638 lines
17 KiB
Python
638 lines
17 KiB
Python
# sympy.external.ntheory
|
|
#
|
|
# This module provides pure Python implementations of some number theory
|
|
# functions that are alternately used from gmpy2 if it is installed.
|
|
|
|
import sys
|
|
import math
|
|
|
|
import mpmath.libmp as mlib
|
|
|
|
|
|
_small_trailing = [0] * 256
|
|
for j in range(1, 8):
|
|
_small_trailing[1 << j :: 1 << (j + 1)] = [j] * (1 << (7 - j))
|
|
|
|
|
|
def bit_scan1(x, n=0):
|
|
if not x:
|
|
return
|
|
x = abs(x >> n)
|
|
low_byte = x & 0xFF
|
|
if low_byte:
|
|
return _small_trailing[low_byte] + n
|
|
|
|
t = 8 + n
|
|
x >>= 8
|
|
# 2**m is quick for z up through 2**30
|
|
z = x.bit_length() - 1
|
|
if x == 1 << z:
|
|
return z + t
|
|
|
|
if z < 300:
|
|
# fixed 8-byte reduction
|
|
while not x & 0xFF:
|
|
x >>= 8
|
|
t += 8
|
|
else:
|
|
# binary reduction important when there might be a large
|
|
# number of trailing 0s
|
|
p = z >> 1
|
|
while not x & 0xFF:
|
|
while x & ((1 << p) - 1):
|
|
p >>= 1
|
|
x >>= p
|
|
t += p
|
|
return t + _small_trailing[x & 0xFF]
|
|
|
|
|
|
def bit_scan0(x, n=0):
|
|
return bit_scan1(x + (1 << n), n)
|
|
|
|
|
|
def remove(x, f):
|
|
if f < 2:
|
|
raise ValueError("factor must be > 1")
|
|
if x == 0:
|
|
return 0, 0
|
|
if f == 2:
|
|
b = bit_scan1(x)
|
|
return x >> b, b
|
|
m = 0
|
|
y, rem = divmod(x, f)
|
|
while not rem:
|
|
x = y
|
|
m += 1
|
|
if m > 5:
|
|
pow_list = [f**2]
|
|
while pow_list:
|
|
_f = pow_list[-1]
|
|
y, rem = divmod(x, _f)
|
|
if not rem:
|
|
m += 1 << len(pow_list)
|
|
x = y
|
|
pow_list.append(_f**2)
|
|
else:
|
|
pow_list.pop()
|
|
y, rem = divmod(x, f)
|
|
return x, m
|
|
|
|
|
|
def factorial(x):
|
|
"""Return x!."""
|
|
return int(mlib.ifac(int(x)))
|
|
|
|
|
|
def sqrt(x):
|
|
"""Integer square root of x."""
|
|
return int(mlib.isqrt(int(x)))
|
|
|
|
|
|
def sqrtrem(x):
|
|
"""Integer square root of x and remainder."""
|
|
s, r = mlib.sqrtrem(int(x))
|
|
return (int(s), int(r))
|
|
|
|
|
|
if sys.version_info[:2] >= (3, 9):
|
|
# As of Python 3.9 these can take multiple arguments
|
|
gcd = math.gcd
|
|
lcm = math.lcm
|
|
|
|
else:
|
|
# Until python 3.8 is no longer supported
|
|
from functools import reduce
|
|
|
|
|
|
def gcd(*args):
|
|
"""gcd of multiple integers."""
|
|
return reduce(math.gcd, args, 0)
|
|
|
|
|
|
def lcm(*args):
|
|
"""lcm of multiple integers."""
|
|
if 0 in args:
|
|
return 0
|
|
return reduce(lambda x, y: x*y//math.gcd(x, y), args, 1)
|
|
|
|
|
|
def _sign(n):
|
|
if n < 0:
|
|
return -1, -n
|
|
return 1, n
|
|
|
|
|
|
def gcdext(a, b):
|
|
if not a or not b:
|
|
g = abs(a) or abs(b)
|
|
if not g:
|
|
return (0, 0, 0)
|
|
return (g, a // g, b // g)
|
|
|
|
x_sign, a = _sign(a)
|
|
y_sign, b = _sign(b)
|
|
x, r = 1, 0
|
|
y, s = 0, 1
|
|
|
|
while b:
|
|
q, c = divmod(a, b)
|
|
a, b = b, c
|
|
x, r = r, x - q*r
|
|
y, s = s, y - q*s
|
|
|
|
return (a, x * x_sign, y * y_sign)
|
|
|
|
|
|
def is_square(x):
|
|
"""Return True if x is a square number."""
|
|
if x < 0:
|
|
return False
|
|
|
|
# Note that the possible values of y**2 % n for a given n are limited.
|
|
# For example, when n=4, y**2 % n can only take 0 or 1.
|
|
# In other words, if x % 4 is 2 or 3, then x is not a square number.
|
|
# Mathematically, it determines if it belongs to the set {y**2 % n},
|
|
# but implementationally, it can be realized as a logical conjunction
|
|
# with an n-bit integer.
|
|
# see https://mersenneforum.org/showpost.php?p=110896
|
|
# def magic(n):
|
|
# s = {y**2 % n for y in range(n)}
|
|
# s = set(range(n)) - s
|
|
# return sum(1 << bit for bit in s)
|
|
# >>> print(hex(magic(128)))
|
|
# 0xfdfdfdedfdfdfdecfdfdfdedfdfcfdec
|
|
# >>> print(hex(magic(99)))
|
|
# 0x5f6f9ffb6fb7ddfcb75befdec
|
|
# >>> print(hex(magic(91)))
|
|
# 0x6fd1bfcfed5f3679d3ebdec
|
|
# >>> print(hex(magic(85)))
|
|
# 0xdef9ae771ffe3b9d67dec
|
|
if 0xfdfdfdedfdfdfdecfdfdfdedfdfcfdec & (1 << (x & 127)):
|
|
return False # e.g. 2, 3
|
|
m = x % 765765 # 765765 = 99 * 91 * 85
|
|
if 0x5f6f9ffb6fb7ddfcb75befdec & (1 << (m % 99)):
|
|
return False # e.g. 17, 68
|
|
if 0x6fd1bfcfed5f3679d3ebdec & (1 << (m % 91)):
|
|
return False # e.g. 97, 388
|
|
if 0xdef9ae771ffe3b9d67dec & (1 << (m % 85)):
|
|
return False # e.g. 793, 1408
|
|
return mlib.sqrtrem(int(x))[1] == 0
|
|
|
|
|
|
def invert(x, m):
|
|
"""Modular inverse of x modulo m.
|
|
|
|
Returns y such that x*y == 1 mod m.
|
|
|
|
Uses ``math.pow`` but reproduces the behaviour of ``gmpy2.invert``
|
|
which raises ZeroDivisionError if no inverse exists.
|
|
"""
|
|
try:
|
|
return pow(x, -1, m)
|
|
except ValueError:
|
|
raise ZeroDivisionError("invert() no inverse exists")
|
|
|
|
|
|
def legendre(x, y):
|
|
"""Legendre symbol (x / y).
|
|
|
|
Following the implementation of gmpy2,
|
|
the error is raised only when y is an even number.
|
|
"""
|
|
if y <= 0 or not y % 2:
|
|
raise ValueError("y should be an odd prime")
|
|
x %= y
|
|
if not x:
|
|
return 0
|
|
if pow(x, (y - 1) // 2, y) == 1:
|
|
return 1
|
|
return -1
|
|
|
|
|
|
def jacobi(x, y):
|
|
"""Jacobi symbol (x / y)."""
|
|
if y <= 0 or not y % 2:
|
|
raise ValueError("y should be an odd positive integer")
|
|
x %= y
|
|
if not x:
|
|
return int(y == 1)
|
|
if y == 1 or x == 1:
|
|
return 1
|
|
if gcd(x, y) != 1:
|
|
return 0
|
|
j = 1
|
|
while x != 0:
|
|
while x % 2 == 0 and x > 0:
|
|
x >>= 1
|
|
if y % 8 in [3, 5]:
|
|
j = -j
|
|
x, y = y, x
|
|
if x % 4 == y % 4 == 3:
|
|
j = -j
|
|
x %= y
|
|
return j
|
|
|
|
|
|
def kronecker(x, y):
|
|
"""Kronecker symbol (x / y)."""
|
|
if gcd(x, y) != 1:
|
|
return 0
|
|
if y == 0:
|
|
return 1
|
|
sign = -1 if y < 0 and x < 0 else 1
|
|
y = abs(y)
|
|
s = bit_scan1(y)
|
|
y >>= s
|
|
if s % 2 and x % 8 in [3, 5]:
|
|
sign = -sign
|
|
return sign * jacobi(x, y)
|
|
|
|
|
|
def iroot(y, n):
|
|
if y < 0:
|
|
raise ValueError("y must be nonnegative")
|
|
if n < 1:
|
|
raise ValueError("n must be positive")
|
|
if y in (0, 1):
|
|
return y, True
|
|
if n == 1:
|
|
return y, True
|
|
if n == 2:
|
|
x, rem = mlib.sqrtrem(y)
|
|
return int(x), not rem
|
|
if n >= y.bit_length():
|
|
return 1, False
|
|
# Get initial estimate for Newton's method. Care must be taken to
|
|
# avoid overflow
|
|
try:
|
|
guess = int(y**(1./n) + 0.5)
|
|
except OverflowError:
|
|
exp = math.log2(y)/n
|
|
if exp > 53:
|
|
shift = int(exp - 53)
|
|
guess = int(2.0**(exp - shift) + 1) << shift
|
|
else:
|
|
guess = int(2.0**exp)
|
|
if guess > 2**50:
|
|
# Newton iteration
|
|
xprev, x = -1, guess
|
|
while 1:
|
|
t = x**(n - 1)
|
|
xprev, x = x, ((n - 1)*x + y//t)//n
|
|
if abs(x - xprev) < 2:
|
|
break
|
|
else:
|
|
x = guess
|
|
# Compensate
|
|
t = x**n
|
|
while t < y:
|
|
x += 1
|
|
t = x**n
|
|
while t > y:
|
|
x -= 1
|
|
t = x**n
|
|
return x, t == y
|
|
|
|
|
|
def is_fermat_prp(n, a):
|
|
if a < 2:
|
|
raise ValueError("is_fermat_prp() requires 'a' greater than or equal to 2")
|
|
if n < 1:
|
|
raise ValueError("is_fermat_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
a %= n
|
|
if gcd(n, a) != 1:
|
|
raise ValueError("is_fermat_prp() requires gcd(n,a) == 1")
|
|
return pow(a, n - 1, n) == 1
|
|
|
|
|
|
def is_euler_prp(n, a):
|
|
if a < 2:
|
|
raise ValueError("is_euler_prp() requires 'a' greater than or equal to 2")
|
|
if n < 1:
|
|
raise ValueError("is_euler_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
a %= n
|
|
if gcd(n, a) != 1:
|
|
raise ValueError("is_euler_prp() requires gcd(n,a) == 1")
|
|
return pow(a, n >> 1, n) == jacobi(a, n) % n
|
|
|
|
|
|
def _is_strong_prp(n, a):
|
|
s = bit_scan1(n - 1)
|
|
a = pow(a, n >> s, n)
|
|
if a == 1 or a == n - 1:
|
|
return True
|
|
for _ in range(s - 1):
|
|
a = pow(a, 2, n)
|
|
if a == n - 1:
|
|
return True
|
|
if a == 1:
|
|
return False
|
|
return False
|
|
|
|
|
|
def is_strong_prp(n, a):
|
|
if a < 2:
|
|
raise ValueError("is_strong_prp() requires 'a' greater than or equal to 2")
|
|
if n < 1:
|
|
raise ValueError("is_strong_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
a %= n
|
|
if gcd(n, a) != 1:
|
|
raise ValueError("is_strong_prp() requires gcd(n,a) == 1")
|
|
return _is_strong_prp(n, a)
|
|
|
|
|
|
def _lucas_sequence(n, P, Q, k):
|
|
r"""Return the modular Lucas sequence (U_k, V_k, Q_k).
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Given a Lucas sequence defined by P, Q, returns the kth values for
|
|
U and V, along with Q^k, all modulo n. This is intended for use with
|
|
possibly very large values of n and k, where the combinatorial functions
|
|
would be completely unusable.
|
|
|
|
.. math ::
|
|
U_k = \begin{cases}
|
|
0 & \text{if } k = 0\\
|
|
1 & \text{if } k = 1\\
|
|
PU_{k-1} - QU_{k-2} & \text{if } k > 1
|
|
\end{cases}\\
|
|
V_k = \begin{cases}
|
|
2 & \text{if } k = 0\\
|
|
P & \text{if } k = 1\\
|
|
PV_{k-1} - QV_{k-2} & \text{if } k > 1
|
|
\end{cases}
|
|
|
|
The modular Lucas sequences are used in numerous places in number theory,
|
|
especially in the Lucas compositeness tests and the various n + 1 proofs.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
n : int
|
|
n is an odd number greater than or equal to 3
|
|
P : int
|
|
Q : int
|
|
D determined by D = P**2 - 4*Q is non-zero
|
|
k : int
|
|
k is a nonnegative integer
|
|
|
|
Returns
|
|
=======
|
|
|
|
U, V, Qk : (int, int, int)
|
|
`(U_k \bmod{n}, V_k \bmod{n}, Q^k \bmod{n})`
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.external.ntheory import _lucas_sequence
|
|
>>> N = 10**2000 + 4561
|
|
>>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol
|
|
(0, 2, 1)
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://en.wikipedia.org/wiki/Lucas_sequence
|
|
|
|
"""
|
|
if k == 0:
|
|
return (0, 2, 1)
|
|
D = P**2 - 4*Q
|
|
U = 1
|
|
V = P
|
|
Qk = Q % n
|
|
if Q == 1:
|
|
# Optimization for extra strong tests.
|
|
for b in bin(k)[3:]:
|
|
U = (U*V) % n
|
|
V = (V*V - 2) % n
|
|
if b == "1":
|
|
U, V = U*P + V, V*P + U*D
|
|
if U & 1:
|
|
U += n
|
|
if V & 1:
|
|
V += n
|
|
U, V = U >> 1, V >> 1
|
|
elif P == 1 and Q == -1:
|
|
# Small optimization for 50% of Selfridge parameters.
|
|
for b in bin(k)[3:]:
|
|
U = (U*V) % n
|
|
if Qk == 1:
|
|
V = (V*V - 2) % n
|
|
else:
|
|
V = (V*V + 2) % n
|
|
Qk = 1
|
|
if b == "1":
|
|
# new_U = (U + V) // 2
|
|
# new_V = (5*U + V) // 2 = 2*U + new_U
|
|
U, V = U + V, U << 1
|
|
if U & 1:
|
|
U += n
|
|
U >>= 1
|
|
V += U
|
|
Qk = -1
|
|
Qk %= n
|
|
elif P == 1:
|
|
for b in bin(k)[3:]:
|
|
U = (U*V) % n
|
|
V = (V*V - 2*Qk) % n
|
|
Qk *= Qk
|
|
if b == "1":
|
|
# new_U = (U + V) // 2
|
|
# new_V = new_U - 2*Q*U
|
|
U, V = U + V, (Q*U) << 1
|
|
if U & 1:
|
|
U += n
|
|
U >>= 1
|
|
V = U - V
|
|
Qk *= Q
|
|
Qk %= n
|
|
else:
|
|
# The general case with any P and Q.
|
|
for b in bin(k)[3:]:
|
|
U = (U*V) % n
|
|
V = (V*V - 2*Qk) % n
|
|
Qk *= Qk
|
|
if b == "1":
|
|
U, V = U*P + V, V*P + U*D
|
|
if U & 1:
|
|
U += n
|
|
if V & 1:
|
|
V += n
|
|
U, V = U >> 1, V >> 1
|
|
Qk *= Q
|
|
Qk %= n
|
|
return (U % n, V % n, Qk)
|
|
|
|
|
|
def is_fibonacci_prp(n, p, q):
|
|
d = p**2 - 4*q
|
|
if d == 0 or p <= 0 or q not in [1, -1]:
|
|
raise ValueError("invalid values for p,q in is_fibonacci_prp()")
|
|
if n < 1:
|
|
raise ValueError("is_fibonacci_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
return _lucas_sequence(n, p, q, n)[1] == p % n
|
|
|
|
|
|
def is_lucas_prp(n, p, q):
|
|
d = p**2 - 4*q
|
|
if d == 0:
|
|
raise ValueError("invalid values for p,q in is_lucas_prp()")
|
|
if n < 1:
|
|
raise ValueError("is_lucas_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
if gcd(n, q*d) not in [1, n]:
|
|
raise ValueError("is_lucas_prp() requires gcd(n,2*q*D) == 1")
|
|
return _lucas_sequence(n, p, q, n - jacobi(d, n))[0] == 0
|
|
|
|
|
|
def _is_selfridge_prp(n):
|
|
"""Lucas compositeness test with the Selfridge parameters for n.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
The Lucas compositeness test checks whether n is a prime number.
|
|
The test can be run with arbitrary parameters ``P`` and ``Q``, which also change the performance of the test.
|
|
So, which parameters are most effective for running the Lucas compositeness test?
|
|
As an algorithm for determining ``P`` and ``Q``, Selfridge proposed method A [1]_ page 1401
|
|
(Since two methods were proposed, referred to simply as A and B in the paper,
|
|
we will refer to one of them as "method A").
|
|
|
|
method A fixes ``P = 1``. Then, ``D`` defined by ``D = P**2 - 4Q`` is varied from 5, -7, 9, -11, 13, and so on,
|
|
with the first ``D`` being ``jacobi(D, n) == -1``. Once ``D`` is determined,
|
|
``Q`` is determined to be ``(P**2 - D)//4``.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes,
|
|
Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417,
|
|
https://doi.org/10.1090%2FS0025-5718-1980-0583518-6
|
|
http://mpqs.free.fr/LucasPseudoprimes.pdf
|
|
|
|
"""
|
|
for D in range(5, 1_000_000, 2):
|
|
if D & 2: # if D % 4 == 3
|
|
D = -D
|
|
j = jacobi(D, n)
|
|
if j == -1:
|
|
return _lucas_sequence(n, 1, (1-D) // 4, n + 1)[0] == 0
|
|
if j == 0 and D % n:
|
|
return False
|
|
# When j == -1 is hard to find, suspect a square number
|
|
if D == 13 and is_square(n):
|
|
return False
|
|
raise ValueError("appropriate value for D cannot be found in is_selfridge_prp()")
|
|
|
|
|
|
def is_selfridge_prp(n):
|
|
if n < 1:
|
|
raise ValueError("is_selfridge_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
return _is_selfridge_prp(n)
|
|
|
|
|
|
def is_strong_lucas_prp(n, p, q):
|
|
D = p**2 - 4*q
|
|
if D == 0:
|
|
raise ValueError("invalid values for p,q in is_strong_lucas_prp()")
|
|
if n < 1:
|
|
raise ValueError("is_selfridge_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
if gcd(n, q*D) not in [1, n]:
|
|
raise ValueError("is_strong_lucas_prp() requires gcd(n,2*q*D) == 1")
|
|
j = jacobi(D, n)
|
|
s = bit_scan1(n - j)
|
|
U, V, Qk = _lucas_sequence(n, p, q, (n - j) >> s)
|
|
if U == 0 or V == 0:
|
|
return True
|
|
for _ in range(s - 1):
|
|
V = (V*V - 2*Qk) % n
|
|
if V == 0:
|
|
return True
|
|
Qk = pow(Qk, 2, n)
|
|
return False
|
|
|
|
|
|
def _is_strong_selfridge_prp(n):
|
|
for D in range(5, 1_000_000, 2):
|
|
if D & 2: # if D % 4 == 3
|
|
D = -D
|
|
j = jacobi(D, n)
|
|
if j == -1:
|
|
s = bit_scan1(n + 1)
|
|
U, V, Qk = _lucas_sequence(n, 1, (1-D) // 4, (n + 1) >> s)
|
|
if U == 0 or V == 0:
|
|
return True
|
|
for _ in range(s - 1):
|
|
V = (V*V - 2*Qk) % n
|
|
if V == 0:
|
|
return True
|
|
Qk = pow(Qk, 2, n)
|
|
return False
|
|
if j == 0 and D % n:
|
|
return False
|
|
# When j == -1 is hard to find, suspect a square number
|
|
if D == 13 and is_square(n):
|
|
return False
|
|
raise ValueError("appropriate value for D cannot be found in is_strong_selfridge_prp()")
|
|
|
|
|
|
def is_strong_selfridge_prp(n):
|
|
if n < 1:
|
|
raise ValueError("is_strong_selfridge_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
return _is_strong_selfridge_prp(n)
|
|
|
|
|
|
def is_bpsw_prp(n):
|
|
if n < 1:
|
|
raise ValueError("is_bpsw_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
return _is_strong_prp(n, 2) and _is_selfridge_prp(n)
|
|
|
|
|
|
def is_strong_bpsw_prp(n):
|
|
if n < 1:
|
|
raise ValueError("is_strong_bpsw_prp() requires 'n' be greater than 0")
|
|
if n == 1:
|
|
return False
|
|
if n % 2 == 0:
|
|
return n == 2
|
|
return _is_strong_prp(n, 2) and _is_strong_selfridge_prp(n)
|