def mcrt(m, v, r, symmetric=False):
assert crt(m, v, symmetric)[0] == r
mm, e, s = crt1(m)
assert crt2(m, v, mm, e, s, symmetric) == (r, mm)
def zzx_mod_gcd(f, g, **flags):
"""Modular small primes polynomial GCD over Z[x].
Given univariate polynomials f and g over Z[x], returns their
GCD and cofactors, i.e. polynomials h, cff and cfg such that:
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm uses modular small primes approach. It works by
computing several GF(p)[x] GCDs for a set of randomly chosen
primes and uses Chinese Remainder Theorem to recover the GCD
over Z[x] from its images.
The algorithm is probabilistic which means it never fails,
however its running time depends on the number of unlucky
primes chosen for computing GF(p)[x] images.
For more details on the implemented algorithm refer to:
[1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 158
"""
n = zzx_degree(f)
m = zzx_degree(g)
cf = zzx_content(f)
cg = zzx_content(g)
h = igcd(cf, cg)
f = [c // h for c in f]
g = [c // h for c in g]
if n <= 0 or m <= 0:
return zzx_strip([h]), f, g
else:
gcd = h
A = max(zzx_abs(f) + zzx_abs(g))
b = igcd(zzx_LC(f), zzx_LC(g))
B = int(ceil(2**n * A * b * sqrt(n + 1)))
k = int(ceil(2 * b * log((n + 1)**n * A**(2 * n), 2)))
l = int(ceil(log(2 * B + 1, 2)))
prime_max = max(int(ceil(2 * k * log(k))), 51)
while True:
while True:
primes = set([])
unlucky = set([])
ff, gg, hh = {}, {}, {}
while len(primes) < l:
p = randprime(3, prime_max + 1)
if (p in primes) and (b % p == 0):
continue
F = gf_from_int_poly(f, p)
G = gf_from_int_poly(g, p)
H = gf_gcd(F, G, p)
primes.add(p)
ff[p] = F
gg[p] = G
hh[p] = H
e = min([gf_degree(h) for h in hh.itervalues()])
for p in set(primes):
if gf_degree(hh[p]) != e:
primes.remove(p)
unlucky.add(p)
del ff[p]
del gg[p]
del hh[p]
if len(primes) < l // 2:
continue
while len(primes) < l:
p = randprime(3, prime_max + 1)
if (p in primes) or (p in unlucky) or (b % p == 0):
continue
F = gf_from_int_poly(f, p)
G = gf_from_int_poly(g, p)
H = gf_gcd(F, G, p)
if gf_degree(H) != e:
unlucky.add(p)
else:
primes.add(p)
ff[p] = F
gg[p] = G
hh[p] = H
break
fff, ggg = {}, {}
for p in primes:
fff[p] = gf_quo(ff[p], hh[p], p)
ggg[p] = gf_quo(gg[p], hh[p], p)
F, G, H = [], [], []
crt_mm, crt_e, crt_s = crt1(primes)
for i in xrange(0, e + 1):
C = [b * zzx_nth(hh[p], i) for p in primes]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
H.insert(0, c)
H = zzx_strip(H)
for i in xrange(0, zzx_degree(f) - e + 1):
C = [zzx_nth(fff[p], i) for p in primes]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
F.insert(0, c)
for i in xrange(0, zzx_degree(g) - e + 1):
C = [zzx_nth(ggg[p], i) for p in primes]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
G.insert(0, c)
H_norm = zzx_l1_norm(H)
F_norm = zzx_l1_norm(F)
G_norm = zzx_l1_norm(G)
if H_norm * F_norm <= B and H_norm * G_norm <= B:
break
return zzx_mul_term(H, gcd, 0), F, G
def mcrt(m, v, r, symmetric=False):
assert crt(m, v, symmetric)[0] == r
mm, e, s = crt1(m)
assert crt2(m, v, mm, e, s, symmetric) == (r, mm)
def zzx_mod_gcd(f, g, **flags):
"""Modular small primes polynomial GCD over Z[x].
Given univariate polynomials f and g over Z[x], returns their
GCD and cofactors, i.e. polynomials h, cff and cfg such that:
h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)
The algorithm uses modular small primes approach. It works by
computing several GF(p)[x] GCDs for a set of randomly chosen
primes and uses Chinese Remainder Theorem to recover the GCD
over Z[x] from its images.
The algorithm is probabilistic which means it never fails,
however its running time depends on the number of unlucky
primes chosen for computing GF(p)[x] images.
For more details on the implemented algorithm refer to:
[1] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
First Edition, Cambridge University Press, 1999, pp. 158
"""
n = zzx_degree(f)
m = zzx_degree(g)
cf = zzx_content(f)
cg = zzx_content(g)
h = igcd(cf, cg)
f = [ c // h for c in f ]
g = [ c // h for c in g ]
if n <= 0 or m <= 0:
return zzx_strip([h]), f, g
else:
gcd = h
A = max(zzx_abs(f) + zzx_abs(g))
b = igcd(zzx_LC(f), zzx_LC(g))
B = int(ceil(2**n*A*b*sqrt(n + 1)))
k = int(ceil(2*b*log((n + 1)**n*A**(2*n), 2)))
l = int(ceil(log(2*B + 1, 2)))
prime_max = max(int(ceil(2*k*log(k))), 51)
while True:
while True:
primes = set([])
unlucky = set([])
ff, gg, hh = {}, {}, {}
while len(primes) < l:
p = randprime(3, prime_max+1)
if (p in primes) and (b % p == 0):
continue
F = gf_from_int_poly(f, p)
G = gf_from_int_poly(g, p)
H = gf_gcd(F, G, p)
primes.add(p)
ff[p] = F
gg[p] = G
hh[p] = H
e = min([ gf_degree(h) for h in hh.itervalues() ])
for p in set(primes):
if gf_degree(hh[p]) != e:
primes.remove(p)
unlucky.add(p)
del ff[p]
del gg[p]
del hh[p]
if len(primes) < l // 2:
continue
while len(primes) < l:
p = randprime(3, prime_max+1)
if (p in primes) or (p in unlucky) or (b % p == 0):
continue
F = gf_from_int_poly(f, p)
G = gf_from_int_poly(g, p)
H = gf_gcd(F, G, p)
if gf_degree(H) != e:
unlucky.add(p)
else:
primes.add(p)
ff[p] = F
gg[p] = G
hh[p] = H
break
fff, ggg = {}, {}
for p in primes:
fff[p] = gf_quo(ff[p], hh[p], p)
ggg[p] = gf_quo(gg[p], hh[p], p)
F, G, H = [], [], []
crt_mm, crt_e, crt_s = crt1(primes)
for i in xrange(0, e + 1):
C = [ b * zzx_nth(hh[p], i) for p in primes ]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
H.insert(0, c)
H = zzx_strip(H)
for i in xrange(0, zzx_degree(f) - e + 1):
C = [ zzx_nth(fff[p], i) for p in primes ]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
F.insert(0, c)
for i in xrange(0, zzx_degree(g) - e + 1):
C = [ zzx_nth(ggg[p], i) for p in primes ]
c = crt2(primes, C, crt_mm, crt_e, crt_s, True)
G.insert(0, c)
H_norm = zzx_l1_norm(H)
F_norm = zzx_l1_norm(F)
G_norm = zzx_l1_norm(G)
if H_norm*F_norm <= B and H_norm*G_norm <= B:
break
return zzx_mul_term(H, gcd, 0), F, G