1 #!/usr/bin/env python3
2 # Copyright (c) 2020 Pieter Wuille
3 # Distributed under the MIT software license, see the accompanying
4 # file LICENSE or http://www.opensource.org/licenses/mit-license.php.
5 6 """Native Python (slow) reimplementation of libminisketch' algorithms."""
7 8 import random
9 import unittest
10 11 # Irreducible polynomials over GF(2) to use (represented as integers).
12 #
13 # Most fields can be defined by multiple such polynomials. Minisketch uses the one with the minimal
14 # number of nonzero coefficients, and tie-breaking by picking the lexicographically first among
15 # those.
16 #
17 # All polynomials for degrees 2 through 64 (inclusive) are given.
18 GF2_MODULI = [
19 None, None,
20 2**2 + 2**1 + 1,
21 2**3 + 2**1 + 1,
22 2**4 + 2**1 + 1,
23 2**5 + 2**2 + 1,
24 2**6 + 2**1 + 1,
25 2**7 + 2**1 + 1,
26 2**8 + 2**4 + 2**3 + 2**1 + 1,
27 2**9 + 2**1 + 1,
28 2**10 + 2**3 + 1,
29 2**11 + 2**2 + 1,
30 2**12 + 2**3 + 1,
31 2**13 + 2**4 + 2**3 + 2**1 + 1,
32 2**14 + 2**5 + 1,
33 2**15 + 2**1 + 1,
34 2**16 + 2**5 + 2**3 + 2**1 + 1,
35 2**17 + 2**3 + 1,
36 2**18 + 2**3 + 1,
37 2**19 + 2**5 + 2**2 + 2**1 + 1,
38 2**20 + 2**3 + 1,
39 2**21 + 2**2 + 1,
40 2**22 + 2**1 + 1,
41 2**23 + 2**5 + 1,
42 2**24 + 2**4 + 2**3 + 2**1 + 1,
43 2**25 + 2**3 + 1,
44 2**26 + 2**4 + 2**3 + 2**1 + 1,
45 2**27 + 2**5 + 2**2 + 2**1 + 1,
46 2**28 + 2**1 + 1,
47 2**29 + 2**2 + 1,
48 2**30 + 2**1 + 1,
49 2**31 + 2**3 + 1,
50 2**32 + 2**7 + 2**3 + 2**2 + 1,
51 2**33 + 2**10 + 1,
52 2**34 + 2**7 + 1,
53 2**35 + 2**2 + 1,
54 2**36 + 2**9 + 1,
55 2**37 + 2**6 + 2**4 + 2**1 + 1,
56 2**38 + 2**6 + 2**5 + 2**1 + 1,
57 2**39 + 2**4 + 1,
58 2**40 + 2**5 + 2**4 + 2**3 + 1,
59 2**41 + 2**3 + 1,
60 2**42 + 2**7 + 1,
61 2**43 + 2**6 + 2**4 + 2**3 + 1,
62 2**44 + 2**5 + 1,
63 2**45 + 2**4 + 2**3 + 2**1 + 1,
64 2**46 + 2**1 + 1,
65 2**47 + 2**5 + 1,
66 2**48 + 2**5 + 2**3 + 2**2 + 1,
67 2**49 + 2**9 + 1,
68 2**50 + 2**4 + 2**3 + 2**2 + 1,
69 2**51 + 2**6 + 2**3 + 2**1 + 1,
70 2**52 + 2**3 + 1,
71 2**53 + 2**6 + 2**2 + 2**1 + 1,
72 2**54 + 2**9 + 1,
73 2**55 + 2**7 + 1,
74 2**56 + 2**7 + 2**4 + 2**2 + 1,
75 2**57 + 2**4 + 1,
76 2**58 + 2**19 + 1,
77 2**59 + 2**7 + 2**4 + 2**2 + 1,
78 2**60 + 2**1 + 1,
79 2**61 + 2**5 + 2**2 + 2**1 + 1,
80 2**62 + 2**29 + 1,
81 2**63 + 2**1 + 1,
82 2**64 + 2**4 + 2**3 + 2**1 + 1
83 ]
84 85 class GF2Ops:
86 """Class to perform GF(2^field_size) operations on elements represented as integers.
87 88 Given that elements are represented as integers, addition is simply xor, and not
89 exposed here.
90 """
91 92 def __init__(self, field_size):
93 """Construct a GF2Ops object for the specified field size."""
94 self.field_size = field_size
95 self._modulus = GF2_MODULI[field_size]
96 assert self._modulus is not None
97 98 def mul2(self, x):
99 """Multiply x by 2 in GF(2^field_size)."""
100 x <<= 1
101 if x >> self.field_size:
102 x ^= self._modulus
103 return x
104 105 def mul(self, x, y):
106 """Multiply x by y in GF(2^field_size)."""
107 ret = 0
108 while y:
109 if y & 1:
110 ret ^= x
111 y >>= 1
112 x = self.mul2(x)
113 return ret
114 115 def sqr(self, x):
116 """Square x in GF(2^field_size)."""
117 return self.mul(x, x)
118 119 def inv(self, x):
120 """Compute the inverse of x in GF(2^field_size)."""
121 assert x != 0
122 # Use the extended polynomial Euclidean GCD algorithm on (modulus, x), over GF(2).
123 # See https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor.
124 t1, t2 = 0, 1
125 r1, r2 = self._modulus, x
126 r1l, r2l = self.field_size + 1, r2.bit_length()
127 while r2:
128 q = r1l - r2l
129 r1 ^= r2 << q
130 t1 ^= t2 << q
131 r1l = r1.bit_length()
132 if r1 < r2:
133 t1, t2 = t2, t1
134 r1, r2 = r2, r1
135 r1l, r2l = r2l, r1l
136 assert r1 == 1
137 return t1
138 139 class TestGF2Ops(unittest.TestCase):
140 """Test class for basic arithmetic properties of GF2Ops."""
141 142 def field_size_test(self, field_size):
143 """Test operations for given field_size."""
144 145 gf = GF2Ops(field_size)
146 for i in range(100):
147 x = random.randrange(1 << field_size)
148 y = random.randrange(1 << field_size)
149 x2 = gf.mul2(x)
150 xy = gf.mul(x, y)
151 self.assertEqual(x2, gf.mul(x, 2)) # mul2(x) == x*2
152 self.assertEqual(x2, gf.mul(2, x)) # mul2(x) == 2*x
153 self.assertEqual(xy == 0, x == 0 or y == 0)
154 self.assertEqual(xy == x, y == 1 or x == 0)
155 self.assertEqual(xy == y, x == 1 or y == 0)
156 self.assertEqual(xy, gf.mul(y, x)) # x*y == y*x
157 if i < 10:
158 xp = x
159 for _ in range(field_size):
160 xp = gf.sqr(xp)
161 self.assertEqual(xp, x) # x^(2^field_size) == x
162 if y != 0:
163 yi = gf.inv(y)
164 self.assertEqual(y == yi, y == 1) # y==1/x iff y==1
165 self.assertEqual(gf.mul(y, yi), 1) # y*(1/y) == 1
166 yii = gf.inv(yi)
167 self.assertEqual(y, yii) # 1/(1/y) == y
168 if x != 0:
169 xi = gf.inv(x)
170 xyi = gf.inv(xy)
171 self.assertEqual(xyi, gf.mul(xi, yi)) # (1/x)*(1/y) == 1/(x*y)
172 173 def test(self):
174 """Run tests."""
175 for field_size in range(2, 65):
176 self.field_size_test(field_size)
177 178 # The operations below operate on polynomials over GF(2^field_size), represented as lists of
179 # integers:
180 #
181 # [a, b, c, ...] = a + b*x + c*x^2 + ...
182 #
183 # As an invariant, there are never any trailing zeroes in the list representation.
184 #
185 # Examples:
186 # * [] = 0
187 # * [3] = 3
188 # * [0, 1] = x
189 # * [2, 0, 5] = 5*x^2 + 2
190 191 def poly_monic(poly, gf):
192 """Return a monic version of the polynomial poly."""
193 # Multiply every coefficient with the inverse of the top coefficient.
194 inv = gf.inv(poly[-1])
195 return [gf.mul(inv, v) for v in poly]
196 197 def poly_divmod(poly, mod, gf):
198 """Return the polynomial (quotient, remainder) of poly divided by mod."""
199 assert len(mod) > 0 and mod[-1] == 1 # Require monic mod.
200 if len(poly) < len(mod):
201 return ([], poly)
202 val = list(poly)
203 div = [0 for _ in range(len(val) - len(mod) + 1)]
204 while len(val) >= len(mod):
205 term = val[-1]
206 div[len(val) - len(mod)] = term
207 # If the highest coefficient in val is nonzero, subtract a multiple of mod from it.
208 val.pop()
209 if term != 0:
210 for x in range(len(mod) - 1):
211 val[1 + x - len(mod)] ^= gf.mul(term, mod[x])
212 # Prune trailing zero coefficients.
213 while len(val) > 0 and val[-1] == 0:
214 val.pop()
215 return div, val
216 217 def poly_gcd(a, b, gf):
218 """Return the polynomial GCD of a and b."""
219 if len(a) < len(b):
220 a, b = b, a
221 # Use Euclid's algorithm to find the GCD of a and b.
222 # see https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Euclid's_algorithm.
223 while len(b) > 0:
224 b = poly_monic(b, gf)
225 (_, b), a = poly_divmod(a, b, gf), b
226 return a
227 228 def poly_sqr(poly, gf):
229 """Return the square of polynomial poly."""
230 if len(poly) == 0:
231 return []
232 # In characteristic-2 fields, thanks to Frobenius' endomorphism ((a + b)^2 = a^2 + b^2),
233 # squaring a polynomial is easy: square all the coefficients and interleave with zeroes.
234 # E.g., (3 + 5*x + 17*x^2)^2 = 3^2 + (5*x)^2 + (17*x^2)^2.
235 # See https://en.wikipedia.org/wiki/Frobenius_endomorphism.
236 return [0 if i & 1 else gf.sqr(poly[i // 2]) for i in range(2 * len(poly) - 1)]
237 238 def poly_tracemod(poly, param, gf):
239 """Compute y + y^2 + y^4 + ... + y^(2^(field_size-1)) mod poly, where y = param*x."""
240 out = [0, param]
241 for _ in range(gf.field_size - 1):
242 # In each loop iteration i, we start with out = y + y^2 + ... + y^(2^i). By squaring that we
243 # transform it into out = y^2 + y^4 + ... + y^(2^(i+1)).
244 out = poly_sqr(out, gf)
245 # Thus, we just need to add y again to it to get out = y + ... + y^(2^(i+1)).
246 while len(out) < 2:
247 out.append(0)
248 out[1] = param
249 # Finally take a modulus to keep the intermediary polynomials small.
250 _, out = poly_divmod(out, poly, gf)
251 return out
252 253 def poly_frobeniusmod(poly, gf):
254 """Compute x^(2^field_size) mod poly."""
255 out = [0, 1]
256 for _ in range(gf.field_size):
257 _, out = poly_divmod(poly_sqr(out, gf), poly, gf)
258 return out
259 260 def poly_find_roots(poly, gf):
261 """Find the roots of poly if fully factorizable with unique roots, [] otherwise."""
262 assert len(poly) > 0
263 # If the polynomial is constant (and nonzero), it has no roots.
264 if len(poly) == 1:
265 return []
266 # Make the polynomial monic (which doesn't change its roots).
267 poly = poly_monic(poly, gf)
268 # If the polynomial is of the form x+a, return a.
269 if len(poly) == 2:
270 return [poly[0]]
271 # Otherwise, first test that poly can be completely factored into unique roots. The polynomial
272 # x^(2^fieldsize)-x has every field element once as root. Thus we want to know that that is a
273 # multiple of poly. Compute x^(field_size) mod poly, which needs to equal x if that is the case
274 # (unless poly has degree <= 1, but that case is handled above).
275 if poly_frobeniusmod(poly, gf) != [0, 1]:
276 return []
277 278 def rec_split(poly, randv):
279 """Recursively split poly using the Berlekamp trace algorithm."""
280 # See https://hal.archives-ouvertes.fr/hal-00626997/document.
281 assert len(poly) > 1 and poly[-1] == 1 # Require a monic poly.
282 # If poly is of the form x+a, its root is a.
283 if len(poly) == 2:
284 return [poly[0]]
285 # Try consecutive randomization factors randv, until one is found that factors poly.
286 while True:
287 # Compute the trace of (randv*x) mod poly. This is a polynomial that maps half of the
288 # domain to 0, and the other half to 1. Which half that is is controlled by randv.
289 # By taking it modulo poly, we only add a multiple of poly. Thus the result has at least
290 # the shared roots of the trace polynomial and poly still, but may have others.
291 trace = poly_tracemod(poly, randv, gf)
292 # Using the set {2^i*a for i=0..fieldsize-1} gives optimally independent randv values
293 # (no more than fieldsize are ever needed).
294 randv = gf.mul2(randv)
295 # Now take the GCD of this trace polynomial with poly. The result is a polynomial
296 # that only has the shared roots of the trace polynomial and poly as roots.
297 gcd = poly_gcd(trace, poly, gf)
298 # If the result has a degree higher than 1, and lower than that of poly, we found a
299 # useful factorization.
300 if len(gcd) != len(poly) and len(gcd) > 1:
301 break
302 # Otherwise, continue with another randv.
303 # Find the actual factors: the monic version of the GCD above, and poly divided by it.
304 factor1 = poly_monic(gcd, gf)
305 factor2, _ = poly_divmod(poly, gcd, gf)
306 # Recurse.
307 return rec_split(factor1, randv) + rec_split(factor2, randv)
308 309 # Invoke the recursive splitting with a random initial factor, and sort the results.
310 return sorted(rec_split(poly, random.randrange(1, 1 << gf.field_size)))
311 312 class TestPolyFindRoots(unittest.TestCase):
313 """Test class for poly_find_roots."""
314 315 def field_size_test(self, field_size):
316 """Run tests for given field_size."""
317 gf = GF2Ops(field_size)
318 for test_size in [0, 1, 2, 3, 10]:
319 roots = [random.randrange(1 << field_size) for _ in range(test_size)]
320 roots_set = set(roots)
321 # Construct a polynomial with all elements of roots as roots (with multiplicity).
322 poly = [1]
323 for root in roots:
324 new_poly = [0] + poly
325 for n, c in enumerate(poly):
326 new_poly[n] ^= gf.mul(c, root)
327 poly = new_poly
328 # Invoke the root finding algorithm.
329 found_roots = poly_find_roots(poly, gf)
330 # The result must match the input, unless any roots were repeated.
331 if len(roots) == len(roots_set):
332 self.assertEqual(found_roots, sorted(roots))
333 else:
334 self.assertEqual(found_roots, [])
335 336 def test(self):
337 """Run tests."""
338 for field_size in range(2, 65):
339 self.field_size_test(field_size)
340 341 def berlekamp_massey(syndromes, gf):
342 """Implement the Berlekamp-Massey algorithm.
343 344 Takes as input a sequence of GF(2^field_size) elements, and returns the shortest LSFR
345 that generates it, represented as a polynomial.
346 """
347 # See https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm.
348 current = [1]
349 prev = [1]
350 b_inv = 1
351 for n, discrepancy in enumerate(syndromes):
352 # Compute discrepancy
353 for i in range(1, len(current)):
354 discrepancy ^= gf.mul(syndromes[n - i], current[i])
355 356 # Correct if discrepancy is nonzero.
357 if discrepancy:
358 x = n + 1 - (len(current) - 1) - (len(prev) - 1)
359 if 2 * (len(current) - 1) <= n:
360 tmp = list(current)
361 current.extend(0 for _ in range(len(prev) + x - len(current)))
362 mul = gf.mul(discrepancy, b_inv)
363 for i, v in enumerate(prev):
364 current[i + x] ^= gf.mul(mul, v)
365 prev = tmp
366 b_inv = gf.inv(discrepancy)
367 else:
368 mul = gf.mul(discrepancy, b_inv)
369 for i, v in enumerate(prev):
370 current[i + x] ^= gf.mul(mul, v)
371 return current
372 373 class Minisketch:
374 """A Minisketch sketch.
375 376 This represents a sketch of a certain capacity, with elements of a certain bit size.
377 """
378 379 def __init__(self, field_size, capacity):
380 """Initialize an empty sketch with the specified field_size size and capacity."""
381 self.field_size = field_size
382 self.capacity = capacity
383 self.odd_syndromes = [0] * capacity
384 self.gf = GF2Ops(field_size)
385 386 def add(self, element):
387 """Add an element to this sketch. 1 <= element < 2**field_size."""
388 sqr = self.gf.sqr(element)
389 for pos in range(self.capacity):
390 self.odd_syndromes[pos] ^= element
391 element = self.gf.mul(sqr, element)
392 393 def serialized_size(self):
394 """Compute how many bytes a serialization of this sketch will be in size."""
395 return (self.capacity * self.field_size + 7) // 8
396 397 def serialize(self):
398 """Serialize this sketch to bytes."""
399 val = 0
400 for i in range(self.capacity):
401 val |= self.odd_syndromes[i] << (self.field_size * i)
402 return val.to_bytes(self.serialized_size(), 'little')
403 404 def deserialize(self, byte_data):
405 """Deserialize a byte array into this sketch, overwriting its contents."""
406 assert len(byte_data) == self.serialized_size()
407 val = int.from_bytes(byte_data, 'little')
408 for i in range(self.capacity):
409 self.odd_syndromes[i] = (val >> (self.field_size * i)) & ((1 << self.field_size) - 1)
410 411 def clone(self):
412 """Return a clone of this sketch."""
413 ret = Minisketch(self.field_size, self.capacity)
414 ret.odd_syndromes = list(self.odd_syndromes)
415 ret.gf = self.gf
416 return ret
417 418 def merge(self, other):
419 """Merge a sketch with another sketch. Corresponds to XOR'ing their serializations."""
420 assert self.capacity == other.capacity
421 assert self.field_size == other.field_size
422 for i in range(self.capacity):
423 self.odd_syndromes[i] ^= other.odd_syndromes[i]
424 425 def decode(self, max_count=None):
426 """Decode the contents of this sketch.
427 428 Returns either a list of elements or None if undecodable.
429 """
430 # We know the odd syndromes s1=x+y+..., s3=x^3+y^3+..., s5=..., and reconstruct the even
431 # syndromes from this:
432 # * s2 = x^2+y^2+.... = (x+y+...)^2 = s1^2
433 # * s4 = x^4+y^4+.... = (x^2+y^2+...)^2 = s2^2
434 # * s6 = x^6+y^6+.... = (x^3+y^3+...)^2 = s3^2
435 all_syndromes = [0 for _ in range(2 * len(self.odd_syndromes))]
436 for i in range(len(self.odd_syndromes)):
437 all_syndromes[i * 2] = self.odd_syndromes[i]
438 all_syndromes[i * 2 + 1] = self.gf.sqr(all_syndromes[i])
439 # Given the syndromes, find the polynomial that generates them.
440 poly = berlekamp_massey(all_syndromes, self.gf)
441 # Deal with failure and trivial cases.
442 if len(poly) == 0:
443 return None
444 if len(poly) == 1:
445 return []
446 if max_count is not None and len(poly) > 1 + max_count:
447 return None
448 # If the polynomial can be factored into (1-m1*x)*(1-m2*x)*...*(1-mn*x), then {m1,m2,...,mn}
449 # is our set. As each factor (1-m*x) has 1/m as root, we're really just looking for the
450 # inverses of the roots. We find these by reversing the order of the coefficients, and
451 # finding the roots.
452 roots = poly_find_roots(list(reversed(poly)), self.gf)
453 if len(roots) == 0:
454 return None
455 return roots
456 457 class TestMinisketch(unittest.TestCase):
458 """Test class for Minisketch."""
459 460 @classmethod
461 def construct_data(cls, field_size, num_a_only, num_b_only, num_both):
462 """Construct two random lists of elements in [1..2**field_size-1].
463 464 Each list will have unique elements that don't appear in the other (num_a_only in the first
465 and num_b_only in the second), and num_both elements will appear in both."""
466 sample = []
467 # Simulate random.sample here (which doesn't work with ranges over 2**63).
468 for _ in range(num_a_only + num_b_only + num_both):
469 while True:
470 r = random.randrange(1, 1 << field_size)
471 if r not in sample:
472 sample.append(r)
473 break
474 full_a = sample[:num_a_only + num_both]
475 full_b = sample[num_a_only:]
476 random.shuffle(full_a)
477 random.shuffle(full_b)
478 return full_a, full_b
479 480 def field_size_capacity_test(self, field_size, capacity):
481 """Test Minisketch methods for a specific field and capacity."""
482 used_capacity = random.randrange(capacity + 1)
483 num_a = random.randrange(used_capacity + 1)
484 num_both = random.randrange(min(2 * capacity, (1 << field_size) - 1 - used_capacity) + 1)
485 full_a, full_b = self.construct_data(field_size, num_a, used_capacity - num_a, num_both)
486 sketch_a = Minisketch(field_size, capacity)
487 sketch_b = Minisketch(field_size, capacity)
488 for v in full_a:
489 sketch_a.add(v)
490 for v in full_b:
491 sketch_b.add(v)
492 sketch_combined = sketch_a.clone()
493 sketch_b_ser = sketch_b.serialize()
494 sketch_b_received = Minisketch(field_size, capacity)
495 sketch_b_received.deserialize(sketch_b_ser)
496 sketch_combined.merge(sketch_b_received)
497 decode = sketch_combined.decode()
498 self.assertEqual(decode, sorted(set(full_a) ^ set(full_b)))
499 500 def test(self):
501 """Run tests."""
502 for field_size in range(2, 65):
503 for capacity in [0, 1, 2, 5, 10, field_size]:
504 self.field_size_capacity_test(field_size, min(capacity, (1 << field_size) - 1))
505 506 if __name__ == '__main__':
507 unittest.main()
508