Spaces:
Sleeping
Sleeping
| from collections import defaultdict | |
| from sympy.core.containers import Tuple | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import (Dummy, Symbol) | |
| from sympy.functions.combinatorial.numbers import totient | |
| from sympy.ntheory import n_order, is_primitive_root, is_quad_residue, \ | |
| legendre_symbol, jacobi_symbol, primerange, sqrt_mod, \ | |
| primitive_root, quadratic_residues, is_nthpow_residue, nthroot_mod, \ | |
| sqrt_mod_iter, mobius, discrete_log, quadratic_congruence, \ | |
| polynomial_congruence, sieve | |
| from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter, \ | |
| _primitive_root_prime_power_iter, _primitive_root_prime_power2_iter, \ | |
| _nthroot_mod_prime_power, _discrete_log_trial_mul, _discrete_log_shanks_steps, \ | |
| _discrete_log_pollard_rho, _discrete_log_index_calculus, _discrete_log_pohlig_hellman, \ | |
| _binomial_mod_prime_power, binomial_mod | |
| from sympy.polys.domains import ZZ | |
| from sympy.testing.pytest import raises | |
| from sympy.core.random import randint, choice | |
| def test_residue(): | |
| assert n_order(2, 13) == 12 | |
| assert [n_order(a, 7) for a in range(1, 7)] == \ | |
| [1, 3, 6, 3, 6, 2] | |
| assert n_order(5, 17) == 16 | |
| assert n_order(17, 11) == n_order(6, 11) | |
| assert n_order(101, 119) == 6 | |
| assert n_order(11, (10**50 + 151)**2) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650 | |
| raises(ValueError, lambda: n_order(6, 9)) | |
| assert is_primitive_root(2, 7) is False | |
| assert is_primitive_root(3, 8) is False | |
| assert is_primitive_root(11, 14) is False | |
| assert is_primitive_root(12, 17) == is_primitive_root(29, 17) | |
| raises(ValueError, lambda: is_primitive_root(3, 6)) | |
| for p in primerange(3, 100): | |
| li = list(_primitive_root_prime_iter(p)) | |
| assert li[0] == min(li) | |
| for g in li: | |
| assert n_order(g, p) == p - 1 | |
| assert len(li) == totient(totient(p)) | |
| for e in range(1, 4): | |
| li_power = list(_primitive_root_prime_power_iter(p, e)) | |
| li_power2 = list(_primitive_root_prime_power2_iter(p, e)) | |
| assert len(li_power) == len(li_power2) == totient(totient(p**e)) | |
| assert primitive_root(97) == 5 | |
| assert n_order(primitive_root(97, False), 97) == totient(97) | |
| assert primitive_root(97**2) == 5 | |
| assert n_order(primitive_root(97**2, False), 97**2) == totient(97**2) | |
| assert primitive_root(40487) == 5 | |
| assert n_order(primitive_root(40487, False), 40487) == totient(40487) | |
| # note that primitive_root(40487) + 40487 = 40492 is a primitive root | |
| # of 40487**2, but it is not the smallest | |
| assert primitive_root(40487**2) == 10 | |
| assert n_order(primitive_root(40487**2, False), 40487**2) == totient(40487**2) | |
| assert primitive_root(82) == 7 | |
| assert n_order(primitive_root(82, False), 82) == totient(82) | |
| p = 10**50 + 151 | |
| assert primitive_root(p) == 11 | |
| assert n_order(primitive_root(p, False), p) == totient(p) | |
| assert primitive_root(2*p) == 11 | |
| assert n_order(primitive_root(2*p, False), 2*p) == totient(2*p) | |
| assert primitive_root(p**2) == 11 | |
| assert n_order(primitive_root(p**2, False), p**2) == totient(p**2) | |
| assert primitive_root(4 * 11) is None and primitive_root(4 * 11, False) is None | |
| assert primitive_root(15) is None and primitive_root(15, False) is None | |
| raises(ValueError, lambda: primitive_root(-3)) | |
| assert is_quad_residue(3, 7) is False | |
| assert is_quad_residue(10, 13) is True | |
| assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139) | |
| assert is_quad_residue(207, 251) is True | |
| assert is_quad_residue(0, 1) is True | |
| assert is_quad_residue(1, 1) is True | |
| assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True | |
| assert is_quad_residue(1, 4) is True | |
| assert is_quad_residue(2, 27) is False | |
| assert is_quad_residue(13122380800, 13604889600) is True | |
| assert [j for j in range(14) if is_quad_residue(j, 14)] == \ | |
| [0, 1, 2, 4, 7, 8, 9, 11] | |
| raises(ValueError, lambda: is_quad_residue(1.1, 2)) | |
| raises(ValueError, lambda: is_quad_residue(2, 0)) | |
| assert quadratic_residues(S.One) == [0] | |
| assert quadratic_residues(1) == [0] | |
| assert quadratic_residues(12) == [0, 1, 4, 9] | |
| assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12] | |
| assert [len(quadratic_residues(i)) for i in range(1, 20)] == \ | |
| [1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10] | |
| assert list(sqrt_mod_iter(6, 2)) == [0] | |
| assert sqrt_mod(3, 13) == 4 | |
| assert sqrt_mod(3, -13) == 4 | |
| assert sqrt_mod(6, 23) == 11 | |
| assert sqrt_mod(345, 690) == 345 | |
| assert sqrt_mod(67, 101) == None | |
| assert sqrt_mod(1020, 104729) == None | |
| for p in range(3, 100): | |
| d = defaultdict(list) | |
| for i in range(p): | |
| d[pow(i, 2, p)].append(i) | |
| for i in range(1, p): | |
| it = sqrt_mod_iter(i, p) | |
| v = sqrt_mod(i, p, True) | |
| if v: | |
| v = sorted(v) | |
| assert d[i] == v | |
| else: | |
| assert not d[i] | |
| assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24] | |
| assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78] | |
| assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240] | |
| assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72] | |
| assert sqrt_mod(81, 3**5, True) == [9, 18, 36, 45, 63, 72, 90, 99, 117,\ | |
| 126, 144, 153, 171, 180, 198, 207, 225, 234] | |
| assert sqrt_mod(81, 3**6, True) == [9, 72, 90, 153, 171, 234, 252, 315,\ | |
| 333, 396, 414, 477, 495, 558, 576, 639, 657, 720] | |
| assert sqrt_mod(81, 3**7, True) == [9, 234, 252, 477, 495, 720, 738, 963,\ | |
| 981, 1206, 1224, 1449, 1467, 1692, 1710, 1935, 1953, 2178] | |
| for a, p in [(26214400, 32768000000), (26214400, 16384000000), | |
| (262144, 1048576), (87169610025, 163443018796875), | |
| (22315420166400, 167365651248000000)]: | |
| assert pow(sqrt_mod(a, p), 2, p) == a | |
| n = 70 | |
| a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+2) | |
| it = sqrt_mod_iter(a, p) | |
| for i in range(10): | |
| assert pow(next(it), 2, p) == a | |
| a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+3) | |
| it = sqrt_mod_iter(a, p) | |
| for i in range(2): | |
| assert pow(next(it), 2, p) == a | |
| n = 100 | |
| a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+1) | |
| it = sqrt_mod_iter(a, p) | |
| for i in range(2): | |
| assert pow(next(it), 2, p) == a | |
| assert type(next(sqrt_mod_iter(9, 27))) is int | |
| assert type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) | |
| assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) | |
| assert is_nthpow_residue(2, 1, 5) | |
| #issue 10816 | |
| assert is_nthpow_residue(1, 0, 1) is False | |
| assert is_nthpow_residue(1, 0, 2) is True | |
| assert is_nthpow_residue(3, 0, 2) is True | |
| assert is_nthpow_residue(0, 1, 8) is True | |
| assert is_nthpow_residue(2, 3, 2) is True | |
| assert is_nthpow_residue(2, 3, 9) is False | |
| assert is_nthpow_residue(3, 5, 30) is True | |
| assert is_nthpow_residue(21, 11, 20) is True | |
| assert is_nthpow_residue(7, 10, 20) is False | |
| assert is_nthpow_residue(5, 10, 20) is True | |
| assert is_nthpow_residue(3, 10, 48) is False | |
| assert is_nthpow_residue(1, 10, 40) is True | |
| assert is_nthpow_residue(3, 10, 24) is False | |
| assert is_nthpow_residue(1, 10, 24) is True | |
| assert is_nthpow_residue(3, 10, 24) is False | |
| assert is_nthpow_residue(2, 10, 48) is False | |
| assert is_nthpow_residue(81, 3, 972) is False | |
| assert is_nthpow_residue(243, 5, 5103) is True | |
| assert is_nthpow_residue(243, 3, 1240029) is False | |
| assert is_nthpow_residue(36010, 8, 87382) is True | |
| assert is_nthpow_residue(28552, 6, 2218) is True | |
| assert is_nthpow_residue(92712, 9, 50026) is True | |
| x = {pow(i, 56, 1024) for i in range(1024)} | |
| assert {a for a in range(1024) if is_nthpow_residue(a, 56, 1024)} == x | |
| x = { pow(i, 256, 2048) for i in range(2048)} | |
| assert {a for a in range(2048) if is_nthpow_residue(a, 256, 2048)} == x | |
| x = { pow(i, 11, 324000) for i in range(1000)} | |
| assert [ is_nthpow_residue(a, 11, 324000) for a in x] | |
| x = { pow(i, 17, 22217575536) for i in range(1000)} | |
| assert [ is_nthpow_residue(a, 17, 22217575536) for a in x] | |
| assert is_nthpow_residue(676, 3, 5364) | |
| assert is_nthpow_residue(9, 12, 36) | |
| assert is_nthpow_residue(32, 10, 41) | |
| assert is_nthpow_residue(4, 2, 64) | |
| assert is_nthpow_residue(31, 4, 41) | |
| assert not is_nthpow_residue(2, 2, 5) | |
| assert is_nthpow_residue(8547, 12, 10007) | |
| assert is_nthpow_residue(Dummy(even=True) + 3, 3, 2) == True | |
| # _nthroot_mod_prime_power | |
| for p in primerange(2, 10): | |
| for a in range(3): | |
| for n in range(3, 5): | |
| ans = _nthroot_mod_prime_power(a, n, p, 1) | |
| assert isinstance(ans, list) | |
| if len(ans) == 0: | |
| for b in range(p): | |
| assert pow(b, n, p) != a % p | |
| for k in range(2, 10): | |
| assert _nthroot_mod_prime_power(a, n, p, k) == [] | |
| else: | |
| for b in range(p): | |
| pred = pow(b, n, p) == a % p | |
| assert not(pred ^ (b in ans)) | |
| for k in range(2, 10): | |
| ans = _nthroot_mod_prime_power(a, n, p, k) | |
| if not ans: | |
| break | |
| for b in ans: | |
| assert pow(b, n , p**k) == a | |
| assert nthroot_mod(Dummy(odd=True), 3, 2) == 1 | |
| assert nthroot_mod(29, 31, 74) == 45 | |
| assert nthroot_mod(1801, 11, 2663) == 44 | |
| for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663), | |
| (26118163, 1303, 33333347), (1499, 7, 2663), (595, 6, 2663), | |
| (1714, 12, 2663), (28477, 9, 33343)]: | |
| r = nthroot_mod(a, q, p) | |
| assert pow(r, q, p) == a | |
| assert nthroot_mod(11, 3, 109) is None | |
| assert nthroot_mod(16, 5, 36, True) == [4, 22] | |
| assert nthroot_mod(9, 16, 36, True) == [3, 9, 15, 21, 27, 33] | |
| assert nthroot_mod(4, 3, 3249000) is None | |
| assert nthroot_mod(36010, 8, 87382, True) == [40208, 47174] | |
| assert nthroot_mod(0, 12, 37, True) == [0] | |
| assert nthroot_mod(0, 7, 100, True) == [0, 10, 20, 30, 40, 50, 60, 70, 80, 90] | |
| assert nthroot_mod(4, 4, 27, True) == [5, 22] | |
| assert nthroot_mod(4, 4, 121, True) == [19, 102] | |
| assert nthroot_mod(2, 3, 7, True) == [] | |
| for p in range(1, 20): | |
| for a in range(p): | |
| for n in range(1, p): | |
| ans = nthroot_mod(a, n, p, True) | |
| assert isinstance(ans, list) | |
| for b in range(p): | |
| pred = pow(b, n, p) == a | |
| assert not(pred ^ (b in ans)) | |
| ans2 = nthroot_mod(a, n, p, False) | |
| if ans2 is None: | |
| assert ans == [] | |
| else: | |
| assert ans2 in ans | |
| x = Symbol('x', positive=True) | |
| i = Symbol('i', integer=True) | |
| assert _discrete_log_trial_mul(587, 2**7, 2) == 7 | |
| assert _discrete_log_trial_mul(941, 7**18, 7) == 18 | |
| assert _discrete_log_trial_mul(389, 3**81, 3) == 81 | |
| assert _discrete_log_trial_mul(191, 19**123, 19) == 123 | |
| assert _discrete_log_shanks_steps(442879, 7**2, 7) == 2 | |
| assert _discrete_log_shanks_steps(874323, 5**19, 5) == 19 | |
| assert _discrete_log_shanks_steps(6876342, 7**71, 7) == 71 | |
| assert _discrete_log_shanks_steps(2456747, 3**321, 3) == 321 | |
| assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6 | |
| assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19 | |
| assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40 | |
| assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333 | |
| raises(ValueError, lambda: _discrete_log_pollard_rho(11, 7, 31, rseed=0)) | |
| raises(ValueError, lambda: _discrete_log_pollard_rho(227, 3**7, 5, rseed=0)) | |
| assert _discrete_log_index_calculus(983, 948, 2, 491) == 183 | |
| assert _discrete_log_index_calculus(633383, 21794, 2, 316691) == 68048 | |
| assert _discrete_log_index_calculus(941762639, 68822582, 2, 470881319) == 338029275 | |
| assert _discrete_log_index_calculus(999231337607, 888188918786, 2, 499615668803) == 142811376514 | |
| assert _discrete_log_index_calculus(47747730623, 19410045286, 43425105668, 645239603) == 590504662 | |
| assert _discrete_log_pohlig_hellman(98376431, 11**9, 11) == 9 | |
| assert _discrete_log_pohlig_hellman(78723213, 11**31, 11) == 31 | |
| assert _discrete_log_pohlig_hellman(32942478, 11**98, 11) == 98 | |
| assert _discrete_log_pohlig_hellman(14789363, 11**444, 11) == 444 | |
| assert discrete_log(587, 2**9, 2) == 9 | |
| assert discrete_log(2456747, 3**51, 3) == 51 | |
| assert discrete_log(32942478, 11**127, 11) == 127 | |
| assert discrete_log(432751500361, 7**324, 7) == 324 | |
| assert discrete_log(265390227570863,184500076053622, 2) == 17835221372061 | |
| assert discrete_log(22708823198678103974314518195029102158525052496759285596453269189798311427475159776411276642277139650833937, | |
| 17463946429475485293747680247507700244427944625055089103624311227422110546803452417458985046168310373075327, | |
| 123456) == 2068031853682195777930683306640554533145512201725884603914601918777510185469769997054750835368413389728895 | |
| args = 5779, 3528, 6215 | |
| assert discrete_log(*args) == 687 | |
| assert discrete_log(*Tuple(*args)) == 687 | |
| assert quadratic_congruence(400, 85, 125, 1600) == [295, 615, 935, 1255, 1575] | |
| assert quadratic_congruence(3, 6, 5, 25) == [3, 20] | |
| assert quadratic_congruence(120, 80, 175, 500) == [] | |
| assert quadratic_congruence(15, 14, 7, 2) == [1] | |
| assert quadratic_congruence(8, 15, 7, 29) == [10, 28] | |
| assert quadratic_congruence(160, 200, 300, 461) == [144, 431] | |
| assert quadratic_congruence(100000, 123456, 7415263, 48112959837082048697) == [30417843635344493501, 36001135160550533083] | |
| assert quadratic_congruence(65, 121, 72, 277) == [249, 252] | |
| assert quadratic_congruence(5, 10, 14, 2) == [0] | |
| assert quadratic_congruence(10, 17, 19, 2) == [1] | |
| assert quadratic_congruence(10, 14, 20, 2) == [0, 1] | |
| assert polynomial_congruence(6*x**5 + 10*x**4 + 5*x**3 + x**2 + x + 1, | |
| 972000) == [220999, 242999, 463999, 485999, 706999, 728999, 949999, 971999] | |
| assert polynomial_congruence(x**3 - 10*x**2 + 12*x - 82, 33075) == [30287] | |
| assert polynomial_congruence(x**2 + x + 47, 2401) == [785, 1615] | |
| assert polynomial_congruence(10*x**2 + 14*x + 20, 2) == [0, 1] | |
| assert polynomial_congruence(x**3 + 3, 16) == [5] | |
| assert polynomial_congruence(65*x**2 + 121*x + 72, 277) == [249, 252] | |
| assert polynomial_congruence(x**4 - 4, 27) == [5, 22] | |
| assert polynomial_congruence(35*x**3 - 6*x**2 - 567*x + 2308, 148225) == [86957, | |
| 111157, 122531, 146731] | |
| assert polynomial_congruence(x**16 - 9, 36) == [3, 9, 15, 21, 27, 33] | |
| assert polynomial_congruence(x**6 - 2*x**5 - 35, 6125) == [3257] | |
| raises(ValueError, lambda: polynomial_congruence(x**x, 6125)) | |
| raises(ValueError, lambda: polynomial_congruence(x**i, 6125)) | |
| raises(ValueError, lambda: polynomial_congruence(0.1*x**2 + 6, 100)) | |
| assert binomial_mod(-1, 1, 10) == 0 | |
| assert binomial_mod(1, -1, 10) == 0 | |
| raises(ValueError, lambda: binomial_mod(2, 1, -1)) | |
| assert binomial_mod(51, 10, 10) == 0 | |
| assert binomial_mod(10**3, 500, 3**6) == 567 | |
| assert binomial_mod(10**18 - 1, 123456789, 4) == 0 | |
| assert binomial_mod(10**18, 10**12, (10**5 + 3)**2) == 3744312326 | |
| def test_binomial_p_pow(): | |
| n, binomials, binomial = 1000, [1], 1 | |
| for i in range(1, n + 1): | |
| binomial *= n - i + 1 | |
| binomial //= i | |
| binomials.append(binomial) | |
| # Test powers of two, which the algorithm treats slightly differently | |
| trials_2 = 100 | |
| for _ in range(trials_2): | |
| m, power = randint(0, n), randint(1, 20) | |
| assert _binomial_mod_prime_power(n, m, 2, power) == binomials[m] % 2**power | |
| # Test against other prime powers | |
| primes = list(sieve.primerange(2*n)) | |
| trials = 1000 | |
| for _ in range(trials): | |
| m, prime, power = randint(0, n), choice(primes), randint(1, 10) | |
| assert _binomial_mod_prime_power(n, m, prime, power) == binomials[m] % prime**power | |
| def test_deprecated_ntheory_symbolic_functions(): | |
| from sympy.testing.pytest import warns_deprecated_sympy | |
| with warns_deprecated_sympy(): | |
| assert mobius(3) == -1 | |
| with warns_deprecated_sympy(): | |
| assert legendre_symbol(2, 3) == -1 | |
| with warns_deprecated_sympy(): | |
| assert jacobi_symbol(2, 3) == -1 | |