Exemplo n.º 1
0
def tonelli(a,p): # Return x with x^2 == a mod p (-1 if no x)
# Pre: a >= 0, p odd prime
# Post: 0 <= x < p, x*x == a mod p (return -1 if no such x)
	if a % p == 0: return 0
	if legendre(a,p) != 1: return -1
	d = 1
	while d <= 1 or legendre(d,p) != -1: 
		d = int(random.random() * p) % p
	# Represent p-1 == 2^s t
	t = p-1
	s = 0 
	while (t % 2) == 0:
		t //= 2
		s += 1
	A = mod_pow(a, t, p)
	D = mod_pow(d, t, p)
	m = 0
	for i in range(0,s):
		ADm = (A * mod_pow(D, m, p)) % p
		q = ADm
		for j in range(s-i-1): 
			q = (q * q) % p 
		if(q == p-1):
			m += 2**i
	x = (mod_pow(a, (t+1)//2, p) * mod_pow(D, m//2, p)) % p
	return x
Exemplo n.º 2
0
def legendre(a,p): # Return Legendre symbol (a|p): either 0, 1, or -1
# Pre: a >= 0, p odd prime
	l = mod_pow(a % p, (p-1)//2, p)
	return [l, -1][l == p-1]