o in items: 129 | h = sha3_256(to_bytes(o)).digest() 130 | 131 | # Derive a prime from the data. 132 | i = bytes_to_int(h[:HASH_BYTES]) 133 | p = int(gmpy2.next_prime(i)) 134 | primes.append(p) 135 | 136 | # Track maxs 137 | maxHash.add(h) 138 | 139 | return primes, maxHash.maxs 140 | 141 | def commit(self): 142 | return [pows(self.G, self.intHashes, self.MOD), self.maxs] 143 | 144 | def proveSubset(self, subset): 145 | # hash(items not in subset) 146 | # Equivalent to commit / hash(subset) 147 | # TODO: Support duplicates 148 | subsetHashes, maxs = self.hashItems(subset) 149 | otherExponents = set(self.intHashes).difference(subsetHashes) 150 | return pows(self.G, otherExponents, self.MOD) 151 | 152 | def verifySubset(self, subset, proof, commit): 153 | subsetHashes, subsetMaxs = self.hashItems(subset) 154 | actual = pows(proof, subsetHashes, self.MOD) 155 | 156 | commitNum, commitMaxs = commit 157 | maxOk = np.all(np.array(subsetMaxs) <= np.array(commitMaxs)) 158 | # TODO: Verify estimated cardinality 159 | 160 | return actual == commitNum and maxOk 161 | 162 | def proveDisjoint(self, disjoint): 163 | # From https://www.cs.purdue.edu/homes/ninghui/papers/accumulator_acns07.pdf 164 | 165 | u = prod(self.intHashes) 166 | # commit == pow(G, u, MOD) 167 | x = prod(self.hashItems(disjoint)[0]) 168 | 169 | gcd, a, b = extended_euclidean_algorithm(u, x); gcd 170 | if gcd != 1: 171 | print("Warning: Some members of X are in the commited set, we cannot prove that they are disjoint!") 172 | return [0, 0] 173 | 174 | # TODO: Bring the value of a under a maximum size: 175 | # k=?; a = a + k * x; b = b - k * u 176 | 177 | d = pow(sp.G, -b, sp.MOD) 178 | return [a, d] 179 | 180 | def verifyDisjoint(self, disjoint, proof, commit): 181 | # TODO: validate proof values explicitely 182 | disjointHashes, _ = self.hashItems(disjoint) 183 | a, d = proof 184 | d_x = (pows(d, disjointHashes, self.MOD) * self.G) % self.MOD 185 | c_a = pow(commit[0], a, self.MOD) 186 | return d_x == c_a 187 | 188 | 189 | SubsetProver = SubsetProverRsa 190 | 191 | 192 | #%% Example of using SubsetProver 193 | if __name__ == "__main__": 194 | import math 195 | import numpy as np 196 | 197 | fullSet = np.array([3, 12, 17, 23, 35, 99]) # + list(range(100,200))) 198 | subset = np.array([ 12, 23 ]) 199 | complement = np.array([3, 17, 35, 99]) # + list(range(100,200))) 200 | disjoint = np.array([ 5, 6]) 201 | mixed = np.array([ 23, 5, 6]) 202 | 203 | sp = SubsetProver(fullSet) 204 | commit = sp.commit(); print("Commitment:", bits(commit[0]), "bits") 205 | 206 | proofSubset = sp.proveSubset(subset); print("Proof:", bits(proofSubset), "bits") 207 | assert sp.verifySubset(subset, proofSubset, commit) 208 | print("Accepted correct proof of subset!") 209 | 210 | cheatDisjoint = sp.proveSubset(disjoint) 211 | assert not sp.verifySubset(disjoint, cheatDisjoint, commit) 212 | print("Rejected incorrect proof for a disjoint set!") 213 | 214 | cheatMixed = sp.proveSubset(mixed) 215 | assert not sp.verifySubset(mixed, cheatMixed, commit) 216 | print("Rejected incorrect proof for an overlapping set!") 217 | 218 | proofNonSubset = sp.proveDisjoint(disjoint) 219 | assert sp.verifyDisjoint(disjoint, proofNonSubset, commit) 220 | print("Accepted correct proof of non-subset!") 221 | 222 | cheatNotSubset = sp.proveDisjoint(subset) 223 | assert not sp.verifyDisjoint(subset, cheatNotSubset, commit) 224 | print("Rejected incorrect proof of non-subset (subset)!") 225 | 226 | cheatNotMixed = sp.proveDisjoint(mixed) 227 | assert not sp.verifyDisjoint(mixed, cheatNotMixed, commit) 228 | print("Rejected incorrect proof of non-subset (mixed)!") 229 | -------------------------------------------------------------------------------- /rsaCommitment.py: -------------------------------------------------------------------------------- 1 | """ 2 | Cryptographic accumulator. 3 | Prove that some items are a subset of a committed set. 4 | 5 | Commit to an array of values, based on the sets of zeros and ones in binary representations of values. 6 | 7 | Treat each prime p as a slot holding a binary number: 8 | 0 = Do not include p it and prove non-membership. 9 | 1 = Include p and prove membership. 10 | 11 | To encode an array of values, use several consecutive slots to 12 | include/exclude the bits of each value. 13 | 14 | It's actually possible to prove memberships and non-memberships in one step: 15 | In the original "disjoint" check, replace the `g` by `g^x`. 16 | If it passes, `gcd(c, x) == x`, meaning that all x belong and no others. 17 | 18 | Now the proving part is beautiful: its inputs do not say which values 19 | belong to the set or not; it doesn't even know what it is proving! 20 | Then the verifier does treat members and non-members specially to see 21 | whether the gcd checks out. 22 | 23 | Alternative for small values v, or to prove range (a <= v < b): 24 | Let each slot p store a value v. Include p^v in the set and: 25 | Prove that p^a is a member, so v >= a. 26 | Prove that p^(a+1) is not a member, so v < a+1 and v = a. 27 | 28 | TODO: Add random decoy value to prevent an attacker from testing set values. 29 | TODO: Optimize the a,b coefficients in proveMixed(). 30 | TODO: Decide which is more performant: to include zeros or ones in the set. 31 | TODO: Implement as a map using primeHash as starting points for successive bits. 32 | TODO: Consider an implementation with elliptic curve pairing. 33 | 34 | """ 35 | 36 | #%% Hash and bytes utilities 37 | import sys 38 | import math 39 | from hashlib import sha3_256 40 | import numpy as np 41 | import gmpy2 42 | 43 | HASH_BYTES = 16 44 | 45 | safe_ord = ord if sys.version[0] == '2' else lambda x: x 46 | 47 | def to_bytes(obj): 48 | return bytes(str(obj), "utf8") 49 | 50 | def bytes_to_int(x): 51 | o = 0 52 | for b in x: 53 | o = (o << 8) + safe_ord(b) 54 | return o 55 | 56 | 57 | def intHash(data): 58 | h = sha3_256(data).digest() 59 | i = bytes_to_int(h[:HASH_BYTES]) 60 | return i 61 | 62 | def primeHash(data): 63 | " Derive a prime from the data. " 64 | i = intHash(data) 65 | p = int(gmpy2.next_prime(i)) 66 | return p 67 | 68 | 69 | def bits(n): 70 | return math.ceil(math.log2(n)) 71 | 72 | 73 | def prod(xs): 74 | y = 1 75 | for x in xs: 76 | y *= int(x) 77 | return y 78 | 79 | def pows(g, exponents, mod): 80 | " Successive exponentiations in a group of unknown order. " 81 | y = g 82 | for e in exponents: 83 | y = pow(y, int(e), mod) 84 | return y 85 | 86 | def extended_euclidean_algorithm(a, b): 87 | """ 88 | Returns a three-tuple (gcd, x, y) such that 89 | a * x + b * y == gcd, where gcd is the greatest 90 | common divisor of a and b. 91 | 92 | This function implements the extended Euclidean 93 | algorithm and runs in O(log b) in the worst case. 94 | """ 95 | s, old_s = 0, 1 96 | t, old_t = 1, 0 97 | r, old_r = b, a 98 | 99 | while r != 0: 100 | quotient = old_r // r 101 | old_r, r = r, old_r - quotient * r 102 | old_s, s = s, old_s - quotient * s 103 | old_t, t = t, old_t - quotient * t 104 | 105 | return old_r, old_s, old_t 106 | 107 | 108 | # Prime helpers 109 | 110 | def primesfrom2to(n): 111 | # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 112 | """ Input n>=6, Returns a array of primes, 2 <= p < n """ 113 | sieve = np.ones(n//3 + (n%6==2), dtype=np.bool) 114 | sieve[0] = False 115 | for i in range(int(n**0.5)//3+1): 116 | if sieve[i]: 117 | k=3*i+1|1 118 | sieve[ ((k*k)//3) ::2*k] = False 119 | sieve[(k*k+4*k-2*k*(i&1))//3::2*k] = False 120 | return np.r_[2,3,((3*np.nonzero(sieve)[0]+1)|1)] 121 | 122 | 123 | firstPrimes = [] 124 | 125 | def initPrimes(maxId): 126 | global firstPrimes 127 | maxN = maxId * int(np.log(maxId) + 5) 128 | firstPrimes = primesfrom2to(maxN) 129 | print("Precomputed %i primes < %i" % (len(firstPrimes), maxN)) 130 | 131 | def toPrimes(indices): 132 | " Map a list of indices to a list of primes. " 133 | maxId = np.max(indices) if len(indices) else 1 134 | if maxId >= len(firstPrimes): 135 | initPrimes(maxId) 136 | return firstPrimes[ indices ] 137 | 138 | def toBitPositions(ids, values, nbits): 139 | assert len(ids) == len(values) 140 | zeros = [] 141 | ones = [] 142 | 143 | for iVal, val in zip(ids, values): 144 | for iBit in range(iVal * nbits, (iVal+1) * nbits): 145 | if val % 2: 146 | ones.append(iBit) 147 | else: 148 | zeros.append(iBit) 149 | val >>= 1 150 | if val != 0: 151 | print("Warning: values[%i] is too big for %i bits!" % (iVal, nbits)) 152 | 153 | return zeros, ones 154 | 155 | assert toBitPositions([0, 1, 2], [5, 2, 15], nbits=4) == ( 156 | [1, 3, 4, 6, 7], [0, 2, 5, 8, 9, 10, 11]) 157 | 158 | 159 | #%% 160 | 161 | class RSACommitment(object): 162 | """ 163 | Commit to an array values. 164 | Reveal parts 165 | """ 166 | 167 | # Using the RSA-2048 challenge modulus. 168 | # The factors and group order, equivalent to the private key, are believed to be unknown! 169 | # https://en.wikipedia.org/wiki/RSA_numbers#RSA-2048 170 | MOD = RSA2048 = 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357; 171 | # Any prime or coprime is a generator. 172 | G = 2**256 - 2**32 - 977 173 | assert gmpy2.is_prime(G) 174 | assert MOD % G != 0 175 | 176 | 177 | def commit(self, indices): 178 | self.committedPrimes = toPrimes(indices) 179 | return pows(self.G, self.committedPrimes, self.MOD) 180 | 181 | def proveMembers(self, claimedIndices): 182 | # hash(items not in subset) 183 | claimedPrimes = toPrimes(claimedIndices) 184 | otherPrimes = set(self.committedPrimes).difference(claimedPrimes) 185 | return pows(self.G, otherPrimes, self.MOD) 186 | 187 | def verifyMembers(self, claimedIndices, proof, commit): 188 | claimedPrimes = toPrimes(claimedIndices) 189 | actual = pows(proof, claimedPrimes, self.MOD) 190 | return actual == commit 191 | 192 | def proveDisjoint(self, disjointIndices): 193 | return self.proveMixed(disjointIndices, checkDisjoint=True) 194 | 195 | def verifyDisjoint(self, disjointIndices, proof, commit): 196 | return self.verifyMixed([], disjointIndices, proof, commit) 197 | 198 | def proveMixed(self, subsetIndices, checkDisjoint=False): 199 | # From https://www.cs.purdue.edu/homes/ninghui/papers/accumulator_acns07.pdf 200 | 201 | u = prod(self.committedPrimes) # All the committed set 202 | # commit == pow(G, u, MOD) 203 | x = prod(toPrimes(subsetIndices)) # Members and non-members to prove 204 | 205 | gcd, a, b = extended_euclidean_algorithm(u, x); gcd 206 | # au + bx == gcd 207 | 208 | if checkDisjoint and gcd != 1: 209 | print("Warning: Some members of X are in the commited set, we cannot prove that they are disjoint!") 210 | 211 | # Bring the coefficients into the right range. 212 | # Find k such that a=a+k*x > 0, and b=b-k*u < 0. 213 | if a < 0 or b > 0: 214 | k = max(-a // x, b // u) + 1 215 | a = a + k * x 216 | b = b - k * u 217 | 218 | d = pow(self.G, -b, self.MOD) 219 | return [a, d] 220 | 221 | def verifyMixed(self, subsetIndices, disjointIndices, proof, commit): 222 | # TODO: validate proof values explicitely 223 | subsetPrimes = list(toPrimes(subsetIndices)) 224 | g_gcd = pows(self.G, subsetPrimes, self.MOD) 225 | disjointPrimes = list(toPrimes(disjointIndices)) 226 | a, d = proof 227 | d_x = (pows(d, disjointPrimes+subsetPrimes, self.MOD) * g_gcd) % self.MOD 228 | c_a = pow(commit, a, self.MOD) 229 | return d_x == c_a 230 | 231 | 232 | #%% 233 | class RSACommitmentValues(RSACommitment): 234 | " Commitment of numerical values. " 235 | 236 | def __init__(self, nbits): 237 | self.nbits = nbits 238 | 239 | def commitValues(self, values): 240 | # Commit to indices where value bits are 0 241 | valueIds = np.arange(len(values)) 242 | self.zeros, self.ones = toBitPositions(valueIds, values, self.nbits) 243 | return self.commit(self.zeros) 244 | 245 | def proveValues(self, valueIds): 246 | # Find indices of binary 0s and 1s of the values to prove. 247 | valueIds = set(valueIds) 248 | zeros = [i for i in self.zeros if i//self.nbits in valueIds] 249 | ones = [i for i in self.ones if i//self.nbits in valueIds] 250 | 251 | # Prove that zeros are members and ones are not members. 252 | return self.proveMixed(zeros + ones) 253 | 254 | def verifyValues(self, valueIds, values, proof, commit): 255 | # Find indices of binary 1s and 0s of the values to check. 256 | zeros, ones = toBitPositions(valueIds, values, self.nbits) 257 | # Verify the zeros are not members. 258 | return self.verifyMixed(zeros, ones, proof, commit) 259 | 260 | 261 | 262 | #%% Example of using SubsetProver 263 | if __name__ == "__main__": 264 | import math 265 | import numpy as np 266 | 267 | fullSet = np.array([3, 12, 17, 23, 35, 99]) # + list(range(50000,50100))) 268 | subset = np.array([ 12, 23 ]) 269 | complement = np.array([3, 17, 35, 99]) 270 | disjoint = np.array([ 5, 6]) # + list(range(50100,50200))) 271 | mixed = np.array([ 12, 23, 5, 6]) 272 | 273 | sp = RSACommitment() 274 | commit = sp.commit(fullSet) 275 | print("Commitment:", bits(commit)//8, "bytes\n") 276 | 277 | proofSubset = sp.proveMembers(subset) 278 | assert sp.verifyMembers(subset, proofSubset, commit) 279 | print("Proof of memberships:", bits(proofSubset)//8, "bytes") 280 | print("Accepted correct proof of subset!\n") 281 | 282 | cheatDisjoint = sp.proveMembers(disjoint) 283 | assert not sp.verifyMembers(disjoint, cheatDisjoint, commit) 284 | print("Rejected incorrect proof for a disjoint set!\n") 285 | 286 | cheatMixed = sp.proveMembers(mixed) 287 | assert not sp.verifyMembers(mixed, cheatMixed, commit) 288 | print("Rejected incorrect proof for an overlapping set!\n") 289 | 290 | proofDisjoint = sp.proveDisjoint(disjoint) 291 | assert sp.verifyDisjoint(disjoint, proofDisjoint, commit) 292 | print("Proof of non-memberships:", 293 | (bits(proofDisjoint[0]) + bits(proofDisjoint[1])) // 8, "bytes") 294 | print("Accepted correct proof of non-subset!\n") 295 | 296 | cheatNotSubset = sp.proveDisjoint(subset) 297 | assert not sp.verifyDisjoint(subset, cheatNotSubset, commit) 298 | print("Rejected incorrect proof of non-subset (subset)!\n") 299 | 300 | cheatNotMixed = sp.proveDisjoint(mixed) 301 | assert not sp.verifyDisjoint(mixed, cheatNotMixed, commit) 302 | print("Rejected incorrect proof of non-subset (mixed)!\n") 303 | 304 | proofMixed = sp.proveMixed(mixed) 305 | assert sp.verifyMixed(subset, disjoint, proofMixed, commit) 306 | print("Accepted correct proof of a mixed set!") 307 | 308 | # Example with values 309 | values = np.array([3, 12, 17, 23, 35, 99]) 310 | revealIds = [1, 3] 311 | 312 | spv = RSACommitmentValues(nbits=8) 313 | commit = spv.commitValues(values) 314 | 315 | proofValues = spv.proveValues(revealIds) 316 | print("Proof of an array of values:", 317 | (bits(proofValues[0]) + bits(proofValues[1])) // 8, "bytes") 318 | 319 | assert spv.verifyValues(revealIds, values[revealIds], proofValues, commit) 320 | print("Accepted correct proof of an array of values!\n") 321 | 322 | assert not spv.verifyValues(revealIds, [12, 42], proofValues, commit) 323 | print("Rejected incorrect proof of an array of values!\n") 324 | -------------------------------------------------------------------------------- /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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No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------