def main2():
print("MAIN 2:")
print(gauss(Integer(2), Integer(13)))
print(legendre_symbol(Integer(2), Integer(13)))
print(gauss(Integer(2), Integer(31)))
print(legendre_symbol(Integer(2), Integer(31)))
print()
def attack(p, h, c1, c2):
"""
Returns the Legendre symbol of the message encrypted using an unsafe generator.
:param p: the prime used in the ElGamal scheme
:param h: the public key
:param c1: the ciphertext
:param c2: the ciphertext
:return: the Legendre symbol
"""
return int(
legendre_symbol(c2, p) //
max(legendre_symbol(h, p), legendre_symbol(c1, p)))
def split_class(label, p):
"""given the label of one curve E in a p-irreducible isogeny class,
return the list of all labels of curves in that class, partitioned
into 2 parts where those in the first part are m-isogenous to E
with legendre(m,p)=+1 and those in the second part (which may be
empty) are m-isogenous to E with legendre(m,p)=-1.
"""
C = CDB.isogeny_class(label)
isodict = {Elabel(E): E.isogeny_degree(C[0]) for E in C}
set1 = [lab for lab in isodict if legendre_symbol(isodict[lab], 7) == +1]
set2 = [lab for lab in isodict if legendre_symbol(isodict[lab], 7) == -1]
return [set1, set2]
def main4():
print("MAIN 4:")
zeta = CDF(exp(Integer(2) * pi * I / Integer(5)))
print(zeta)
v = [legendre_symbol(n, Integer(5)) * zeta**(Integer(2) * n)
for n in range(Integer(1), Integer(5))]
S = sum([point(tuple(z), pointsize=Integer(100)) for z in v])
G = point(tuple(sum(v)), pointsize=Integer(100), rgbcolor='red')
(S + G).save(filename="gauss_sum.png")
print()
def satisfies_condition_CC_uniform(possible_odd_f, p):
"""Determine whether degrees,p satisfies condition CC.
Args:
K ([NumberField]): the number field
p ([Prime]): the prime p
Returns: boolean
"""
if p % 4 == 1 or p == 2:
return False
for q in prime_range((p / 4) ^ (1 / max(possible_odd_f)) + 1):
if legendre_symbol(q, p) == 1:
if all((q**(2 * f) + q**f + 1) % p != 0 for f in possible_odd_f):
return False
return True
def satisfies_condition_CC(K, p):
"""Determine whether K,p satisfies condition CC.
Args:
K ([NumberField]): the number field
p ([Prime]): the prime p
Returns: boolean
"""
for q in prime_range(p / 4):
if (q ** 2 + q + 1) % p != 0:
if not K.ideal(q).is_prime():
if legendre_symbol(q, p) == 1: # i.e. not inert
return False
return True
def test_unsafe_generator(self):
# Safe prime.
p = 16902648776703029279
# Unsafe generator, generates the entire group.
g = 7
for i in range(100):
x = randint(1, p - 1)
h = pow(g, x, p)
y = randint(1, p - 1)
s = pow(h, y, p)
c1 = pow(g, y, p)
m = randint(1, p - 1)
c2 = m * s % p
k = self.unsafe_generator.attack(p, h, c1, c2)
self.assertIsInstance(k, int)
self.assertEqual(legendre_symbol(m, p), k)
def satisfies_condition_CC(K, p):
"""Determine whether K,p satisfies condition CC.
Args:
K ([NumberField]): the number field
p ([Prime]): the prime p
Returns: boolean
"""
for q in prime_range(p / 4):
for frak_q in K.primes_above(q):
f = frak_q.residue_class_degree()
if f % 2 == 1 and q**f < p / 4:
if (q**(2 * f) + q**f + 1) % p != 0:
if legendre_symbol(q, p) == 1: # i.e. not inert
return False
return True
def jordan_blocks_odd(S, p):
'''
p: odd prime,
S: half integral matrix.
Let S ~ sum p**a_i * b_i * x_i^2 be a jordan decomposition.
Returns the instance of JordanBlocks attached to
a list of (a_i, b_i) sorted so that a_0 >= a_1 >= ...
'''
p = Integer(p)
res = []
u = least_quadratic_nonresidue(p)
for e, ls in groupby(_jordan_decomposition_odd_p(S, p),
key=lambda x: valuation(x, p)):
ls = [x // p ** e for x in ls]
part_res = [(e, ZZ(1)) for _ in ls]
if legendre_symbol(mul(ls), p) == -1:
part_res[-1] = (e, u)
res.extend(part_res)
return JordanBlocks(list(reversed(res)), p)
def find_sqrt(a, p):
assert (p - Integer(1)) % Integer(4) == Integer(0)
assert legendre_symbol(a, p) == Integer(1)
S = PolynomialRing(GF(p), names=('x',))
(x,) = S.gens()
R = S.quotient(x**Integer(2) - a, names=('alpha',))
(alpha,) = R.gens()
while True:
z = GF(p).random_element()
w = (Integer(1) + z * alpha)**((p - Integer(1)) // Integer(2))
(u, v) = (w[Integer(0)], w[Integer(1)])
if v != Integer(0):
break
if (-u / v)**Integer(2) == a:
return -u / v
if ((Integer(1) - u) / v)**Integer(2) == a:
return (Integer(1) - u) / v
if ((-Integer(1) - u) / v)**Integer(2) == a:
return (-Integer(1) - u) / v
def _generate_s(bases, k2, k3):
s = []
for b in bases:
s_b = set()
for p in range(1, 4 * b, 2):
if legendre_symbol(b, p) == -1:
s_b.add(p)
s.append(s_b)
for i in range(len(s)):
mod = 4 * bases[i]
inv2 = inverse_mod(k2, mod)
inv3 = inverse_mod(k3, mod)
s2 = set()
s3 = set()
for z in s[i]:
s2.add(inv2 * (z + k2 - 1) % mod)
s3.add(inv3 * (z + k3 - 1) % mod)
s[i] &= s2 & s3
return s
def t(m):
if p.divides(m):
return 2 + (1 - legendre_symbol(m // p, p)) // 2
else:
return (1 - legendre_symbol(m, p)) // 2
def get_type_1_primes(K, aux_prime_count=3, loop_curves=False):
"""Compute the type 1 primes"""
C_K = K.class_group()
aux_primes = [Q_2]
prime_to_append = Q_2
for _ in range(1, aux_prime_count):
prime_to_append = next_prime(prime_to_append)
aux_primes.append(prime_to_append)
# We need at least one aux prime that splits to have provable termination
my_legendre_symbols = set([legendre_symbol(K._D, p) for p in aux_primes])
if 1 not in my_legendre_symbols:
prime_to_append = next_prime(prime_to_append)
while legendre_symbol(K._D, prime_to_append) != 1:
prime_to_append = next_prime(prime_to_append)
aux_primes.append(prime_to_append)
D_dict = {}
for q in aux_primes:
frak_q = K.primes_above(q)[0]
residue_field = frak_q.residue_field(names="z")
residue_field_card = residue_field.cardinality()
frak_q_class_group_order = C_K(frak_q).multiplicative_order()
exponent = 12 * frak_q_class_group_order
running_D = q
if loop_curves:
weil_polys = get_weil_polys(residue_field)
else:
weil_polys = R.weil_polynomials(2, residue_field_card)
# First add the C((0,0), frak_q) terms
for wp in weil_polys:
D = get_C00(wp, residue_field_card, exponent)
D = Integer(D)
# If q splits then we know C((0,0),frak_q) is non-zero
# so we can insist on only adding D if it is non-zero
if legendre_symbol(K._D, q) == 1:
if D != 0:
running_D = lcm(running_D, D)
else:
# Here we do not provably know that the integer is non-zero
# so add it directly to the running_D
running_D = lcm(running_D, D)
# Put the other term in the lcm before adding to dictionary
running_D = lcm(residue_field_card ** exponent - 1, running_D)
D_dict[q] = running_D
output = gcd(list(D_dict.values()))
output = set(output.prime_divisors())
# Add the bad formal immersion primes
output = output.union(set(prime_range(62)), {71})
# Sort and return
output = list(output)
output.sort()
return output
def gauss_sum(a, p):
K = CyclotomicField(p, names=('zeta',))
(zeta,) = K.gens()
return sum(legendre_symbol(n, p) * zeta**(a * n)
for n in range(Integer(1), p))