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@primus-lab

primus-lab/chebyshev.py

Last active Oct 28, 2019
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Primality test using Chebyshev polynomials of the first kind
# Authors: Peter Taylor and Pedja Terzic
import sys
print(" ***** CHEBYSHEV *****\n\n\n")
while True:
n=int(input("Enter a number : "))
def is_prime_naive(n):
if n <= 2 or n % 2 == 0:
return n == 2
return all(n % i != 0 for i in range(3, int(n ** 0.5 + 1), 2))
def is_pseudoprime_chebyshev(n):
r = smallest_r(n)
if r == 0:
return n == 2
# We work with polynomials represented in little-endian format
# (poly[i] is the coefficient of x^i)
xp, _ = tpair_mod_n_xpowrm1(n, n, r)
return normalise_poly(xp) == [0] * (n % r) + [1]
def smallest_r(n):
if n <= 1 or n % 2 == 0:
return 0
for r in range(5, sys.maxsize**10, 2):
if not is_prime_naive(r):
continue
u = n % r
if u == 0 and r < u:
return 0
if u > 1 and u < r - 1:
return r
def tpair_mod_n_xpowrm1(m, n, r):
"""Returns the tuple (T_m, T_{m+1}) modulo (n, x^r - 1)
where T_i is the ith Chebyshev polynomial"""
if m == 0:
return ([1], [0, 1])
T_k, T_kinc = tpair_mod_n_xpowrm1(m >> 1, n, r)
# T_2kinc = 2 * T_k * T_kinc - x
T_2kinc = [2 * coeff % n
for coeff in convolve_mod_n_xpowrm1(T_k, T_kinc, n, r)]
T_2kinc = poly_add_mod_n(T_2kinc, [0, n - 1], n)
if m & 1:
# m = 2k+1, m+1 = 2k+2
T_m = T_2kinc
# T_minc = 2 * T_kinc * T_kinc - 1
# TODO Optimised polynomial squaring
T_minc = [2 * coeff % n
for coeff in convolve_mod_n_xpowrm1(T_kinc, T_kinc, n, r)]
T_minc = poly_add_mod_n(T_minc, [n - 1], n)
else:
# m = 2k, m+1 = 2k+1
# T_m = 2 * T_k * T_k - 1
T_m = [2 * coeff % n
for coeff in convolve_mod_n_xpowrm1(T_k, T_k, n, r)]
T_m = poly_add_mod_n(T_m, [n - 1], n)
T_minc = T_2kinc
return (T_m, T_minc)
def convolve_mod_n_xpowrm1(p, q, n, r):
""" Calculates the convolution p * q modulo (n, x^r - 1) """
if p == [] or q == []:
return []
conv = [0] * min(len(p) + len(q) - 1, r)
for i, p_i in enumerate(p):
for j, q_j in enumerate(q):
idx = (i + j) % r
conv[idx] = (conv[idx] + p_i * q_j) % n
return conv
def poly_add_mod_n(p, q, n):
r = [0] * max(len(p), len(q))
for i, p_i in enumerate(p):
r[i] = p_i
for j, q_j in enumerate(q):
r[j] = (r[j] + q_j) % n
return r
def normalise_poly(p):
while len(p) > 0 and p[-1] == 0:
p = p[:-1]
return p
if n < 2:
print("Number must be greater than one")
elif is_pseudoprime_chebyshev(n):
print("prime")
else:
print("composite")
try_again = ""
# Loop until users opts to go again or quit
while not(try_again == "1") and not(try_again == "0"):
try_again = input("Press 1 to try again, 0 to exit. ")
if try_again in ["1", "0"]:
continue # a valid entry found
else:
print("Invalid input- Press 1 to try again, 0 to exit.")
# at this point, try_again must be "0" or "1"
if try_again == "0":
break
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