def __init__(self,a,b,f):
assert elliptic_nonsingular(a,b,f), "Elliptic curves must be non-singular"
assert is_prime(f), "f must be prime"
self.a = a
self.b = b
self.f = f
self.identity = Elliptic_Point(float('inf'),float('inf'),self)
def kronecker_symbol(a, n):
"""The Kronecker Symbol"""
if n == 2:
if a % 2 == 0:
return 0
if a % 8 == 1 or a % 8 == 7:
return 1
return -1
elif n == 0:
if a == 1 or a == -1:
return 1
return 0
elif n == -1:
if a < 0:
return -1
return 1
elif n == 1 or a == 1:
return 1
elif is_prime(n):
return legendre_symbol(a, n)
else:
fac = prime_factorization(n)
if n < 0:
out = kronecker_symbol(a, -1)
for f in fac:
out *= kronecker_symbol(a, f)
return out
else:
out = kronecker_symbol(a, 1)
for f in fac:
out *= kronecker_symbol(a, f)
return out
def legendre_symbol(a, p):
"""The Legendre Symbol"""
assert is_prime(p)
assert p != 2
out = pow(a, (p - 1) // 2, p)
if out == 1:
return 1
if out == 0:
return 0
else:
return -1
def egyptian_form_prime(rational):
N = rational.n
D = rational.d
assert N < D
if N == 2 and is_prime(D):
A = Rational(2, D + 1)
B = Rational(2, D * (D + 1))
return [A, B]
else:
return []
def shamir(S, n, k, p):
assert p > S, "p must be strictly greater than S"
assert p > n, "p must be strictly greater than n"
assert is_prime(p), "p must be prime"
A = [S] + [randint(1, p) for i in range(k - 1)]
PA = Polynomial(A, p)
ps = [(i, PA.evaluate(i)) for i in range(1, n + 1)]
return ps
def quad_residue(q,m):
"""Check if q is a quadratic residue modulo m"""
if is_prime(m) and gcd(q,m) == 1:
e = q**((m-1)//2) % m
if e == 1:
return True
return False
else:
for i in range(m):
if i**2 % m == q:
return True
return False