#!/usr/bin/env python3 # Copyright (c) 2020 Pieter Wuille # Distributed under the MIT software license, see the accompanying # file LICENSE or http://www.opensource.org/licenses/mit-license.php. """Native Python (slow) reimplementation of libminisketch' algorithms.""" import random import unittest # Irreducible polynomials over GF(2) to use (represented as integers). # # Most fields can be defined by multiple such polynomials. Minisketch uses the one with the minimal # number of nonzero coefficients, and tie-breaking by picking the lexicographically first among # those. # # All polynomials for degrees 2 through 64 (inclusive) are given. GF2_MODULI = [ None, None, 2**2 + 2**1 + 1, 2**3 + 2**1 + 1, 2**4 + 2**1 + 1, 2**5 + 2**2 + 1, 2**6 + 2**1 + 1, 2**7 + 2**1 + 1, 2**8 + 2**4 + 2**3 + 2**1 + 1, 2**9 + 2**1 + 1, 2**10 + 2**3 + 1, 2**11 + 2**2 + 1, 2**12 + 2**3 + 1, 2**13 + 2**4 + 2**3 + 2**1 + 1, 2**14 + 2**5 + 1, 2**15 + 2**1 + 1, 2**16 + 2**5 + 2**3 + 2**1 + 1, 2**17 + 2**3 + 1, 2**18 + 2**3 + 1, 2**19 + 2**5 + 2**2 + 2**1 + 1, 2**20 + 2**3 + 1, 2**21 + 2**2 + 1, 2**22 + 2**1 + 1, 2**23 + 2**5 + 1, 2**24 + 2**4 + 2**3 + 2**1 + 1, 2**25 + 2**3 + 1, 2**26 + 2**4 + 2**3 + 2**1 + 1, 2**27 + 2**5 + 2**2 + 2**1 + 1, 2**28 + 2**1 + 1, 2**29 + 2**2 + 1, 2**30 + 2**1 + 1, 2**31 + 2**3 + 1, 2**32 + 2**7 + 2**3 + 2**2 + 1, 2**33 + 2**10 + 1, 2**34 + 2**7 + 1, 2**35 + 2**2 + 1, 2**36 + 2**9 + 1, 2**37 + 2**6 + 2**4 + 2**1 + 1, 2**38 + 2**6 + 2**5 + 2**1 + 1, 2**39 + 2**4 + 1, 2**40 + 2**5 + 2**4 + 2**3 + 1, 2**41 + 2**3 + 1, 2**42 + 2**7 + 1, 2**43 + 2**6 + 2**4 + 2**3 + 1, 2**44 + 2**5 + 1, 2**45 + 2**4 + 2**3 + 2**1 + 1, 2**46 + 2**1 + 1, 2**47 + 2**5 + 1, 2**48 + 2**5 + 2**3 + 2**2 + 1, 2**49 + 2**9 + 1, 2**50 + 2**4 + 2**3 + 2**2 + 1, 2**51 + 2**6 + 2**3 + 2**1 + 1, 2**52 + 2**3 + 1, 2**53 + 2**6 + 2**2 + 2**1 + 1, 2**54 + 2**9 + 1, 2**55 + 2**7 + 1, 2**56 + 2**7 + 2**4 + 2**2 + 1, 2**57 + 2**4 + 1, 2**58 + 2**19 + 1, 2**59 + 2**7 + 2**4 + 2**2 + 1, 2**60 + 2**1 + 1, 2**61 + 2**5 + 2**2 + 2**1 + 1, 2**62 + 2**29 + 1, 2**63 + 2**1 + 1, 2**64 + 2**4 + 2**3 + 2**1 + 1 ] class GF2Ops: """Class to perform GF(2^field_size) operations on elements represented as integers. Given that elements are represented as integers, addition is simply xor, and not exposed here. """ def __init__(self, field_size): """Construct a GF2Ops object for the specified field size.""" self.field_size = field_size self._modulus = GF2_MODULI[field_size] assert self._modulus is not None def mul2(self, x): """Multiply x by 2 in GF(2^field_size).""" x <<= 1 if x >> self.field_size: x ^= self._modulus return x def mul(self, x, y): """Multiply x by y in GF(2^field_size).""" ret = 0 while y: if y & 1: ret ^= x y >>= 1 x = self.mul2(x) return ret def sqr(self, x): """Square x in GF(2^field_size).""" return self.mul(x, x) def inv(self, x): """Compute the inverse of x in GF(2^field_size).""" assert x != 0 # Use the extended polynomial Euclidean GCD algorithm on (modulus, x), over GF(2). # See https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor. t1, t2 = 0, 1 r1, r2 = self._modulus, x r1l, r2l = self.field_size + 1, r2.bit_length() while r2: q = r1l - r2l r1 ^= r2 << q t1 ^= t2 << q r1l = r1.bit_length() if r1 < r2: t1, t2 = t2, t1 r1, r2 = r2, r1 r1l, r2l = r2l, r1l assert r1 == 1 return t1 class TestGF2Ops(unittest.TestCase): """Test class for basic arithmetic properties of GF2Ops.""" def field_size_test(self, field_size): """Test operations for given field_size.""" gf = GF2Ops(field_size) for i in range(100): x = random.randrange(1 << field_size) y = random.randrange(1 << field_size) x2 = gf.mul2(x) xy = gf.mul(x, y) self.assertEqual(x2, gf.mul(x, 2)) # mul2(x) == x*2 self.assertEqual(x2, gf.mul(2, x)) # mul2(x) == 2*x self.assertEqual(xy == 0, x == 0 or y == 0) self.assertEqual(xy == x, y == 1 or x == 0) self.assertEqual(xy == y, x == 1 or y == 0) self.assertEqual(xy, gf.mul(y, x)) # x*y == y*x if i < 10: xp = x for _ in range(field_size): xp = gf.sqr(xp) self.assertEqual(xp, x) # x^(2^field_size) == x if y != 0: yi = gf.inv(y) self.assertEqual(y == yi, y == 1) # y==1/x iff y==1 self.assertEqual(gf.mul(y, yi), 1) # y*(1/y) == 1 yii = gf.inv(yi) self.assertEqual(y, yii) # 1/(1/y) == y if x != 0: xi = gf.inv(x) xyi = gf.inv(xy) self.assertEqual(xyi, gf.mul(xi, yi)) # (1/x)*(1/y) == 1/(x*y) def test(self): """Run tests.""" for field_size in range(2, 65): self.field_size_test(field_size) # The operations below operate on polynomials over GF(2^field_size), represented as lists of # integers: # # [a, b, c, ...] = a + b*x + c*x^2 + ... # # As an invariant, there are never any trailing zeroes in the list representation. # # Examples: # * [] = 0 # * [3] = 3 # * [0, 1] = x # * [2, 0, 5] = 5*x^2 + 2 def poly_monic(poly, gf): """Return a monic version of the polynomial poly.""" # Multiply every coefficient with the inverse of the top coefficient. inv = gf.inv(poly[-1]) return [gf.mul(inv, v) for v in poly] def poly_divmod(poly, mod, gf): """Return the polynomial (quotient, remainder) of poly divided by mod.""" assert len(mod) > 0 and mod[-1] == 1 # Require monic mod. if len(poly) < len(mod): return ([], poly) val = list(poly) div = [0 for _ in range(len(val) - len(mod) + 1)] while len(val) >= len(mod): term = val[-1] div[len(val) - len(mod)] = term # If the highest coefficient in val is nonzero, subtract a multiple of mod from it. val.pop() if term != 0: for x in range(len(mod) - 1): val[1 + x - len(mod)] ^= gf.mul(term, mod[x]) # Prune trailing zero coefficients. while len(val) > 0 and val[-1] == 0: val.pop() return div, val def poly_gcd(a, b, gf): """Return the polynomial GCD of a and b.""" if len(a) < len(b): a, b = b, a # Use Euclid's algorithm to find the GCD of a and b. # see https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Euclid's_algorithm. while len(b) > 0: b = poly_monic(b, gf) (_, b), a = poly_divmod(a, b, gf), b return a def poly_sqr(poly, gf): """Return the square of polynomial poly.""" if len(poly) == 0: return [] # In characteristic-2 fields, thanks to Frobenius' endomorphism ((a + b)^2 = a^2 + b^2), # squaring a polynomial is easy: square all the coefficients and interleave with zeroes. # E.g., (3 + 5*x + 17*x^2)^2 = 3^2 + (5*x)^2 + (17*x^2)^2. # See https://en.wikipedia.org/wiki/Frobenius_endomorphism. return [0 if i & 1 else gf.sqr(poly[i // 2]) for i in range(2 * len(poly) - 1)] def poly_tracemod(poly, param, gf): """Compute y + y^2 + y^4 + ... + y^(2^(field_size-1)) mod poly, where y = param*x.""" out = [0, param] for _ in range(gf.field_size - 1): # In each loop iteration i, we start with out = y + y^2 + ... + y^(2^i). By squaring that we # transform it into out = y^2 + y^4 + ... + y^(2^(i+1)). out = poly_sqr(out, gf) # Thus, we just need to add y again to it to get out = y + ... + y^(2^(i+1)). while len(out) < 2: out.append(0) out[1] = param # Finally take a modulus to keep the intermediary polynomials small. _, out = poly_divmod(out, poly, gf) return out def poly_frobeniusmod(poly, gf): """Compute x^(2^field_size) mod poly.""" out = [0, 1] for _ in range(gf.field_size): _, out = poly_divmod(poly_sqr(out, gf), poly, gf) return out def poly_find_roots(poly, gf): """Find the roots of poly if fully factorizable with unique roots, [] otherwise.""" assert len(poly) > 0 # If the polynomial is constant (and nonzero), it has no roots. if len(poly) == 1: return [] # Make the polynomial monic (which doesn't change its roots). poly = poly_monic(poly, gf) # If the polynomial is of the form x+a, return a. if len(poly) == 2: return [poly[0]] # Otherwise, first test that poly can be completely factored into unique roots. The polynomial # x^(2^fieldsize)-x has every field element once as root. Thus we want to know that that is a # multiple of poly. Compute x^(field_size) mod poly, which needs to equal x if that is the case # (unless poly has degree <= 1, but that case is handled above). if poly_frobeniusmod(poly, gf) != [0, 1]: return [] def rec_split(poly, randv): """Recursively split poly using the Berlekamp trace algorithm.""" # See https://hal.archives-ouvertes.fr/hal-00626997/document. assert len(poly) > 1 and poly[-1] == 1 # Require a monic poly. # If poly is of the form x+a, its root is a. if len(poly) == 2: return [poly[0]] # Try consecutive randomization factors randv, until one is found that factors poly. while True: # Compute the trace of (randv*x) mod poly. This is a polynomial that maps half of the # domain to 0, and the other half to 1. Which half that is is controlled by randv. # By taking it modulo poly, we only add a multiple of poly. Thus the result has at least # the shared roots of the trace polynomial and poly still, but may have others. trace = poly_tracemod(poly, randv, gf) # Using the set {2^i*a for i=0..fieldsize-1} gives optimally independent randv values # (no more than fieldsize are ever needed). randv = gf.mul2(randv) # Now take the GCD of this trace polynomial with poly. The result is a polynomial # that only has the shared roots of the trace polynomial and poly as roots. gcd = poly_gcd(trace, poly, gf) # If the result has a degree higher than 1, and lower than that of poly, we found a # useful factorization. if len(gcd) != len(poly) and len(gcd) > 1: break # Otherwise, continue with another randv. # Find the actual factors: the monic version of the GCD above, and poly divided by it. factor1 = poly_monic(gcd, gf) factor2, _ = poly_divmod(poly, gcd, gf) # Recurse. return rec_split(factor1, randv) + rec_split(factor2, randv) # Invoke the recursive splitting with a random initial factor, and sort the results. return sorted(rec_split(poly, random.randrange(1, 1 << gf.field_size))) class TestPolyFindRoots(unittest.TestCase): """Test class for poly_find_roots.""" def field_size_test(self, field_size): """Run tests for given field_size.""" gf = GF2Ops(field_size) for test_size in [0, 1, 2, 3, 10]: roots = [random.randrange(1 << field_size) for _ in range(test_size)] roots_set = set(roots) # Construct a polynomial with all elements of roots as roots (with multiplicity). poly = [1] for root in roots: new_poly = [0] + poly for n, c in enumerate(poly): new_poly[n] ^= gf.mul(c, root) poly = new_poly # Invoke the root finding algorithm. found_roots = poly_find_roots(poly, gf) # The result must match the input, unless any roots were repeated. if len(roots) == len(roots_set): self.assertEqual(found_roots, sorted(roots)) else: self.assertEqual(found_roots, []) def test(self): """Run tests.""" for field_size in range(2, 65): self.field_size_test(field_size) def berlekamp_massey(syndromes, gf): """Implement the Berlekamp-Massey algorithm. Takes as input a sequence of GF(2^field_size) elements, and returns the shortest LSFR that generates it, represented as a polynomial. """ # See https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm. current = [1] prev = [1] b_inv = 1 for n, discrepancy in enumerate(syndromes): # Compute discrepancy for i in range(1, len(current)): discrepancy ^= gf.mul(syndromes[n - i], current[i]) # Correct if discrepancy is nonzero. if discrepancy: x = n + 1 - (len(current) - 1) - (len(prev) - 1) if 2 * (len(current) - 1) <= n: tmp = list(current) current.extend(0 for _ in range(len(prev) + x - len(current))) mul = gf.mul(discrepancy, b_inv) for i, v in enumerate(prev): current[i + x] ^= gf.mul(mul, v) prev = tmp b_inv = gf.inv(discrepancy) else: mul = gf.mul(discrepancy, b_inv) for i, v in enumerate(prev): current[i + x] ^= gf.mul(mul, v) return current class Minisketch: """A Minisketch sketch. This represents a sketch of a certain capacity, with elements of a certain bit size. """ def __init__(self, field_size, capacity): """Initialize an empty sketch with the specified field_size size and capacity.""" self.field_size = field_size self.capacity = capacity self.odd_syndromes = [0] * capacity self.gf = GF2Ops(field_size) def add(self, element): """Add an element to this sketch. 1 <= element < 2**field_size.""" sqr = self.gf.sqr(element) for pos in range(self.capacity): self.odd_syndromes[pos] ^= element element = self.gf.mul(sqr, element) def serialized_size(self): """Compute how many bytes a serialization of this sketch will be in size.""" return (self.capacity * self.field_size + 7) // 8 def serialize(self): """Serialize this sketch to bytes.""" val = 0 for i in range(self.capacity): val |= self.odd_syndromes[i] << (self.field_size * i) return val.to_bytes(self.serialized_size(), 'little') def deserialize(self, byte_data): """Deserialize a byte array into this sketch, overwriting its contents.""" assert len(byte_data) == self.serialized_size() val = int.from_bytes(byte_data, 'little') for i in range(self.capacity): self.odd_syndromes[i] = (val >> (self.field_size * i)) & ((1 << self.field_size) - 1) def clone(self): """Return a clone of this sketch.""" ret = Minisketch(self.field_size, self.capacity) ret.odd_syndromes = list(self.odd_syndromes) ret.gf = self.gf return ret def merge(self, other): """Merge a sketch with another sketch. Corresponds to XOR'ing their serializations.""" assert self.capacity == other.capacity assert self.field_size == other.field_size for i in range(self.capacity): self.odd_syndromes[i] ^= other.odd_syndromes[i] def decode(self, max_count=None): """Decode the contents of this sketch. Returns either a list of elements or None if undecodable. """ # We know the odd syndromes s1=x+y+..., s3=x^3+y^3+..., s5=..., and reconstruct the even # syndromes from this: # * s2 = x^2+y^2+.... = (x+y+...)^2 = s1^2 # * s4 = x^4+y^4+.... = (x^2+y^2+...)^2 = s2^2 # * s6 = x^6+y^6+.... = (x^3+y^3+...)^2 = s3^2 all_syndromes = [0 for _ in range(2 * len(self.odd_syndromes))] for i in range(len(self.odd_syndromes)): all_syndromes[i * 2] = self.odd_syndromes[i] all_syndromes[i * 2 + 1] = self.gf.sqr(all_syndromes[i]) # Given the syndromes, find the polynomial that generates them. poly = berlekamp_massey(all_syndromes, self.gf) # Deal with failure and trivial cases. if len(poly) == 0: return None if len(poly) == 1: return [] if max_count is not None and len(poly) > 1 + max_count: return None # If the polynomial can be factored into (1-m1*x)*(1-m2*x)*...*(1-mn*x), then {m1,m2,...,mn} # is our set. As each factor (1-m*x) has 1/m as root, we're really just looking for the # inverses of the roots. We find these by reversing the order of the coefficients, and # finding the roots. roots = poly_find_roots(list(reversed(poly)), self.gf) if len(roots) == 0: return None return roots class TestMinisketch(unittest.TestCase): """Test class for Minisketch.""" @classmethod def construct_data(cls, field_size, num_a_only, num_b_only, num_both): """Construct two random lists of elements in [1..2**field_size-1]. Each list will have unique elements that don't appear in the other (num_a_only in the first and num_b_only in the second), and num_both elements will appear in both.""" sample = [] # Simulate random.sample here (which doesn't work with ranges over 2**63). for _ in range(num_a_only + num_b_only + num_both): while True: r = random.randrange(1, 1 << field_size) if r not in sample: sample.append(r) break full_a = sample[:num_a_only + num_both] full_b = sample[num_a_only:] random.shuffle(full_a) random.shuffle(full_b) return full_a, full_b def field_size_capacity_test(self, field_size, capacity): """Test Minisketch methods for a specific field and capacity.""" used_capacity = random.randrange(capacity + 1) num_a = random.randrange(used_capacity + 1) num_both = random.randrange(min(2 * capacity, (1 << field_size) - 1 - used_capacity) + 1) full_a, full_b = self.construct_data(field_size, num_a, used_capacity - num_a, num_both) sketch_a = Minisketch(field_size, capacity) sketch_b = Minisketch(field_size, capacity) for v in full_a: sketch_a.add(v) for v in full_b: sketch_b.add(v) sketch_combined = sketch_a.clone() sketch_b_ser = sketch_b.serialize() sketch_b_received = Minisketch(field_size, capacity) sketch_b_received.deserialize(sketch_b_ser) sketch_combined.merge(sketch_b_received) decode = sketch_combined.decode() self.assertEqual(decode, sorted(set(full_a) ^ set(full_b))) def test(self): """Run tests.""" for field_size in range(2, 65): for capacity in [0, 1, 2, 5, 10, field_size]: self.field_size_capacity_test(field_size, min(capacity, (1 << field_size) - 1)) if __name__ == '__main__': unittest.main()