def test_jacobi_prime(module):
sqrs = set()
for a in xrange(module):
sqrs.add((a * a) % module)
for a in xrange(module):
if gcd(a, module) == 1:
real = 1 if a in sqrs else -1
else:
real = 0
test = jacobi(a, module)
assertEqual(real, test)
def test_sqrt_pp_rand():
print("\nTesting random residues by random modules")
for size, maxpow in [(2, 500), (10, 100), (64, 15), (128, 5), (129, 5),
(256, 2)]:
for i in xrange(10):
p = generate_prime(size, k=25)
print(" Testing %s-bit prime with max power %s: %s..." %
(size, maxpow, str(p)[:32]))
for j in xrange(10):
k = random.randint(1, maxpow)
x = random.randint(0, p**k - 1)
a = pow(x, 2, p**k)
check_valid_sqrt_pp(x, a, p, k)
def test_jacobi():
def test_jacobi_prime(module):
sqrs = set()
for a in xrange(module):
sqrs.add((a * a) % module)
for a in xrange(module):
if gcd(a, module) == 1:
real = 1 if a in sqrs else -1
else:
real = 0
test = jacobi(a, module)
assertEqual(real, test)
plist = primes(100) + [293, 1993, 2969, 2971, 9973, 11311]
for module in plist[2:]:
test_jacobi_prime(module)
plist = primes(10)[2:]
lezhs = {}
for p in plist:
lezhs[p] = [jacobi(a, p) for a in xrange(p)]
for pnum in xrange(2, 4):
for f in combinations_with_replacement(plist, pnum):
n = reduce(operator.mul, f)
for a in xrange(n):
real = reduce(operator.mul, [lezhs[p][a % p] for p in f])
test = jacobi(a, n)
if real != test:
print("")
print("%d | %d %s" % (a, n, repr(f)))
print("Lezhandre symbols: %s" %
repr([lezhs[p][a % p] for p in f]))
for p in f:
print(lezhs[p])
print("real %s" % repr(real))
print("test %s" % repr(test))
print()
assertEqual(real, test)
assertRaises(ValueError, jacobi, 1, 2)
assertRaises(ValueError, jacobi, 0, 6)
assertRaises(ValueError, jacobi, 0, 0)
assertRaises(Exception, jacobi, "qwe", 1024)
assertRaises(Exception, jacobi, 123, "qwe")
def test_sqrt_composite_rand():
print("\nTesting all residues by random composite modules")
for size, ntries in [(2, 2), (3, 3), (5, 10), (7, 20), (10, 20)]:
for i in xrange(ntries):
n = randint_bits(size)
f = factorize(n)
print(" Testing %s-bit number: %s..." % (size, str(n)[:32]))
for x in xrange(n):
a = pow(x, 2, n)
is_sqrt = has_sqrtmod(a, f)
if is_sqrt:
check_valid_sqrt_composite(None, a, f)
check_jacobi(a, n, is_sqrt)
assertTrue(has_sqrtmod(a, f))
check_valid_sqrt_composite(x, a, f)
def test_crt():
for module in [2, 3, 5, 7, 1993]:
for a in xrange(module):
assertEqual(solve_crt([a], [module]), a)
assertEqual(solve_crt([a, 0], [module, 1]), a)
modules = [2, 3, 5, 19, 137]
for i in xrange(1000):
rems = []
a = 7
for m in modules:
rems.append(a % m)
a += 31337
a = solve_crt(rems, modules)
for i in xrange(len(modules)):
assertEqual(rems[i], a % modules[i])
assertRaises(TypeError, solve_crt, [1, 2, 3], [1, 2])
assertRaises(ValueError, solve_crt, [], [])
def test_factorial_mod():
print("\nTesting factorial mod prime powers")
for p, max_e in [(2, 8), (3, 4), (5, 3), (7, 3), (11, 2)]:
print(" prime %s pow up to %s" % (repr(p), repr(max_e)))
for i in xrange(250):
n = random.randint(1, 3000)
e = random.randint(1, max_e)
my = factorial_mod(n, {p: e})
real = factorial(n) % (p ** e)
assertEqual(my, real)
print("\nTesting factorial mod small composites")
for i in xrange(150):
n = random.randint(1, 8000)
x = random.randint(0, n * 2)
my = factorial_mod(x, factorize(n))
real = factorial(x) % n
assertEqual(my, real)
def test_nck():
for n in (2, 5, 7, 100):
csum = 0
for x in xrange(n + 1):
csum += nCk(n, x)
assertEqual(csum, 2**n)
row = [1]
for n in xrange(1, 200):
row = [0] + row + [0]
row = [row[i - 1] + row[i] for i in xrange(1, len(row))]
for i in xrange(len(row)):
assertEqual(row[i], nCk(n, i))
assertEqual(nCk(10, -1), 0)
assertEqual(nCk(10, 11), 0)
assertEqual(nCk(0, 0), 1)
assertEqual(nCk(0, 1), 0)
assertRaises(ValueError, nCk, -1, 0)
def test_nCk_mod_pp():
print("\nTesting nCk mod prime powers")
for p, max_e in [(2, 8), (3, 4), (5, 3), (7, 3), (11, 2), (13, 2)]:
print(" prime %s pow up to %s" % (repr(p), repr(max_e)))
for i in xrange(100):
k = random.randint(1, 10000)
n = k + random.randint(0, 10000)
e = random.randint(1, max_e)
my = nCk_mod_prime_power(n, k, p, e)
real = nCk(n, k) % (p ** e)
assertEqual(my, real)
def test_sqrt_pp_all():
print("\nTesting all residues by small modules")
for prime, maxpow in [(2, 11), (3, 7), (5, 5), (7, 4), (11, 3), (13, 3),
(97, 2)]:
for k in xrange(1, maxpow + 1):
n = prime**k
print(" Testing %s**%s" % (prime, k))
for x in xrange(n):
a = pow(x, 2, n)
is_sqrt = has_sqrtmod(a, {prime: k})
is_sqrt2 = has_sqrtmod_prime_power(a, prime, k)
assertEqual(is_sqrt, is_sqrt2)
if is_sqrt:
check_valid_sqrt_pp(None, a, prime, k)
check_jacobi(a, n, is_sqrt)
assertTrue(has_sqrtmod_prime_power(a, prime, k))
assertTrue(has_sqrtmod(a, {prime: k}))
check_valid_sqrt_pp(x, a, prime, k)
def test_genprime():
for size in (2, 10, 64, 128, 129, 256):
for ntry in xrange(10):
p = generate_prime(size, k=25)
assertEqual(len_in_bits(p), size)
assertTrue(prime_test_ferma(p, k=25))
assertTrue(prime_test_solovay_strassen(p, k=25))
assertTrue(prime_test_miller_rabin(p, k=25))
assertTrue(prime_test(p, k=25))
assertRaises(ValueError, generate_prime, 1)
assertRaises(TypeError, generate_prime, "")
def test_sqrt_composite_all():
print("\nTesting all residues by small composite modules")
for n in [
10, 30, 50, 99, 100, 655, 1025, 1337, 7**3 * 3, 2**6 * 13,
2**4 * 3**3 * 5, 3 * 3 * 5 * 7, 1024
]:
f = factorize(n)
print(" Testing %s = %s" % (n, f))
for x in xrange(n):
a = pow(x, 2, n)
is_sqrt = has_sqrtmod(a, f)
if is_sqrt:
check_valid_sqrt_composite(None, a, f)
check_jacobi(a, n, is_sqrt)
assertTrue(has_sqrtmod(a, f))
check_valid_sqrt_composite(x, a, f)