def padic_lift_list(f, factors, p, q):
"""
padicLift(f, factors, p, q) -> lifted_factors
Find a lifted integer coefficient polynomials such that:
f = G1*G2*...*Gm (mod q*p),
Gi = gi (mod q),
from f and gi's of integer coefficient polynomials such that:
f = g1*g2*...*gm (mod q),
gi's are pairwise coprime
with positive integers p dividing q.
"""
ZpZx = poly_ring.PolynomialRing(
intresidue.IntegerResidueClassRing.getInstance(p))
gg = arith1.product(factors)
h = ZpZx.createElement([(d, c // q) for (d, c) in (f - gg).iterterms()])
lifted = []
for g in factors:
gg = gg.pseudo_floordiv(g)
g_mod = ZpZx.createElement(g)
if gg.degree() == 0:
break
u, v, w = extgcdp(g, gg, p)
if w.degree() > 0:
raise ValueError("factors must be pairwise coprime.")
v_mod = ZpZx.createElement(v)
t = v_mod * h // g_mod
lifted.append(g +
minimum_absolute_injection(v_mod * h - g_mod * t) * q)
u_mod = ZpZx.createElement(u)
gg_mod = ZpZx.createElement(gg)
h = u_mod * h + gg_mod * t
lifted.append(g + minimum_absolute_injection(h) * q)
return lifted
def find_combination(f, d, factors, q):
"""
find_combination(f, d, factors, q) -> g, list
Find a combination of d factors which divides f (or its
complement). The returned values are: the product g of the
combination and a list consisting of the combination itself.
If there is no combination, return (0,[]).
"""
lf = f.leading_coefficient()
ZqZX = poly_ring.PolynomialRing(
intresidue.IntegerResidueClassRing.getInstance(q))
if d == 1:
for g in factors:
product = minimum_absolute_injection(ZqZX.createElement(lf * g))
if divisibility_test(lf * f, product):
return (product.primitive_part(), [g])
else:
for idx in combinatorial.combinationIndexGenerator(len(factors), d):
picked = [factors[i] for i in idx]
product = lf * arith1.product(picked)
product = minimum_absolute_injection(ZqZX.createElement(product))
if divisibility_test(lf * f, product):
return (product.primitive_part(), picked)
return 0, [] # nothing found
def testSubring(self):
self.assertTrue(self.F17.issubring(self.F17))
self.assertTrue(self.F17.issuperring(self.F17))
# polynomial ring
import nzmath.poly.ring as ring
F17X = ring.PolynomialRing(self.F17, 1)
self.assertTrue(self.F17.issubring(F17X))
self.assertFalse(self.F17.issuperring(F17X))
# rational field
self.assertFalse(self.F17.issuperring(theRationalField))
self.assertFalse(self.F17.issubring(theRationalField))
of integer coefficient polynomials
If you need lifted factors only once, you may want to use lift_upto
function. On the other hand, if you might happen to need another
lift, you would directly use one of the lifters and its lift method
for consecutive lifts.
"""
import sys
import nzmath.arith1 as arith1
import nzmath.rational as rational
import nzmath.poly.ring as polyring
# module globals
the_ring = polyring.PolynomialRing(rational.theIntegerRing)
the_one = the_ring.one
the_zero = the_ring.zero
def _extgcdp(f, g, p):
"""
_extgcdp(f,g,p) -> u,v,w
Find u,v,w such that f*u + g*v = w = gcd(f,g) mod p.
p should be a prime number.
This is a private function.
"""
modp = lambda c: c % p
u, v, w, x, y, z = the_one, the_zero, f, the_zero, the_one, g
def setUp(self):
Z = rational.theIntegerRing
self.zx = poly_ring.PolynomialRing(Z)
def setUp(self):
self.Z = rational.theIntegerRing
self.Q = rational.theRationalField
self.Z2 = poly_ring.PolynomialRing(self.Z, 2)
self.Q3 = poly_ring.PolynomialRing(self.Q, 3)