def sqrt_mod_iter(a, p, domain=int):
"""
Iterate over solutions to ``x**2 = a mod p``
Parameters
==========
a : integer
p : positive integer
domain : integer domain, ``int``, ``ZZ`` or ``Integer``
Examples
========
>>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter
>>> list(sqrt_mod_iter(11, 43))
[21, 22]
"""
from sympy.polys.galoistools import gf_crt1, gf_crt2
from sympy.polys.domains import ZZ
a, p = as_int(a), abs(as_int(p))
if isprime(p):
a = a % p
if a == 0:
res = _sqrt_mod1(a, p, 1)
else:
res = _sqrt_mod_prime_power(a, p, 1)
if res:
if domain is ZZ:
for x in res:
yield x
else:
for x in res:
yield domain(x)
else:
f = factorint(p)
v = []
pv = []
for px, ex in f.items():
if a % px == 0:
rx = _sqrt_mod1(a, px, ex)
if not rx:
return
else:
rx = _sqrt_mod_prime_power(a, px, ex)
if not rx:
return
v.append(rx)
pv.append(px ** ex)
mm, e, s = gf_crt1(pv, ZZ)
if domain is ZZ:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield r
else:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield domain(r)
def sqrt_mod_iter(a, p, domain=int):
"""
iterate over solutions to ``x**2 = a mod p``
Parameters
==========
a : integer
p : positive integer
domain : integer domain, ``int``, ``ZZ`` or ``Integer``
Examples
========
>>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter
>>> list(sqrt_mod_iter(11, 43))
[21, 22]
"""
from sympy.polys.galoistools import gf_crt1, gf_crt2
from sympy.polys.domains import ZZ
a, p = as_int(a), abs(as_int(p))
if isprime(p):
a = a % p
if a == 0:
res = _sqrt_mod1(a, p, 1)
else:
res = _sqrt_mod_prime_power(a, p, 1)
if res:
if domain is ZZ:
for x in res:
yield x
else:
for x in res:
yield domain(x)
else:
f = factorint(p)
v = []
pv = []
for px, ex in f.items():
if a % px == 0:
rx = _sqrt_mod1(a, px, ex)
if not rx:
raise StopIteration
else:
rx = _sqrt_mod_prime_power(a, px, ex)
if not rx:
raise StopIteration
v.append(rx)
pv.append(px**ex)
mm, e, s = gf_crt1(pv, ZZ)
if domain is ZZ:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield r
else:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield domain(r)
def crt2(m, v, mm, e, s, symmetric=False):
"""Second part of Chinese Remainder Theorem, for multiple application. """
result = gf_crt2(v, m, mm, e, s, ZZ)
if symmetric:
return symmetric_residue(result, mm), mm
return result, mm
def test_gf_crt():
U = [49, 76, 65]
M = [99, 97, 95]
p = 912285
u = 639985
assert gf_crt(U, M, ZZ) == u
E = [9215, 9405, 9603]
S = [62, 24, 12]
assert gf_crt1(M, ZZ) == (p, E, S)
assert gf_crt2(U, M, p, E, S, ZZ) == u
def crt2(m, v, mm, e, s, symmetric=False):
"""Second part of Chinese Remainder Theorem, for multiple application.
Examples
========
>>> from sympy.ntheory.modular import crt1, crt2
>>> mm, e, s = crt1([18, 42, 6])
>>> crt2([18, 42, 6], [0, 0, 0], mm, e, s)
(0, 4536)
"""
result = gf_crt2(v, m, mm, e, s, ZZ)
if symmetric:
return symmetric_residue(result, mm), mm
return result, mm
def crt2(m, v, mm, e, s, symmetric=False):
"""Second part of Chinese Remainder Theorem, for multiple application.
Examples
========
>>> from sympy.ntheory.modular import crt1, crt2
>>> mm, e, s = crt1([18, 42, 6])
>>> crt2([18, 42, 6], [0, 0, 0], mm, e, s)
(0, 4536)
"""
result = gf_crt2(v, m, mm, e, s, ZZ)
if symmetric:
return symmetric_residue(result, mm), mm
return result, mm