from __future__ import annotations from sympy.core.function import Function from sympy.core.numbers import igcd, igcdex, mod_inverse from sympy.core.power import isqrt from sympy.core.singleton import S from sympy.polys import Poly from sympy.polys.domains import ZZ from sympy.polys.galoistools import gf_crt1, gf_crt2, linear_congruence from .primetest import isprime from .factor_ import factorint, trailing, totient, multiplicity, perfect_power from sympy.utilities.misc import as_int from sympy.core.random import _randint, randint from itertools import cycle, product def n_order(a, n): """Returns the order of ``a`` modulo ``n``. The order of ``a`` modulo ``n`` is the smallest integer ``k`` such that ``a**k`` leaves a remainder of 1 with ``n``. Parameters ========== a : integer n : integer, n > 1. a and n should be relatively prime Examples ======== >>> from sympy.ntheory import n_order >>> n_order(3, 7) 6 >>> n_order(4, 7) 3 """ from collections import defaultdict a, n = as_int(a), as_int(n) if n <= 1: raise ValueError("n should be an integer greater than 1") a = a % n # Trivial if a == 1: return 1 if igcd(a, n) != 1: raise ValueError("The two numbers should be relatively prime") # We want to calculate # order = totient(n), factors = factorint(order) factors = defaultdict(int) for px, kx in factorint(n).items(): if kx > 1: factors[px] += kx - 1 for py, ky in factorint(px - 1).items(): factors[py] += ky order = 1 for px, kx in factors.items(): order *= px**kx # Now the `order` is the order of the group. # The order of `a` divides the order of the group. for p, e in factors.items(): for _ in range(e): if pow(a, order // p, n) == 1: order //= p else: break return order def _primitive_root_prime_iter(p): """ Generates the primitive roots for a prime ``p`` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter >>> list(_primitive_root_prime_iter(19)) [2, 3, 10, 13, 14, 15] References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 """ # it is assumed that p is an int v = [(p - 1) // i for i in factorint(p - 1).keys()] a = 2 while a < p: for pw in v: # a TypeError below may indicate that p was not an int if pow(a, pw, p) == 1: break else: yield a a += 1 def primitive_root(p): """ Returns the smallest primitive root or None. Parameters ========== p : positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import primitive_root >>> primitive_root(19) 2 References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 .. [2] P. Hackman "Elementary Number Theory" (2009), Chapter C """ p = as_int(p) if p < 1: raise ValueError('p is required to be positive') if p <= 2: return 1 f = factorint(p) if len(f) > 2: return None if len(f) == 2: if 2 not in f or f[2] > 1: return None # case p = 2*p1**k, p1 prime for p1, e1 in f.items(): if p1 != 2: break i = 1 while i < p: i += 2 if i % p1 == 0: continue if is_primitive_root(i, p): return i else: if 2 in f: if p == 4: return 3 return None p1, n = list(f.items())[0] if n > 1: # see Ref [2], page 81 g = primitive_root(p1) if is_primitive_root(g, p1**2): return g else: for i in range(2, g + p1 + 1): if igcd(i, p) == 1 and is_primitive_root(i, p): return i return next(_primitive_root_prime_iter(p)) def is_primitive_root(a, p): """ Returns True if ``a`` is a primitive root of ``p``. ``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and totient(p) is the smallest positive number s.t. a**totient(p) cong 1 mod(p) Parameters ========== a : integer p : integer, p > 1. a and p should be relatively prime Examples ======== >>> from sympy.ntheory import is_primitive_root, n_order, totient >>> is_primitive_root(3, 10) True >>> is_primitive_root(9, 10) False >>> n_order(3, 10) == totient(10) True >>> n_order(9, 10) == totient(10) False """ a, p = as_int(a), as_int(p) if p <= 1: raise ValueError("p should be an integer greater than 1") a = a % p if igcd(a, p) != 1: raise ValueError("The two numbers should be relatively prime") # Primitive root of p exist only for # p = 2, 4, q**e, 2*q**e (q is odd prime) if p <= 4: # The primitive root is only p-1. return a == p - 1 t = trailing(p) if t > 1: return False q = p >> t if isprime(q): group_order = q - 1 factors = set(factorint(q - 1).keys()) else: m = perfect_power(q) if not m: return False q, e = m if not isprime(q): return False group_order = q**(e - 1)*(q - 1) factors = set(factorint(q - 1).keys()) factors.add(q) return all(pow(a, group_order // prime, p) != 1 for prime in factors) def _sqrt_mod_tonelli_shanks(a, p): """ Returns the square root in the case of ``p`` prime with ``p == 1 (mod 8)`` References ========== .. [1] R. Crandall and C. Pomerance "Prime Numbers", 2nt Ed., page 101 """ s = trailing(p - 1) t = p >> s # find a non-quadratic residue while 1: d = randint(2, p - 1) r = legendre_symbol(d, p) if r == -1: break #assert legendre_symbol(d, p) == -1 A = pow(a, t, p) D = pow(d, t, p) m = 0 for i in range(s): adm = A*pow(D, m, p) % p adm = pow(adm, 2**(s - 1 - i), p) if adm % p == p - 1: m += 2**i #assert A*pow(D, m, p) % p == 1 x = pow(a, (t + 1)//2, p)*pow(D, m//2, p) % p return x def sqrt_mod(a, p, all_roots=False): """ Find a root of ``x**2 = a mod p``. Parameters ========== a : integer p : positive integer all_roots : if True the list of roots is returned or None Notes ===== If there is no root it is returned None; else the returned root is less or equal to ``p // 2``; in general is not the smallest one. It is returned ``p // 2`` only if it is the only root. Use ``all_roots`` only when it is expected that all the roots fit in memory; otherwise use ``sqrt_mod_iter``. Examples ======== >>> from sympy.ntheory import sqrt_mod >>> sqrt_mod(11, 43) 21 >>> sqrt_mod(17, 32, True) [7, 9, 23, 25] """ if all_roots: return sorted(sqrt_mod_iter(a, p)) try: p = abs(as_int(p)) it = sqrt_mod_iter(a, p) r = next(it) if r > p // 2: return p - r elif r < p // 2: return r else: try: r = next(it) if r > p // 2: return p - r except StopIteration: pass return r except StopIteration: return None def _product(*iters): """ Cartesian product generator Notes ===== Unlike itertools.product, it works also with iterables which do not fit in memory. See https://bugs.python.org/issue10109 Author: Fernando Sumudu with small changes """ inf_iters = tuple(cycle(enumerate(it)) for it in iters) num_iters = len(inf_iters) cur_val = [None]*num_iters first_v = True while True: i, p = 0, num_iters while p and not i: p -= 1 i, cur_val[p] = next(inf_iters[p]) if not p and not i: if first_v: first_v = False else: break yield cur_val def sqrt_mod_iter(a, p, domain=int): """ Iterate over solutions to ``x**2 = a mod p``. Parameters ========== a : integer p : positive integer domain : integer domain, ``int``, ``ZZ`` or ``Integer`` Examples ======== >>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter >>> list(sqrt_mod_iter(11, 43)) [21, 22] """ a, p = as_int(a), abs(as_int(p)) if isprime(p): a = a % p if a == 0: res = _sqrt_mod1(a, p, 1) else: res = _sqrt_mod_prime_power(a, p, 1) if res: if domain is ZZ: yield from res else: for x in res: yield domain(x) else: f = factorint(p) v = [] pv = [] for px, ex in f.items(): if a % px == 0: rx = _sqrt_mod1(a, px, ex) if not rx: return else: rx = _sqrt_mod_prime_power(a, px, ex) if not rx: return v.append(rx) pv.append(px**ex) mm, e, s = gf_crt1(pv, ZZ) if domain is ZZ: for vx in _product(*v): r = gf_crt2(vx, pv, mm, e, s, ZZ) yield r else: for vx in _product(*v): r = gf_crt2(vx, pv, mm, e, s, ZZ) yield domain(r) def _sqrt_mod_prime_power(a, p, k): """ Find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0`` Parameters ========== a : integer p : prime number k : positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power >>> _sqrt_mod_prime_power(11, 43, 1) [21, 22] References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 160 .. [2] http://www.numbertheory.org/php/squareroot.html .. [3] [Gathen99]_ """ pk = p**k a = a % pk if k == 1: if p == 2: return [ZZ(a)] if not (a % p < 2 or pow(a, (p - 1) // 2, p) == 1): return None if p % 4 == 3: res = pow(a, (p + 1) // 4, p) elif p % 8 == 5: sign = pow(a, (p - 1) // 4, p) if sign == 1: res = pow(a, (p + 3) // 8, p) else: b = pow(4*a, (p - 5) // 8, p) x = (2*a*b) % p if pow(x, 2, p) == a: res = x else: res = _sqrt_mod_tonelli_shanks(a, p) # ``_sqrt_mod_tonelli_shanks(a, p)`` is not deterministic; # sort to get always the same result return sorted([ZZ(res), ZZ(p - res)]) if k > 1: # see Ref.[2] if p == 2: if a % 8 != 1: return None if k <= 3: s = set() for i in range(0, pk, 4): s.add(1 + i) s.add(-1 + i) return list(s) # according to Ref.[2] for k > 2 there are two solutions # (mod 2**k-1), that is four solutions (mod 2**k), which can be # obtained from the roots of x**2 = 0 (mod 8) rv = [ZZ(1), ZZ(3), ZZ(5), ZZ(7)] # hensel lift them to solutions of x**2 = 0 (mod 2**k) # if r**2 - a = 0 mod 2**nx but not mod 2**(nx+1) # then r + 2**(nx - 1) is a root mod 2**(nx+1) n = 3 res = [] for r in rv: nx = n while nx < k: r1 = (r**2 - a) >> nx if r1 % 2: r = r + (1 << (nx - 1)) #assert (r**2 - a)% (1 << (nx + 1)) == 0 nx += 1 if r not in res: res.append(r) x = r + (1 << (k - 1)) #assert (x**2 - a) % pk == 0 if x < (1 << nx) and x not in res: if (x**2 - a) % pk == 0: res.append(x) return res rv = _sqrt_mod_prime_power(a, p, 1) if not rv: return None r = rv[0] fr = r**2 - a # hensel lifting with Newton iteration, see Ref.[3] chapter 9 # with f(x) = x**2 - a; one has f'(a) != 0 (mod p) for p != 2 n = 1 px = p while 1: n1 = n n1 *= 2 if n1 > k: break n = n1 px = px**2 frinv = igcdex(2*r, px)[0] r = (r - fr*frinv) % px fr = r**2 - a if n < k: px = p**k frinv = igcdex(2*r, px)[0] r = (r - fr*frinv) % px return [r, px - r] def _sqrt_mod1(a, p, n): """ Find solution to ``x**2 == a mod p**n`` when ``a % p == 0`` see http://www.numbertheory.org/php/squareroot.html """ pn = p**n a = a % pn if a == 0: # case gcd(a, p**k) = p**n m = n // 2 if n % 2 == 1: pm1 = p**(m + 1) def _iter0a(): i = 0 while i < pn: yield i i += pm1 return _iter0a() else: pm = p**m def _iter0b(): i = 0 while i < pn: yield i i += pm return _iter0b() # case gcd(a, p**k) = p**r, r < n f = factorint(a) r = f[p] if r % 2 == 1: return None m = r // 2 a1 = a >> r if p == 2: if n - r == 1: pnm1 = 1 << (n - m + 1) pm1 = 1 << (m + 1) def _iter1(): k = 1 << (m + 2) i = 1 << m while i < pnm1: j = i while j < pn: yield j j += k i += pm1 return _iter1() if n - r == 2: res = _sqrt_mod_prime_power(a1, p, n - r) if res is None: return None pnm = 1 << (n - m) def _iter2(): s = set() for r in res: i = 0 while i < pn: x = (r << m) + i if x not in s: s.add(x) yield x i += pnm return _iter2() if n - r > 2: res = _sqrt_mod_prime_power(a1, p, n - r) if res is None: return None pnm1 = 1 << (n - m - 1) def _iter3(): s = set() for r in res: i = 0 while i < pn: x = ((r << m) + i) % pn if x not in s: s.add(x) yield x i += pnm1 return _iter3() else: m = r // 2 a1 = a // p**r res1 = _sqrt_mod_prime_power(a1, p, n - r) if res1 is None: return None pm = p**m pnr = p**(n-r) pnm = p**(n-m) def _iter4(): s = set() pm = p**m for rx in res1: i = 0 while i < pnm: x = ((rx + i) % pn) if x not in s: s.add(x) yield x*pm i += pnr return _iter4() def is_quad_residue(a, p): """ Returns True if ``a`` (mod ``p``) is in the set of squares mod ``p``, i.e a % p in set([i**2 % p for i in range(p)]). Examples ======== If ``p`` is an odd prime, an iterative method is used to make the determination: >>> from sympy.ntheory import is_quad_residue >>> sorted(set([i**2 % 7 for i in range(7)])) [0, 1, 2, 4] >>> [j for j in range(7) if is_quad_residue(j, 7)] [0, 1, 2, 4] See Also ======== legendre_symbol, jacobi_symbol """ a, p = as_int(a), as_int(p) if p < 1: raise ValueError('p must be > 0') if a >= p or a < 0: a = a % p if a < 2 or p < 3: return True if not isprime(p): if p % 2 and jacobi_symbol(a, p) == -1: return False r = sqrt_mod(a, p) if r is None: return False else: return True return pow(a, (p - 1) // 2, p) == 1 def is_nthpow_residue(a, n, m): """ Returns True if ``x**n == a (mod m)`` has solutions. References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 """ a = a % m a, n, m = as_int(a), as_int(n), as_int(m) if m <= 0: raise ValueError('m must be > 0') if n < 0: raise ValueError('n must be >= 0') if n == 0: if m == 1: return False return a == 1 if a == 0: return True if n == 1: return True if n == 2: return is_quad_residue(a, m) return _is_nthpow_residue_bign(a, n, m) def _is_nthpow_residue_bign(a, n, m): r"""Returns True if `x^n = a \pmod{n}` has solutions for `n > 2`.""" # assert n > 2 # assert a > 0 and m > 0 if primitive_root(m) is None or igcd(a, m) != 1: # assert m >= 8 for prime, power in factorint(m).items(): if not _is_nthpow_residue_bign_prime_power(a, n, prime, power): return False return True f = totient(m) k = int(f // igcd(f, n)) return pow(a, k, int(m)) == 1 def _is_nthpow_residue_bign_prime_power(a, n, p, k): r"""Returns True/False if a solution for `x^n = a \pmod{p^k}` does/does not exist.""" # assert a > 0 # assert n > 2 # assert p is prime # assert k > 0 if a % p: if p != 2: return _is_nthpow_residue_bign(a, n, pow(p, k)) if n & 1: return True c = trailing(n) return a % pow(2, min(c + 2, k)) == 1 else: a %= pow(p, k) if not a: return True mu = multiplicity(p, a) if mu % n: return False pm = pow(p, mu) return _is_nthpow_residue_bign_prime_power(a//pm, n, p, k - mu) def _nthroot_mod2(s, q, p): f = factorint(q) v = [] for b, e in f.items(): v.extend([b]*e) for qx in v: s = _nthroot_mod1(s, qx, p, False) return s def _nthroot_mod1(s, q, p, all_roots): """ Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1`` References ========== .. [1] A. M. Johnston "A Generalized qth Root Algorithm" """ g = primitive_root(p) if not isprime(q): r = _nthroot_mod2(s, q, p) else: f = p - 1 assert (p - 1) % q == 0 # determine k k = 0 while f % q == 0: k += 1 f = f // q # find z, x, r1 f1 = igcdex(-f, q)[0] % q z = f*f1 x = (1 + z) // q r1 = pow(s, x, p) s1 = pow(s, f, p) h = pow(g, f*q, p) t = discrete_log(p, s1, h) g2 = pow(g, z*t, p) g3 = igcdex(g2, p)[0] r = r1*g3 % p #assert pow(r, q, p) == s res = [r] h = pow(g, (p - 1) // q, p) #assert pow(h, q, p) == 1 hx = r for i in range(q - 1): hx = (hx*h) % p res.append(hx) if all_roots: res.sort() return res return min(res) def _help(m, prime_modulo_method, diff_method, expr_val): """ Helper function for _nthroot_mod_composite and polynomial_congruence. Parameters ========== m : positive integer prime_modulo_method : function to calculate the root of the congruence equation for the prime divisors of m diff_method : function to calculate derivative of expression at any given point expr_val : function to calculate value of the expression at any given point """ from sympy.ntheory.modular import crt f = factorint(m) dd = {} for p, e in f.items(): tot_roots = set() if e == 1: tot_roots.update(prime_modulo_method(p)) else: for root in prime_modulo_method(p): diff = diff_method(root, p) if diff != 0: ppow = p m_inv = mod_inverse(diff, p) for j in range(1, e): ppow *= p root = (root - expr_val(root, ppow) * m_inv) % ppow tot_roots.add(root) else: new_base = p roots_in_base = {root} while new_base < pow(p, e): new_base *= p new_roots = set() for k in roots_in_base: if expr_val(k, new_base)!= 0: continue while k not in new_roots: new_roots.add(k) k = (k + (new_base // p)) % new_base roots_in_base = new_roots tot_roots = tot_roots | roots_in_base if tot_roots == set(): return [] dd[pow(p, e)] = tot_roots a = [] m = [] for x, y in dd.items(): m.append(x) a.append(list(y)) return sorted({crt(m, list(i))[0] for i in product(*a)}) def _nthroot_mod_composite(a, n, m): """ Find the solutions to ``x**n = a mod m`` when m is not prime. """ return _help(m, lambda p: nthroot_mod(a, n, p, True), lambda root, p: (pow(root, n - 1, p) * (n % p)) % p, lambda root, p: (pow(root, n, p) - a) % p) def nthroot_mod(a, n, p, all_roots=False): """ Find the solutions to ``x**n = a mod p``. Parameters ========== a : integer n : positive integer p : positive integer all_roots : if False returns the smallest root, else the list of roots Examples ======== >>> from sympy.ntheory.residue_ntheory import nthroot_mod >>> nthroot_mod(11, 4, 19) 8 >>> nthroot_mod(11, 4, 19, True) [8, 11] >>> nthroot_mod(68, 3, 109) 23 """ a = a % p a, n, p = as_int(a), as_int(n), as_int(p) if n == 2: return sqrt_mod(a, p, all_roots) # see Hackman "Elementary Number Theory" (2009), page 76 if not isprime(p): return _nthroot_mod_composite(a, n, p) if a % p == 0: return [0] if not is_nthpow_residue(a, n, p): return [] if all_roots else None if (p - 1) % n == 0: return _nthroot_mod1(a, n, p, all_roots) # The roots of ``x**n - a = 0 (mod p)`` are roots of # ``gcd(x**n - a, x**(p - 1) - 1) = 0 (mod p)`` pa = n pb = p - 1 b = 1 if pa < pb: a, pa, b, pb = b, pb, a, pa while pb: # x**pa - a = 0; x**pb - b = 0 # x**pa - a = x**(q*pb + r) - a = (x**pb)**q * x**r - a = # b**q * x**r - a; x**r - c = 0; c = b**-q * a mod p q, r = divmod(pa, pb) c = pow(b, q, p) c = igcdex(c, p)[0] c = (c * a) % p pa, pb = pb, r a, b = b, c if pa == 1: if all_roots: res = [a] else: res = a elif pa == 2: return sqrt_mod(a, p, all_roots) else: res = _nthroot_mod1(a, pa, p, all_roots) return res def quadratic_residues(p) -> list[int]: """ Returns the list of quadratic residues. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_residues >>> quadratic_residues(7) [0, 1, 2, 4] """ p = as_int(p) r = {pow(i, 2, p) for i in range(p // 2 + 1)} return sorted(r) def legendre_symbol(a, p): r""" Returns the Legendre symbol `(a / p)`. For an integer ``a`` and an odd prime ``p``, the Legendre symbol is defined as .. math :: \genfrac(){}{}{a}{p} = \begin{cases} 0 & \text{if } p \text{ divides } a\\ 1 & \text{if } a \text{ is a quadratic residue modulo } p\\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p \end{cases} Parameters ========== a : integer p : odd prime Examples ======== >>> from sympy.ntheory import legendre_symbol >>> [legendre_symbol(i, 7) for i in range(7)] [0, 1, 1, -1, 1, -1, -1] >>> sorted(set([i**2 % 7 for i in range(7)])) [0, 1, 2, 4] See Also ======== is_quad_residue, jacobi_symbol """ a, p = as_int(a), as_int(p) if not isprime(p) or p == 2: raise ValueError("p should be an odd prime") a = a % p if not a: return 0 if pow(a, (p - 1) // 2, p) == 1: return 1 return -1 def jacobi_symbol(m, n): r""" Returns the Jacobi symbol `(m / n)`. For any integer ``m`` and any positive odd integer ``n`` the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of ``n``: .. math :: \genfrac(){}{}{m}{n} = \genfrac(){}{}{m}{p^{1}}^{\alpha_1} \genfrac(){}{}{m}{p^{2}}^{\alpha_2} ... \genfrac(){}{}{m}{p^{k}}^{\alpha_k} \text{ where } n = p_1^{\alpha_1} p_2^{\alpha_2} ... p_k^{\alpha_k} Like the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = -1` then ``m`` is a quadratic nonresidue modulo ``n``. But, unlike the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = 1` then ``m`` may or may not be a quadratic residue modulo ``n``. Parameters ========== m : integer n : odd positive integer Examples ======== >>> from sympy.ntheory import jacobi_symbol, legendre_symbol >>> from sympy import S >>> jacobi_symbol(45, 77) -1 >>> jacobi_symbol(60, 121) 1 The relationship between the ``jacobi_symbol`` and ``legendre_symbol`` can be demonstrated as follows: >>> L = legendre_symbol >>> S(45).factors() {3: 2, 5: 1} >>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1 True See Also ======== is_quad_residue, legendre_symbol """ m, n = as_int(m), as_int(n) if n < 0 or not n % 2: raise ValueError("n should be an odd positive integer") if m < 0 or m > n: m %= n if not m: return int(n == 1) if n == 1 or m == 1: return 1 if igcd(m, n) != 1: return 0 j = 1 while m != 0: while m % 2 == 0 and m > 0: m >>= 1 if n % 8 in [3, 5]: j = -j m, n = n, m if m % 4 == n % 4 == 3: j = -j m %= n return j class mobius(Function): """ Mobius function maps natural number to {-1, 0, 1} It is defined as follows: 1) `1` if `n = 1`. 2) `0` if `n` has a squared prime factor. 3) `(-1)^k` if `n` is a square-free positive integer with `k` number of prime factors. It is an important multiplicative function in number theory and combinatorics. It has applications in mathematical series, algebraic number theory and also physics (Fermion operator has very concrete realization with Mobius Function model). Parameters ========== n : positive integer Examples ======== >>> from sympy.ntheory import mobius >>> mobius(13*7) 1 >>> mobius(1) 1 >>> mobius(13*7*5) -1 >>> mobius(13**2) 0 References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_function .. [2] Thomas Koshy "Elementary Number Theory with Applications" """ @classmethod def eval(cls, n): if n.is_integer: if n.is_positive is not True: raise ValueError("n should be a positive integer") else: raise TypeError("n should be an integer") if n.is_prime: return S.NegativeOne elif n is S.One: return S.One elif n.is_Integer: a = factorint(n) if any(i > 1 for i in a.values()): return S.Zero return S.NegativeOne**len(a) def _discrete_log_trial_mul(n, a, b, order=None): """ Trial multiplication algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm finds the discrete logarithm using exhaustive search. This naive method is used as fallback algorithm of ``discrete_log`` when the group order is very small. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_trial_mul >>> _discrete_log_trial_mul(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n x = 1 for i in range(order): if x == a: return i x = x * b % n raise ValueError("Log does not exist") def _discrete_log_shanks_steps(n, a, b, order=None): """ Baby-step giant-step algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm is a time-memory trade-off of the method of exhaustive search. It uses `O(sqrt(m))` memory, where `m` is the group order. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps >>> _discrete_log_shanks_steps(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n_order(b, n) m = isqrt(order) + 1 T = {} x = 1 for i in range(m): T[x] = i x = x * b % n z = mod_inverse(b, n) z = pow(z, m, n) x = a for i in range(m): if x in T: return i * m + T[x] x = x * z % n raise ValueError("Log does not exist") def _discrete_log_pollard_rho(n, a, b, order=None, retries=10, rseed=None): """ Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. It is a randomized algorithm with the same expected running time as ``_discrete_log_shanks_steps``, but requires a negligible amount of memory. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho >>> _discrete_log_pollard_rho(227, 3**7, 3) 7 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ a %= n b %= n if order is None: order = n_order(b, n) randint = _randint(rseed) for i in range(retries): aa = randint(1, order - 1) ba = randint(1, order - 1) xa = pow(b, aa, n) * pow(a, ba, n) % n c = xa % 3 if c == 0: xb = a * xa % n ab = aa bb = (ba + 1) % order elif c == 1: xb = xa * xa % n ab = (aa + aa) % order bb = (ba + ba) % order else: xb = b * xa % n ab = (aa + 1) % order bb = ba for j in range(order): c = xa % 3 if c == 0: xa = a * xa % n ba = (ba + 1) % order elif c == 1: xa = xa * xa % n aa = (aa + aa) % order ba = (ba + ba) % order else: xa = b * xa % n aa = (aa + 1) % order c = xb % 3 if c == 0: xb = a * xb % n bb = (bb + 1) % order elif c == 1: xb = xb * xb % n ab = (ab + ab) % order bb = (bb + bb) % order else: xb = b * xb % n ab = (ab + 1) % order c = xb % 3 if c == 0: xb = a * xb % n bb = (bb + 1) % order elif c == 1: xb = xb * xb % n ab = (ab + ab) % order bb = (bb + bb) % order else: xb = b * xb % n ab = (ab + 1) % order if xa == xb: r = (ba - bb) % order try: e = mod_inverse(r, order) * (ab - aa) % order if (pow(b, e, n) - a) % n == 0: return e except ValueError: pass break raise ValueError("Pollard's Rho failed to find logarithm") def _discrete_log_pohlig_hellman(n, a, b, order=None): """ Pohlig-Hellman algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. In order to compute the discrete logarithm, the algorithm takes advantage of the factorization of the group order. It is more efficient when the group order factors into many small primes. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pohlig_hellman >>> _discrete_log_pohlig_hellman(251, 210, 71) 197 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ from .modular import crt a %= n b %= n if order is None: order = n_order(b, n) f = factorint(order) l = [0] * len(f) for i, (pi, ri) in enumerate(f.items()): for j in range(ri): gj = pow(b, l[i], n) aj = pow(a * mod_inverse(gj, n), order // pi**(j + 1), n) bj = pow(b, order // pi, n) cj = discrete_log(n, aj, bj, pi, True) l[i] += cj * pi**j d, _ = crt([pi**ri for pi, ri in f.items()], l) return d def discrete_log(n, a, b, order=None, prime_order=None): """ Compute the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. This is a recursive function to reduce the discrete logarithm problem in cyclic groups of composite order to the problem in cyclic groups of prime order. It employs different algorithms depending on the problem (subgroup order size, prime order or not): * Trial multiplication * Baby-step giant-step * Pollard's Rho * Pohlig-Hellman Examples ======== >>> from sympy.ntheory import discrete_log >>> discrete_log(41, 15, 7) 3 References ========== .. [1] https://mathworld.wolfram.com/DiscreteLogarithm.html .. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). """ n, a, b = as_int(n), as_int(a), as_int(b) if order is None: order = n_order(b, n) if prime_order is None: prime_order = isprime(order) if order < 1000: return _discrete_log_trial_mul(n, a, b, order) elif prime_order: if order < 1000000000000: return _discrete_log_shanks_steps(n, a, b, order) return _discrete_log_pollard_rho(n, a, b, order) return _discrete_log_pohlig_hellman(n, a, b, order) def quadratic_congruence(a, b, c, p): """ Find the solutions to ``a x**2 + b x + c = 0 mod p. Parameters ========== a : int b : int c : int p : int A positive integer. """ a = as_int(a) b = as_int(b) c = as_int(c) p = as_int(p) a = a % p b = b % p c = c % p if a == 0: return linear_congruence(b, -c, p) if p == 2: roots = [] if c % 2 == 0: roots.append(0) if (a + b + c) % 2 == 0: roots.append(1) return roots if isprime(p): inv_a = mod_inverse(a, p) b *= inv_a c *= inv_a if b % 2 == 1: b = b + p d = ((b * b) // 4 - c) % p y = sqrt_mod(d, p, all_roots=True) res = set() for i in y: res.add((i - b // 2) % p) return sorted(res) y = sqrt_mod(b * b - 4 * a * c, 4 * a * p, all_roots=True) res = set() for i in y: root = linear_congruence(2 * a, i - b, 4 * a * p) for j in root: res.add(j % p) return sorted(res) def _polynomial_congruence_prime(coefficients, p): """A helper function used by polynomial_congruence. It returns the root of a polynomial modulo prime number by naive search from [0, p). Parameters ========== coefficients : list of integers p : prime number """ roots = [] rank = len(coefficients) for i in range(0, p): f_val = 0 for coeff in range(0,rank - 1): f_val = (f_val + pow(i, int(rank - coeff - 1), p) * coefficients[coeff]) % p f_val = f_val + coefficients[-1] if f_val % p == 0: roots.append(i) return roots def _diff_poly(root, coefficients, p): """A helper function used by polynomial_congruence. It returns the derivative of the polynomial evaluated at the root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer """ diff = 0 rank = len(coefficients) for coeff in range(0, rank - 1): if not coefficients[coeff]: continue diff = (diff + pow(root, rank - coeff - 2, p)*(rank - coeff - 1)* coefficients[coeff]) % p return diff % p def _val_poly(root, coefficients, p): """A helper function used by polynomial_congruence. It returns value of the polynomial at root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer """ rank = len(coefficients) f_val = 0 for coeff in range(0, rank - 1): f_val = (f_val + pow(root, rank - coeff - 1, p)* coefficients[coeff]) % p f_val = f_val + coefficients[-1] return f_val % p def _valid_expr(expr): """ return coefficients of expr if it is a univariate polynomial with integer coefficients else raise a ValueError. """ if not expr.is_polynomial(): raise ValueError("The expression should be a polynomial") polynomial = Poly(expr) if not polynomial.is_univariate: raise ValueError("The expression should be univariate") if not polynomial.domain == ZZ: raise ValueError("The expression should should have integer coefficients") return polynomial.all_coeffs() def polynomial_congruence(expr, m): """ Find the solutions to a polynomial congruence equation modulo m. Parameters ========== coefficients : Coefficients of the Polynomial m : positive integer Examples ======== >>> from sympy.ntheory import polynomial_congruence >>> from sympy.abc import x >>> expr = x**6 - 2*x**5 -35 >>> polynomial_congruence(expr, 6125) [3257] """ coefficients = _valid_expr(expr) coefficients = [num % m for num in coefficients] rank = len(coefficients) if rank == 3: return quadratic_congruence(*coefficients, m) if rank == 2: return quadratic_congruence(0, *coefficients, m) if coefficients[0] == 1 and 1 + coefficients[-1] == sum(coefficients): return nthroot_mod(-coefficients[-1], rank - 1, m, True) if isprime(m): return _polynomial_congruence_prime(coefficients, m) return _help(m, lambda p: _polynomial_congruence_prime(coefficients, p), lambda root, p: _diff_poly(root, coefficients, p), lambda root, p: _val_poly(root, coefficients, p))