def te_s_t(a, b):
g, s, t = gmpy.gcdext(a, b)
if abs(a) == abs(b):
test_s = s == 0
elif b == 0 or abs(b) == 2 * g:
test_s = s == bool(a > 0) - bool(a < 0) # sign of a
else:
test_s = abs(s) < abs(b) / (2 * g)
if abs(a) == abs(b) or a == 0 or abs(a) == 2 * g:
test_t = t == bool(b > 0) - bool(b < 0) # sign of b
else:
test_t = abs(t) < abs(a) / (2 * g)
return g == a * s + b * t and test_s and test_t
def operation2(cls, f, /): # Cohen: Algorithm 5.4.8 (NUDUPL)
a, b, c = f
d1, u, _ = gcdext(b, a)
assert d1 == 1 # because -discriminant is prime
A = a // d1
B = b // d1
C = (-c * u) % A
C1 = A - C
if C1 < C:
C = -C1
d, v3 = A, C
v2, v = 1, 0
z = 0
L = iroot(-cls.discriminant // 4, 4)[0]
while abs(v3) > L: # partial Euclid
d, (q, v3) = v3, divmod(d, v3)
v, v2, = v2, v - q * v2
z += 1
if z % 2:
v2, v3 = -v2, -v3
if z == 0:
g = (B * v3 + c) // d
a2 = d**2
b2 = b + 2 * d * v3
c2 = v3**2 + g * d1
else:
e = (c * v + B * d) // A
h = e * v2
g = (h - B) // v
a2 = d**2 + d1 * e * v
b2 = d1 * (h + v * g) + 2 * d * v3
c2 = v3**2 + d1 * g * v2
f2 = int(a2), int(b2), int(
c2) # NB: convert from gmpy2.mpz, if gmpy2 is used for gcdext()
return cls(cls._reduce(f2), check=False)
def operation(cls, f1, f2): # Cohen: Algorithm 5.4.9 (NUCOMP)
if f1[0] < f2[0]:
f1, f2 = f2, f1
a1, b1, c1 = f1
a2, b2, c2 = f2
s = (b1 + b2) // 2
n = b2 - s
d, u, v = gcdext(a2, a1)
if d == 1:
A = -u * n
d1 = d
elif s % d == 0:
A = -u * n
d1 = d
a1 //= d1
a2 //= d1
s //= d1
else:
d1, u1, _ = gcdext(s, d)
if d1 > 1:
a1 //= d1
a2 //= d1
s //= d1
d //= d1
l = (-u1 * (u * (c1 % d) + v * (c2 % d))) % d
A = -u * (n // d) + l * (a1 // d)
A = A % a1
A1 = a1 - A
if A1 < A:
A = - A1
d, v3 = a1, A
v2, v = 1, 0
z = 0
L = iroot(-cls.discriminant//4, 4)[0]
while abs(v3) > L: # partial Euclid
d, (q, v3) = v3, divmod(d, v3)
v, v2, = v2, v - q * v2
z += 1
if z%2:
v2, v3 = -v2, -v3
if z == 0:
Q1 = a2 * v3
f = (Q1 + n) // d
g = (v3 * s + c2) // d
a3 = d * a2
b3 = 2*Q1 + b2
c3 = v3 * f + g * d1 # erratum Cohen (step 6)
else:
b = (a2 * d + n * v) // a1
Q1 = b * v3
Q2 = Q1 + n
f = Q2 // d
e = (s * d + c2 * v) // a1
Q3 = e * v2
Q4 = Q3 - s
g = Q4 // v
a3 = d * b + d1 * e * v
b3 = Q1 + Q2 + d1 * (Q3 + Q4)
c3 = v3 * f + d1 * g * v2
f3 = int(a3), int(b3), int(c3) # NB: convert from gmpy2.mpz, if gmpy2 is used for gcdext()
return cls(cls._reduce(f3), check=False)
def test_basic(self):
self.assertFalse(gmpy.is_prime(1))
self.assertTrue(gmpy.is_prime(2))
self.assertTrue(gmpy.is_prime(101))
self.assertFalse(gmpy.is_prime(561))
self.assertTrue(gmpy.is_prime(2**16 + 1))
self.assertFalse(gmpy.is_prime(41041))
self.assertEqual(gmpy.next_prime(1), 2)
self.assertEqual(gmpy.next_prime(2), 3)
self.assertEqual(gmpy.next_prime(256), 257)
self.assertEqual(gmpy.powmod(3, 256, 257), 1)
self.assertEqual(gmpy.gcdext(3, 257), (1, 86, -1))
self.assertEqual(gmpy.gcdext(1234, 257), (1, -126, 605))
self.assertEqual(gmpy.gcdext(-1234 * 3, -257 * 3), (3, 126, -605))
def te_s_t(a, b):
g, s, t = gmpy.gcdext(a, b)
if abs(a) == abs(b):
test_s = s == 0
elif b == 0 or abs(b) == 2 * g:
test_s = s == bool(a > 0) - bool(a < 0) # sign of a
else:
test_s = abs(s) < abs(b) / (2 * g)
if abs(a) == abs(b) or a == 0 or abs(a) == 2 * g:
test_t = t == bool(b > 0) - bool(b < 0) # sign of b
else:
test_t = abs(t) < abs(a) / (2 * g)
return g == a * s + b * t and test_s and test_t
self.assertTrue(
all((te_s_t(0, 0), te_s_t(0, -1), te_s_t(1, 0), te_s_t(-1, 1))))
self.assertTrue(te_s_t(-1234, 257))
self.assertTrue(te_s_t(-12537, -257))
self.assertTrue(te_s_t(-11 * 1234, -11 * 2567))
self.assertTrue(te_s_t(1234, -2 * 1234))
self.assertTrue(te_s_t(-2 * 12364, 12364))
# self.assertEqual(gmpy.invert(3, -1), 0) # pending gmpy2 issue if modulus is 1 or -1
self.assertEqual(gmpy.invert(3, 257), 86)
self.assertRaises(ZeroDivisionError, gmpy.invert, 2, 0)
self.assertRaises(ZeroDivisionError, gmpy.invert, 2, 4)
self.assertEqual(gmpy.legendre(0, 101), 0)
self.assertEqual(gmpy.legendre(42, 101), -1)
self.assertEqual(gmpy.legendre(54, 101), 1)
self.assertRaises(ValueError, gmpy.legendre, 1, 2)
self.assertEqual(gmpy.jacobi(9, 99), 0)
self.assertEqual(gmpy.jacobi(43, 99), -1)
self.assertEqual(gmpy.jacobi(53, 99), 1)
self.assertRaises(ValueError, gmpy.jacobi, 1, 20)
self.assertEqual(gmpy.kronecker(0, 0), 0)
self.assertEqual(gmpy.kronecker(-1, 0), 1)
self.assertEqual(gmpy.kronecker(43, -98), -1)
self.assertEqual(gmpy.kronecker(-53, -98), 1)
self.assertEqual(gmpy.kronecker(-54, -98), 0)
self.assertTrue(gmpy.is_square(625))
self.assertFalse(gmpy.is_square(652))
self.assertEqual(gmpy.isqrt(0), 0)
self.assertEqual(gmpy.isqrt(1225), 35)
self.assertTrue(gmpy.iroot(0, 10)[1])
self.assertFalse(gmpy.iroot(1226, 2)[1])
self.assertEqual(gmpy.iroot(1226, 2)[0], 35)
self.assertEqual(gmpy.iroot(3**10 + 42, 10)[0], 3)