def register(self, divisor, isprime=Unknown):
"""
Register a divisor of the number, if the divisor is a true
divisor of the number. The number is divided by the divisor
as many times as possible.
"""
for base, index in self.factors:
if base == divisor:
if isprime and not self.primality[base]:
self.setPrimality(base, isprime)
break
common_divisor = gcd.gcd(base, divisor)
if common_divisor == 1:
continue
# common_divisor > 1:
if common_divisor == divisor:
if isprime:
self.setPrimality(divisor, isprime)
k, coprime = arith1.vp(base, common_divisor)
while not gcd.coprime(common_divisor, coprime):
# try a smaller factor
common_divisor = gcd.gcd(common_divisor, coprime)
k, coprime = arith1.vp(base, common_divisor)
if k:
if coprime > 1:
self.replace(base, [(common_divisor, k), (coprime, 1)])
else:
self.replace(base, [(common_divisor, k)])
else: # common_divisor properly divides divisor.
self.register(common_divisor)
self.register(divisor // common_divisor)
def _lucas_test_sequence(n, a, b):
"""
Return x_0, x_1, x_m, x_{m+1} of Lucas sequence of parameter a, b,
where m = (n - (a**2 - 4*b / n)) // 2.
"""
d = a**2 - 4*b
if (d >= 0 and arith1.floorsqrt(d) ** 2 == d) \
or not(gcd.coprime(n, 2*a*b*d)):
raise ValueError("Choose another parameters.")
x_0 = 2
inv_b = arith1.inverse(b, n)
x_1 = ((a ** 2) * inv_b - 2) % n
# Chain functions
def even_step(u):
"""
'double' u.
"""
return (u**2 - x_0) % n
def odd_step(u, v):
"""
'add' u and v.
"""
return (u*v - x_1) % n
m = (n - arith1.legendre(d, n)) // 2
x_m, x_mplus1 = Lucas_chain(m, even_step, odd_step, x_0, x_1)
return x_0, x_1, x_m, x_mplus1
def _lucas_test_sequence(n, a, b):
"""
Return x_0, x_1, x_m, x_{m+1} of Lucas sequence of parameter a, b,
where m = (n - (a**2 - 4*b / n)) // 2.
"""
d = a**2 - 4 * b
if (d >= 0 and arith1.issquare(d) or not (gcd.coprime(n, 2 * a * b * d))):
raise ValueError("Choose another parameters.")
x_0 = 2
inv_b = arith1.inverse(b, n)
x_1 = ((a**2) * inv_b - 2) % n
# Chain functions
def even_step(u):
"""
'double' u.
"""
return (u**2 - x_0) % n
def odd_step(u, v):
"""
'add' u and v.
"""
return (u * v - x_1) % n
m = (n - arith1.legendre(d, n)) // 2
x_m, x_mplus1 = _lucas_chain(m, even_step, odd_step, x_0, x_1)
return x_0, x_1, x_m, x_mplus1
def register(self, divisor, isprime=Unknown):
"""
Register a divisor of the number, if the divisor is a true
divisor of the number. The number is divided by the divisor
as many times as possible.
"""
for base, index in self.factors:
if base == divisor:
if isprime and not self.primality[base]:
self.setPrimality(base, isprime)
break
common_divisor = gcd.gcd(base, divisor)
if common_divisor == 1:
continue
# common_divisor > 1:
if common_divisor == divisor:
if isprime:
self.setPrimality(divisor, isprime)
k, coprime = arith1.vp(base, common_divisor)
while not gcd.coprime(common_divisor, coprime):
# try a smaller factor
common_divisor = gcd.gcd(common_divisor, coprime)
k, coprime = arith1.vp(base, common_divisor)
if k:
if coprime > 1:
self.replace(base, [(common_divisor, k), (coprime, 1)])
else:
self.replace(base, [(common_divisor, k)])
else: # common_divisor properly divides divisor.
self.register(common_divisor)
self.register(divisor // common_divisor)
def next_probable_prime(n, times=20):
"""
Return the smallest probable prime bigger than the given integer.
n ought to be greater than, say, 1000. There is no penalty to use
prime.nextPrime(n) for n less than 10**12.
This implementation uses miller_rabin to determine the
pseudoprimality, and 'times' parameter can be specified.
"""
n += (1 + (n & 1)) # make n be odd.
while not gcd.coprime(n, 510510) or not miller_rabin(n, times):
n += 2
return n
def embedding(f_q1, f_q2):
"""
Return embedding homomorphism function from f_q1 to f_q2,
where q1 = p ** k1, q2 = p ** k2 and k1 divides k2.
"""
if card(f_q1) == card(f_q2):
return fqiso(f_q1, f_q2)
# search multiplicative generators of both fields and relate them.
# 0. initialize basic variables
p = f_q2.getCharacteristic()
q1, q2 = card(f_q1), card(f_q2)
q1_mult_order, q2_mult_order = q1 - 1, q2 - 1
# 1. find a multiplicative generator of f_q2
for i in bigrange.range(p, q2):
f_q2_gen = f_q2.createElement(i)
if f_q2.order(f_q2_gen) == q2_mult_order:
break
f_q2_subgen = f_q2_gen**((q2 - 1) // (q1 - 1))
# 2.1 minimal polynomial of g = f_q2_gen ** ((q2 - 1) // (q1 - 1))
# minpoly = (X - g)(X - g**p)...(X - g**(p**(k1 - 1)))
q, gpow = 1, f_q2_subgen
linear_factors = []
while q < q2:
linear_factors.append(univar.BasicPolynomial({0: gpow, 1: f_q2.one}))
# finally, raise power (Frobenius map).
q *= p
gpow **= p
minpoly = arith1.product(linear_factors)
# 2.2 minpoly has f_q2 coefficints but they are indeed f_p elements
minpoly = univar.BasicPolynomial([(d, c.rep[0]) for (d, c) in minpoly])
# 3. find a multiplicative generator of f_q1
for i in bigrange.range(p, q1):
f_q1_gen = f_q1.createElement(i)
if f_q1.order(f_q1_gen) == q1_mult_order:
break
# 4. find f_q1_gen ** c of a root of minpoly and let it f_q1_gen
for c in bigrange.range(1, q1):
if gcd.coprime(c, q1_mult_order):
if not minpoly(f_q1_gen**c):
break
f_q1_gen = f_q1_gen**c
# 5. solve DLP:
# x_1 = f_q1_gen ** t
# (Here uses "brute force" method)
x_1 = f_q1.createElement(p)
gpow = f_q1_gen
for i in bigrange.range(1, q):
if gpow == x_1:
image_of_x_1 = f_q2_subgen**i
break
gpow *= f_q1_gen
# finally, define a function
def f_q1_to_f_q2_homo(f_q1_elem):
"""
Return the image of the isomorphism of the given element.
"""
if not f_q1_elem:
return f_q2.zero
if f_q1_elem.rep.degree() == 0:
# F_p elements
return f_q2.createElement(f_q1_elem.rep)
return f_q1_elem.rep(image_of_x_1)
return f_q1_to_f_q2_homo
