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Diffstat (limited to 'frozen_deps/ecdsa/numbertheory.py')
| -rw-r--r-- | frozen_deps/ecdsa/numbertheory.py | 810 |
1 files changed, 810 insertions, 0 deletions
diff --git a/frozen_deps/ecdsa/numbertheory.py b/frozen_deps/ecdsa/numbertheory.py new file mode 100644 index 0000000..e5cc888 --- /dev/null +++ b/frozen_deps/ecdsa/numbertheory.py @@ -0,0 +1,810 @@ +#! /usr/bin/env python +# +# Provide some simple capabilities from number theory. +# +# Version of 2008.11.14. +# +# Written in 2005 and 2006 by Peter Pearson and placed in the public domain. +# Revision history: +# 2008.11.14: Use pow(base, exponent, modulus) for modular_exp. +# Make gcd and lcm accept arbitrarly many arguments. + +from __future__ import division + +from six import integer_types, PY2 +from six.moves import reduce + +try: + xrange +except NameError: + xrange = range +try: + from gmpy2 import powmod + + GMPY2 = True + GMPY = False +except ImportError: + GMPY2 = False + try: + from gmpy import mpz + + GMPY = True + except ImportError: + GMPY = False + +import math +import warnings + + +class Error(Exception): + """Base class for exceptions in this module.""" + + pass + + +class SquareRootError(Error): + pass + + +class NegativeExponentError(Error): + pass + + +def modular_exp(base, exponent, modulus): # pragma: no cover + """Raise base to exponent, reducing by modulus""" + # deprecated in 0.14 + warnings.warn( + "Function is unused in library code. If you use this code, " + "change to pow() builtin.", + DeprecationWarning, + ) + if exponent < 0: + raise NegativeExponentError( + "Negative exponents (%d) not allowed" % exponent + ) + return pow(base, exponent, modulus) + + +def polynomial_reduce_mod(poly, polymod, p): + """Reduce poly by polymod, integer arithmetic modulo p. + + Polynomials are represented as lists of coefficients + of increasing powers of x.""" + + # This module has been tested only by extensive use + # in calculating modular square roots. + + # Just to make this easy, require a monic polynomial: + assert polymod[-1] == 1 + + assert len(polymod) > 1 + + while len(poly) >= len(polymod): + if poly[-1] != 0: + for i in xrange(2, len(polymod) + 1): + poly[-i] = (poly[-i] - poly[-1] * polymod[-i]) % p + poly = poly[0:-1] + + return poly + + +def polynomial_multiply_mod(m1, m2, polymod, p): + """Polynomial multiplication modulo a polynomial over ints mod p. + + Polynomials are represented as lists of coefficients + of increasing powers of x.""" + + # This is just a seat-of-the-pants implementation. + + # This module has been tested only by extensive use + # in calculating modular square roots. + + # Initialize the product to zero: + + prod = (len(m1) + len(m2) - 1) * [0] + + # Add together all the cross-terms: + + for i in xrange(len(m1)): + for j in xrange(len(m2)): + prod[i + j] = (prod[i + j] + m1[i] * m2[j]) % p + + return polynomial_reduce_mod(prod, polymod, p) + + +def polynomial_exp_mod(base, exponent, polymod, p): + """Polynomial exponentiation modulo a polynomial over ints mod p. + + Polynomials are represented as lists of coefficients + of increasing powers of x.""" + + # Based on the Handbook of Applied Cryptography, algorithm 2.227. + + # This module has been tested only by extensive use + # in calculating modular square roots. + + assert exponent < p + + if exponent == 0: + return [1] + + G = base + k = exponent + if k % 2 == 1: + s = G + else: + s = [1] + + while k > 1: + k = k // 2 + G = polynomial_multiply_mod(G, G, polymod, p) + if k % 2 == 1: + s = polynomial_multiply_mod(G, s, polymod, p) + + return s + + +def jacobi(a, n): + """Jacobi symbol""" + + # Based on the Handbook of Applied Cryptography (HAC), algorithm 2.149. + + # This function has been tested by comparison with a small + # table printed in HAC, and by extensive use in calculating + # modular square roots. + + assert n >= 3 + assert n % 2 == 1 + a = a % n + if a == 0: + return 0 + if a == 1: + return 1 + a1, e = a, 0 + while a1 % 2 == 0: + a1, e = a1 // 2, e + 1 + if e % 2 == 0 or n % 8 == 1 or n % 8 == 7: + s = 1 + else: + s = -1 + if a1 == 1: + return s + if n % 4 == 3 and a1 % 4 == 3: + s = -s + return s * jacobi(n % a1, a1) + + +def square_root_mod_prime(a, p): + """Modular square root of a, mod p, p prime.""" + + # Based on the Handbook of Applied Cryptography, algorithms 3.34 to 3.39. + + # This module has been tested for all values in [0,p-1] for + # every prime p from 3 to 1229. + + assert 0 <= a < p + assert 1 < p + + if a == 0: + return 0 + if p == 2: + return a + + jac = jacobi(a, p) + if jac == -1: + raise SquareRootError("%d has no square root modulo %d" % (a, p)) + + if p % 4 == 3: + return pow(a, (p + 1) // 4, p) + + if p % 8 == 5: + d = pow(a, (p - 1) // 4, p) + if d == 1: + return pow(a, (p + 3) // 8, p) + if d == p - 1: + return (2 * a * pow(4 * a, (p - 5) // 8, p)) % p + raise RuntimeError("Shouldn't get here.") + + if PY2: + # xrange on python2 can take integers representable as C long only + range_top = min(0x7FFFFFFF, p) + else: + range_top = p + for b in xrange(2, range_top): + if jacobi(b * b - 4 * a, p) == -1: + f = (a, -b, 1) + ff = polynomial_exp_mod((0, 1), (p + 1) // 2, f, p) + assert ff[1] == 0 + return ff[0] + raise RuntimeError("No b found.") + + +if GMPY2: + + def inverse_mod(a, m): + """Inverse of a mod m.""" + if a == 0: + return 0 + return powmod(a, -1, m) + + +elif GMPY: + + def inverse_mod(a, m): + """Inverse of a mod m.""" + # while libgmp likely does support inverses modulo, it is accessible + # only using the native `pow()` function, and `pow()` sanity checks + # the parameters before passing them on to underlying implementation + # on Python2 + if a == 0: + return 0 + a = mpz(a) + m = mpz(m) + + lm, hm = mpz(1), mpz(0) + low, high = a % m, m + while low > 1: + r = high // low + lm, low, hm, high = hm - lm * r, high - low * r, lm, low + + return lm % m + + +else: + + def inverse_mod(a, m): + """Inverse of a mod m.""" + + if a == 0: + return 0 + + lm, hm = 1, 0 + low, high = a % m, m + while low > 1: + r = high // low + lm, low, hm, high = hm - lm * r, high - low * r, lm, low + + return lm % m + + +try: + gcd2 = math.gcd +except AttributeError: + + def gcd2(a, b): + """Greatest common divisor using Euclid's algorithm.""" + while a: + a, b = b % a, a + return b + + +def gcd(*a): + """Greatest common divisor. + + Usage: gcd([ 2, 4, 6 ]) + or: gcd(2, 4, 6) + """ + + if len(a) > 1: + return reduce(gcd2, a) + if hasattr(a[0], "__iter__"): + return reduce(gcd2, a[0]) + return a[0] + + +def lcm2(a, b): + """Least common multiple of two integers.""" + + return (a * b) // gcd(a, b) + + +def lcm(*a): + """Least common multiple. + + Usage: lcm([ 3, 4, 5 ]) + or: lcm(3, 4, 5) + """ + + if len(a) > 1: + return reduce(lcm2, a) + if hasattr(a[0], "__iter__"): + return reduce(lcm2, a[0]) + return a[0] + + +def factorization(n): + """Decompose n into a list of (prime,exponent) pairs.""" + + assert isinstance(n, integer_types) + + if n < 2: + return [] + + result = [] + d = 2 + + # Test the small primes: + + for d in smallprimes: + if d > n: + break + q, r = divmod(n, d) + if r == 0: + count = 1 + while d <= n: + n = q + q, r = divmod(n, d) + if r != 0: + break + count = count + 1 + result.append((d, count)) + + # If n is still greater than the last of our small primes, + # it may require further work: + + if n > smallprimes[-1]: + if is_prime(n): # If what's left is prime, it's easy: + result.append((n, 1)) + else: # Ugh. Search stupidly for a divisor: + d = smallprimes[-1] + while 1: + d = d + 2 # Try the next divisor. + q, r = divmod(n, d) + if q < d: # n < d*d means we're done, n = 1 or prime. + break + if r == 0: # d divides n. How many times? + count = 1 + n = q + while d <= n: # As long as d might still divide n, + q, r = divmod(n, d) # see if it does. + if r != 0: + break + n = q # It does. Reduce n, increase count. + count = count + 1 + result.append((d, count)) + if n > 1: + result.append((n, 1)) + + return result + + +def phi(n): # pragma: no cover + """Return the Euler totient function of n.""" + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + assert isinstance(n, integer_types) + + if n < 3: + return 1 + + result = 1 + ff = factorization(n) + for f in ff: + e = f[1] + if e > 1: + result = result * f[0] ** (e - 1) * (f[0] - 1) + else: + result = result * (f[0] - 1) + return result + + +def carmichael(n): # pragma: no cover + """Return Carmichael function of n. + + Carmichael(n) is the smallest integer x such that + m**x = 1 mod n for all m relatively prime to n. + """ + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + return carmichael_of_factorized(factorization(n)) + + +def carmichael_of_factorized(f_list): # pragma: no cover + """Return the Carmichael function of a number that is + represented as a list of (prime,exponent) pairs. + """ + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + if len(f_list) < 1: + return 1 + + result = carmichael_of_ppower(f_list[0]) + for i in xrange(1, len(f_list)): + result = lcm(result, carmichael_of_ppower(f_list[i])) + + return result + + +def carmichael_of_ppower(pp): # pragma: no cover + """Carmichael function of the given power of the given prime.""" + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + p, a = pp + if p == 2 and a > 2: + return 2 ** (a - 2) + else: + return (p - 1) * p ** (a - 1) + + +def order_mod(x, m): # pragma: no cover + """Return the order of x in the multiplicative group mod m.""" + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + # Warning: this implementation is not very clever, and will + # take a long time if m is very large. + + if m <= 1: + return 0 + + assert gcd(x, m) == 1 + + z = x + result = 1 + while z != 1: + z = (z * x) % m + result = result + 1 + return result + + +def largest_factor_relatively_prime(a, b): # pragma: no cover + """Return the largest factor of a relatively prime to b.""" + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + while 1: + d = gcd(a, b) + if d <= 1: + break + b = d + while 1: + q, r = divmod(a, d) + if r > 0: + break + a = q + return a + + +def kinda_order_mod(x, m): # pragma: no cover + """Return the order of x in the multiplicative group mod m', + where m' is the largest factor of m relatively prime to x. + """ + # deprecated in 0.14 + warnings.warn( + "Function is unused by library code. If you use this code, " + "please open an issue in " + "https://github.com/warner/python-ecdsa", + DeprecationWarning, + ) + + return order_mod(x, largest_factor_relatively_prime(m, x)) + + +def is_prime(n): + """Return True if x is prime, False otherwise. + + We use the Miller-Rabin test, as given in Menezes et al. p. 138. + This test is not exact: there are composite values n for which + it returns True. + + In testing the odd numbers from 10000001 to 19999999, + about 66 composites got past the first test, + 5 got past the second test, and none got past the third. + Since factors of 2, 3, 5, 7, and 11 were detected during + preliminary screening, the number of numbers tested by + Miller-Rabin was (19999999 - 10000001)*(2/3)*(4/5)*(6/7) + = 4.57 million. + """ + + # (This is used to study the risk of false positives:) + global miller_rabin_test_count + + miller_rabin_test_count = 0 + + if n <= smallprimes[-1]: + if n in smallprimes: + return True + else: + return False + + if gcd(n, 2 * 3 * 5 * 7 * 11) != 1: + return False + + # Choose a number of iterations sufficient to reduce the + # probability of accepting a composite below 2**-80 + # (from Menezes et al. Table 4.4): + + t = 40 + n_bits = 1 + int(math.log(n, 2)) + for k, tt in ( + (100, 27), + (150, 18), + (200, 15), + (250, 12), + (300, 9), + (350, 8), + (400, 7), + (450, 6), + (550, 5), + (650, 4), + (850, 3), + (1300, 2), + ): + if n_bits < k: + break + t = tt + + # Run the test t times: + + s = 0 + r = n - 1 + while (r % 2) == 0: + s = s + 1 + r = r // 2 + for i in xrange(t): + a = smallprimes[i] + y = pow(a, r, n) + if y != 1 and y != n - 1: + j = 1 + while j <= s - 1 and y != n - 1: + y = pow(y, 2, n) + if y == 1: + miller_rabin_test_count = i + 1 + return False + j = j + 1 + if y != n - 1: + miller_rabin_test_count = i + 1 + return False + return True + + +def next_prime(starting_value): + """Return the smallest prime larger than the starting value.""" + + if starting_value < 2: + return 2 + result = (starting_value + 1) | 1 + while not is_prime(result): + result = result + 2 + return result + + +smallprimes = [ + 2, + 3, + 5, + 7, + 11, + 13, + 17, + 19, + 23, + 29, + 31, + 37, + 41, + 43, + 47, + 53, + 59, + 61, + 67, + 71, + 73, + 79, + 83, + 89, + 97, + 101, + 103, + 107, + 109, + 113, + 127, + 131, + 137, + 139, + 149, + 151, + 157, + 163, + 167, + 173, + 179, + 181, + 191, + 193, + 197, + 199, + 211, + 223, + 227, + 229, + 233, + 239, + 241, + 251, + 257, + 263, + 269, + 271, + 277, + 281, + 283, + 293, + 307, + 311, + 313, + 317, + 331, + 337, + 347, + 349, + 353, + 359, + 367, + 373, + 379, + 383, + 389, + 397, + 401, + 409, + 419, + 421, + 431, + 433, + 439, + 443, + 449, + 457, + 461, + 463, + 467, + 479, + 487, + 491, + 499, + 503, + 509, + 521, + 523, + 541, + 547, + 557, + 563, + 569, + 571, + 577, + 587, + 593, + 599, + 601, + 607, + 613, + 617, + 619, + 631, + 641, + 643, + 647, + 653, + 659, + 661, + 673, + 677, + 683, + 691, + 701, + 709, + 719, + 727, + 733, + 739, + 743, + 751, + 757, + 761, + 769, + 773, + 787, + 797, + 809, + 811, + 821, + 823, + 827, + 829, + 839, + 853, + 857, + 859, + 863, + 877, + 881, + 883, + 887, + 907, + 911, + 919, + 929, + 937, + 941, + 947, + 953, + 967, + 971, + 977, + 983, + 991, + 997, + 1009, + 1013, + 1019, + 1021, + 1031, + 1033, + 1039, + 1049, + 1051, + 1061, + 1063, + 1069, + 1087, + 1091, + 1093, + 1097, + 1103, + 1109, + 1117, + 1123, + 1129, + 1151, + 1153, + 1163, + 1171, + 1181, + 1187, + 1193, + 1201, + 1213, + 1217, + 1223, + 1229, +] + +miller_rabin_test_count = 0 |