def generate_smooth(number, factor_base):
result = []
start = calculate_start(number)
end = calculate_end(number)
sieve = np.array([x**2-number for x in range(start,end)],
copy = False,
order = 'C')
xs = []
ys = []
for factor in factor_base:
roots = []
if 2 == factor:
roots = [modsqrt(number, factor)]
# ring formed by mod 2 only has one root
else:
first_root = modsqrt(number, factor)
roots = [first_root, factor - first_root]
for root in roots:
s = math.ceil(start / factor) * factor - start
for ind in [s-root,s+root]:
index = int(ind) # for python2
while index < end - start:
if 0 <= index:
sieve[index] = factor_out(sieve[index], factor)
if sieve[index] == 1:
xs.append(index+start)
ys.append((index+start)**2-number)
if len(xs) > len(factor_base) + 30:
return (xs, ys)
index = index + factor
return (xs,ys)
def roots_loop(g, deg_g, p, Fp):
if deg_g == 0:
return []
if deg_g == 1:
return [(-g[0]/g[1]).toInteger()]
elif deg_g == 2:
d = g[1]*g[1]-4*g[0]
e = arith1.modsqrt(d.toInteger(), p)
g1 = -g[1]
g2 = 2*g[2]
return [((g1 - e) / g2).toInteger(), ((g1 + e) / g2).toInteger()]
deg_h = 0
x = uniutil.FinitePrimeFieldPolynomial({0:-1, (p-1)//2:1}, Fp)
a = 0
while deg_h == 0 or deg_h == deg_g:
b = uniutil.FinitePrimeFieldPolynomial({0:-a, 1:1}, Fp)
v = g(b)
h = x.gcd(v)
a = a + 1
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0:a-1, 1:1}, Fp)
s = h(b)
deg_s = deg_h
t = g.exact_division(s)
deg_t = t.degree()
return roots_loop(s, deg_s, p, Fp) + roots_loop(t, deg_t, p, Fp)
def roots_loop(g, deg_g, p, Fp):
if deg_g == 0:
return []
if deg_g == 1:
return [(-g[0] / g[1]).toInteger()]
elif deg_g == 2:
d = g[1] * g[1] - 4 * g[0]
e = arith1.modsqrt(d.toInteger(), p)
g1 = -g[1]
g2 = 2 * g[2]
return [((g1 - e) / g2).toInteger(), ((g1 + e) / g2).toInteger()]
deg_h = 0
x = uniutil.FinitePrimeFieldPolynomial({0: -1, (p - 1) >> 1: 1}, Fp)
a = 0
while deg_h == 0 or deg_h == deg_g:
b = uniutil.FinitePrimeFieldPolynomial({0: -a, 1: 1}, Fp)
v = g(b)
h = x.gcd(v)
a = a + 1
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0: a - 1, 1: 1}, Fp)
s = h(b)
deg_s = deg_h
t = g.exact_division(s)
deg_t = t.degree()
return roots_loop(s, deg_s, p, Fp) + roots_loop(t, deg_t, p, Fp)
def cornacchia(d, p):
"""
Return the solution of x^2 + d * y^2 = p .
p be a prime and d be an integer such that 0 < d < p.
"""
if (d <= 0) or (d >= p):
raise ValueError("invalid input")
k = arith1.legendre(-d, p)
if k == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(-d, p)
if x0 < (p / 2):
x0 = p - x0
a = p
b = x0
l = arith1.floorsqrt(p)
while b > l:
a, b = b, a % b
c, r = divmod(p - b * b, d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t == 0:
raise ValueError("no solution")
else:
return (b, t)
def cornacchia(d, p):
"""
Return the solution of x^2 + d * y^2 = p .
p be a prime and d be an integer such that 0 < d < p.
"""
if (d <= 0) or (d >= p):
raise ValueError("invalid input")
k = arith1.legendre(-d, p)
if k == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(-d, p)
if x0 < (p / 2):
x0 = p - x0
a = p
b = x0
l = arith1.floorsqrt(p)
while b > l:
a, b = b, a % b
c, r = divmod(p - b * b, d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t == 0:
raise ValueError("no solution")
else:
return (b, t)
def cornacchiamodify(d, p):
"""
Algorithm 26 (Modified cornacchia)
Input : p be a prime and d be an integer such that d < 0 and d > -4p with
d = 0, 1 (mod 4)
Output : the solution of u^2 -d * v^2 = 4p.
"""
q = 4 * p
if (d >= 0) or (d <= -q):
raise ValueError("invalid input")
if p == 2:
b = arith1.issquare(d + 8)
if b:
return (b, 1)
else:
raise ValueError("no solution")
if arith1.legendre(d, p) == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(d, p)
if (x0 - d) & 1:
x0 = p - x0
a = 2 * p
b = x0
l = arith1.floorsqrt(q)
while b > l:
a, b = b, a % b
c, r = divmod(q - b * b, -d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t:
return (b, t)
else:
raise ValueError("no solution")
def cornacchiamodify(d, p):
"""
Algorithm 26 (Modified cornacchia)
Input : p be a prime and d be an integer such that d < 0 and d > -4p with
d = 0, 1 (mod 4)
Output : the solution of u^2 -d * v^2 = 4p.
"""
q = 4 * p
if (d >= 0) or (d <= -q):
raise ValueError("invalid input")
if p == 2:
b = arith1.issquare(d + 8)
if b:
return (b, 1)
else:
raise ValueError("no solution")
if arith1.legendre(d, p) == -1:
raise ValueError("no solution")
x0 = arith1.modsqrt(d, p)
if (x0 - d) % 2 != 0:
x0 = p - x0
a = 2 * p
b = x0
l = arith1.floorsqrt(q)
while b > l:
a, b = b, a % b
c, r = divmod(q - b * b, -d)
if r:
raise ValueError("no solution")
t = arith1.issquare(c)
if t:
return (b, t)
else:
raise ValueError("no solution")
def SquareRoot(self, element):
""" Return square root if exist.
"""
if not element or element.n == 1:
return element # trivial case
if element.m == 2:
return element.getRing().one
return arith1.modsqrt(element.n, element.m)
def SquareRoot(self, element):
""" Return square root if exist.
"""
if not element or element.n == 1:
return element # trivial case
if element.m == 2:
return element.getRing().one
return arith1.modsqrt(element.n, element.m)
def random_point(self):
"""
Returns:
A random point on the curve.
"""
x = random.randrange(self.p)
y_square = x**3 + self.a * x + self.b
while jacobi(y_square, self.p) == -1:
x = random.randrange(self.p)
y_square = x**3 + self.a * x + self.b
y = modsqrt(y_square, self.p)
return x,
def random_point(self):
"""
Returns:
A random point on the curve.
"""
x = random.randrange(self.p)
y_square = x**3 + self.a * x + self.b
while jacobi(y_square, self.p) == -1:
x = random.randrange(self.p)
y_square = x**3 + self.a * x + self.b
y = modsqrt(y_square, self.p)
return x,
def choose_point(self):
"""
Choose point on E_{a,b}(Z_n)
Algorithm 27 (Atkin-morain ECPP) Step5
"""
n, f = self.modulus, self.f
x = bigrandom.randrange(n)
Q = f(x)
while arith1.legendre(Q, n) == -1:
x = bigrandom.randrange(n)
Q = f(x)
y = arith1.modsqrt(Q, n)
return [intresidue.IntegerResidueClass(t, n) for t in (x, y)]
def choose_point(self):
"""
Choose point on E_{a,b}(Z_n)
Algorithm 27 (Atkin-morain ECPP) Step5
"""
n, f = self.modulus, self.f
x = bigrandom.randrange(n)
Q = f(x)
while arith1.legendre(Q, n) == -1:
x = bigrandom.randrange(n)
Q = f(x)
y = arith1.modsqrt(Q, n)
return [intresidue.IntegerResidueClass(t, n) for t in (x, y)]
def root_Fp(g, p, flag=True):
"""
Return a root over F_p of nonzero polynomial g.
p must be prime.
If flag = False, return a root randomly
"""
if isinstance(g, list):
if not isinstance(g[0], tuple):
g = zip(range(len(g)), g)
Fp = finitefield.FinitePrimeField(p)
g = uniutil.FinitePrimeFieldPolynomial(g, Fp)
h = uniutil.FinitePrimeFieldPolynomial({1: -1, p: 1}, Fp)
g = g.gcd(h)
deg_g = g.degree()
if g[0] == 0:
deg_g = deg_g - 1
g = g.shift_degree_to(deg_g)
while True:
if deg_g == 0:
return None
if deg_g == 1:
return (-g[0] / g[1]).toInteger()
elif deg_g == 2:
d = g[1] * g[1] - 4 * g[0]
e = arith1.modsqrt(d.toInteger(), p)
return ((-g[1] - e) / (2 * g[2])).toInteger()
deg_h = 0
x = uniutil.FinitePrimeFieldPolynomial({0: -1, (p - 1) >> 1: 1}, Fp)
if flag:
a = 0
while deg_h == 0 or deg_h == deg_g:
b = uniutil.FinitePrimeFieldPolynomial({0: -a, 1: 1}, Fp)
v = g(b)
h = x.gcd(v)
a = a + 1
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0: a - 1, 1: 1}, Fp)
else:
while deg_h == 0 or deg_h == deg_g:
a = bigrandom.randrange(p)
b = uniutil.FinitePrimeFieldPolynomial({0: -a, 1: 1}, Fp)
v = g(b)
h = x.gcd(v)
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0: a, 1: 1}, Fp)
g = h(b)
deg_g = deg_h
def root_Fp(g, p, flag=True):
"""
Return a root over F_p of nonzero polynomial g.
p must be prime.
If flag = False, return a root randomly
"""
if isinstance(g, list):
if not isinstance(g[0], tuple):
g = zip(range(len(g)), g)
Fp = finitefield.FinitePrimeField(p)
g = uniutil.FinitePrimeFieldPolynomial(g, Fp)
h = uniutil.FinitePrimeFieldPolynomial({1:-1, p:1}, Fp)
g = g.gcd(h)
deg_g = g.degree()
if g[0] == 0:
deg_g = deg_g - 1
g = g.shift_degree_to(deg_g)
while True:
if deg_g == 0:
return None
if deg_g == 1:
return (-g[0]/g[1]).toInteger()
elif deg_g == 2:
d = g[1]*g[1] - 4*g[0]
e = arith1.modsqrt(d.toInteger(), p)
return ((-g[1]-e)/(2*g[2])).toInteger()
deg_h = 0
x = uniutil.FinitePrimeFieldPolynomial({0:-1, (p-1)//2:1}, Fp)
if flag:
a = 0
while deg_h == 0 or deg_h == deg_g:
b = uniutil.FinitePrimeFieldPolynomial({0:-a, 1:1}, Fp)
v = g(b)
h = x.gcd(v)
a = a + 1
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0:a-1, 1:1}, Fp)
else:
while deg_h == 0 or deg_h == deg_g:
a = bigrandom.randrange(p)
b = uniutil.FinitePrimeFieldPolynomial({0:-a, 1:1}, Fp)
v = g(b)
h = x.gcd(v)
deg_h = h.degree()
b = uniutil.FinitePrimeFieldPolynomial({0:a, 1:1}, Fp)
g = h(b)
deg_g = deg_h
def cornacchia_smith(p, d):
'''
modified Cornacchia's Algorithm to solve a^2 + b^2 |D| = 4p for a and b
Args:
p:
d:
Returns:
a, b such that a^2 + b^2 |D| = 4p
'''
# check input
if not -4 * p < d < 0:
raise ValueError(" -4p < D < 0 not true.")
elif not (d % 4 in {0, 1}):
raise ValueError(" D = 0, 1 (mod 4) not true.")
# case where p=2
if p == 2:
r = sqrt(d + 8)
if r != -1:
return r, 1
else:
return None
# test for solvability
if jacobi(d % p, p) < 1:
return None
x = modsqrt(d, p)
if (x % 2) != (d % 2):
x = p - x
# euclid chain
a, b = (2 * p, x)
c = floorsqrt(4 * p)
while b > c:
a, b = b, a % b
t = 4 * p - b * b
if t % (-d) != 0:
return None
if not issquare(t / (-d)):
return None
return b, int(mpmath.sqrt(t / -d))
def choose_point(ec):
"""
Choose a random point on EC.
Adapted from random_point function. With the additional check for modsqrt.
Args:
ec: an elliptic curve by the equation y^2 = x^3 + a * x + b
Returns:
a valid point on the curve.
"""
x = random.randrange(ec.p)
y_square = (x**3 + ec.a * x + ec.b) % ec.p
while jacobi(y_square, ec.p) == -1:
x = random.randrange(ec.p)
y_square = (x**3 + ec.a * x + ec.b) % ec.p
y = modsqrt(y_square, ec.p)
if (y**2 % ec.p) != y_square:
raise ValueError("Error computing square root.")
return x, y
def sqroot_power(a, p, n):
"""
return squareroot of a mod p^k for k = 2,3,...,n
"""
r = arith1.modsqrt(a, p)
x = (r, p-r)
i = 2
answer = [x]
ppower = p
while i <= n:
b_1 = (x[0]**2-a) // ppower
b_2 = (x[1]**2-a) // ppower
x_1 = -b_1 * arith1.inverse(2*x[0], p)
x_2 = -b_2 * arith1.inverse(2*x[1], p)
X_1 = x[0] + x_1*ppower % (p*ppower)
X_2 = x[1] + x_2*ppower % (p*ppower)
x = [X_1, X_2]
answer.append(x)
i += 1
ppower *= p
return answer
def sqroot(disc, p):
"""
Return a reduced quadratic form with the given discriminant.
'disc' is a quadratic residue mod 'p'.
"""
if p == 2: # if 8 | disc => (disc / 8) = 0, 8 not | disc but 4 | disc => 2
if (disc % 8) == 0:
bp = disc
elif (disc % 4) == 0: # 4 - 4 * odd % 8 => 0
bp = 2
elif (disc % 8) == 1: # disc is odd and disc % 8 is 1
bp = disc
else: # disc is odd and disc % 4 is 1 => impossible (-5 / 2) = -1
raise ValueError("disc is odd and disc % 4 is 1 => impossible (-5 / 2) = -1")
else:
bpf1 = arith1.modsqrt(disc, p)
bpf2 = disc
bp = crt([(bpf1, p), (bpf2, 4)])
if bp > p:
bp = 2 * p - bp
fpt = reducePDF((p, bp, ((bp ** 2) - disc) // (4 * p)))
return fpt
def e2_Fp(x, p):
"""
p is prime
f = x[0] + x[1]*t + x[2]*t**2
"""
c, b, a = [_x % p for _x in x]
if a == 0:
return [e1_ZnZ([c, b], p)]
if p == 2:
solutions = []
if x[0] & 1 == 0:
solutions.append(0)
if (x[0] + x[1] + x[2]) & 1 == 0:
solutions.append(1)
if len(solutions) == 1:
return solutions * 2
return solutions
d = b**2 - 4 * a * c
if arith1.legendre(d, p) == -1:
return []
sqrtd = arith1.modsqrt(d, p)
a = arith1.inverse(2 * a, p)
return [((-b + sqrtd) * a) % p, ((-b - sqrtd) * a) % p]
def e2_Fp(x, p):
"""
p is prime
f = x[0] + x[1]*t + x[2]*t**2
"""
c, b, a = [_x % p for _x in x]
if a == 0:
return [e1_ZnZ([c, b], p)]
if p == 2:
solutions = []
if x[0] % 2 == 0:
solutions.append(0)
if (x[0] + x[1] + x[2]) % 2 == 0:
solutions.append(1)
if len(solutions) == 1:
return solutions * 2
return solutions
d = b**2 - 4*a*c
if arith1.legendre(d, p) == -1:
return []
sqrtd = arith1.modsqrt(d, p)
a = arith1.inverse(2*a, p)
return [((-b+sqrtd)*a)%p, ((-b-sqrtd)*a)%p]
def sqroot(disc, p):
"""
Return a reduced quadratic form with the given discriminant.
'disc' is a quadratic residue mod 'p'.
"""
if p == 2: # if 8 | disc => (disc / 8) = 0, 8 not | disc but 4 | disc => 2
if (disc & 7) == 0:
bp = disc
elif (disc & 3) == 0: # 4 - 4 * odd % 8 => 0
bp = 2
elif (disc & 7) == 1: # disc is odd and disc % 8 is 1
bp = disc
else: # disc is odd and disc & 3 is 1 => impossible (-5 / 2) = -1
raise ValueError(
"disc is odd and disc & 3 is 1 => impossible (-5 / 2) = -1")
else:
bpf1 = arith1.modsqrt(disc, p)
bpf2 = disc
bp = crt([(bpf1, p), (bpf2, 4)])
if bp > p:
bp = 2 * p - bp
fpt = reducePDF((p, bp, ((bp**2) - disc) // (4 * p)))
return fpt
def testModsqrt(self):
self.assertTrue(arith1.modsqrt(2, 17) in (6, 11))
self.assertTrue(arith1.modsqrt(124413, 2**17 - 1) in (3998, 127073))
self.assertEqual(1, arith1.modsqrt(1, 2**13 - 1))
self.assertTrue(arith1.modsqrt(2, 7, 2) in (10, 39))
self.assertTrue(arith1.modsqrt(12, 97, 3) in (448799, 463874))