def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [dup_trunc(F, p**l, K)]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, _ = gf_gcdex(g, h, p, K)
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
for _ in range(1, d + 1):
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def dup_zz_hensel_lift(p, f, f_list, l, K):
"""
Multifactor Hensel lifting in `Z[x]`.
Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`
is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`
over `Z[x]` satisfying::
f = lc(f) f_1 ... f_r (mod p)
and a positive integer `l`, returns a list of monic polynomials
`F_1`, `F_2`, ..., `F_r` satisfying::
f = lc(f) F_1 ... F_r (mod p**l)
F_i = f_i (mod p), i = 1..r
References
==========
1. [Gathen99]_
"""
r = len(f_list)
lc = dup_LC(f, K)
if r == 1:
F = dup_mul_ground(f, K.gcdex(lc, p**l)[0], K)
return [ dup_trunc(F, p**l, K) ]
m = p
k = r // 2
d = int(_ceil(_log(l, 2)))
g = gf_from_int_poly([lc], p)
for f_i in f_list[:k]:
g = gf_mul(g, gf_from_int_poly(f_i, p), p, K)
h = gf_from_int_poly(f_list[k], p)
for f_i in f_list[k + 1:]:
h = gf_mul(h, gf_from_int_poly(f_i, p), p, K)
s, t, _ = gf_gcdex(g, h, p, K)
g = gf_to_int_poly(g, p)
h = gf_to_int_poly(h, p)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
for _ in range(1, d + 1):
(g, h, s, t), m = dup_zz_hensel_step(m, f, g, h, s, t, K), m**2
return dup_zz_hensel_lift(p, g, f_list[:k], l, K) \
+ dup_zz_hensel_lift(p, h, f_list[k:], l, K)
def test_gf_gcdex():
assert gf_gcdex([], [], 11, ZZ) == ([1], [], [])
assert gf_gcdex([2], [], 11, ZZ) == ([6], [], [1])
assert gf_gcdex([], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([2], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([], [3,0], 11, ZZ) == ([], [4], [1,0])
assert gf_gcdex([3,0], [], 11, ZZ) == ([4], [], [1,0])
assert gf_gcdex([3,0], [3,0], 11, ZZ) == ([], [4], [1,0])
assert gf_gcdex([1,8,7], [1,7,1,7], 11, ZZ) == ([5,6], [6], [1,7])
def inverse(f):
# Fast inverse method
global phid
global q
p = ZZ.map(f)
mod = ZZ.map(phid)
s, t, g = gf_gcdex(p, mod, q, ZZ)
if len(g) == 1 and g[0] == 1:
return s
else:
return [-1]
def find_inv(poly, mod, constant = -a):
xN_minus_a = np.zeros(N+1)
xN_minus_a[0] = 1
xN_minus_a[-1] = constant
f_poly = ZZ.map(poly)
x_mod = ZZ.map(xN_minus_a)
s, t, g = gf_gcdex(f_poly, x_mod, mod, ZZ)
# Recall that s*f + t*g = h.
# If g = 1, then s is the inverse of f mod t (= x^n-a)
if len(g) == 1 and g[0] == 1:
return trunc_polynomial(s)
return trunc_polynomial(0)
def dup_zz_diophantine(F, m, p, K):
"""Wang/EEZ: Solve univariate Diophantine equations. """
if len(F) == 2:
a, b = F
f = gf_from_int_poly(a, p)
g = gf_from_int_poly(b, p)
s, t, G = gf_gcdex(g, f, p, K)
s = gf_lshift(s, m, K)
t = gf_lshift(t, m, K)
q, s = gf_div(s, f, p, K)
t = gf_add_mul(t, q, g, p, K)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
result = [s, t]
else:
G = [F[-1]]
for f in reversed(F[1:-1]):
G.insert(0, dup_mul(f, G[0], K))
S, T = [], [[1]]
for f, g in zip(F, G):
t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K)
T.append(t)
S.append(s)
result, S = [], S + [T[-1]]
for s, f in zip(S, F):
s = gf_from_int_poly(s, p)
f = gf_from_int_poly(f, p)
r = gf_rem(gf_lshift(s, m, K), f, p, K)
s = gf_to_int_poly(r, p)
result.append(s)
return result
def dup_zz_diophantine(F, m, p, K):
"""Wang/EEZ: Solve univariate Diophantine equations. """
if len(F) == 2:
a, b = F
f = gf_from_int_poly(a, p)
g = gf_from_int_poly(b, p)
s, t, G = gf_gcdex(g, f, p, K)
s = gf_lshift(s, m, K)
t = gf_lshift(t, m, K)
q, s = gf_div(s, f, p, K)
t = gf_add_mul(t, q, g, p, K)
s = gf_to_int_poly(s, p)
t = gf_to_int_poly(t, p)
result = [s, t]
else:
G = [F[-1]]
for f in reversed(F[1:-1]):
G.insert(0, dup_mul(f, G[0], K))
S, T = [], [[1]]
for f, g in zip(F, G):
t, s = dmp_zz_diophantine([g, f], T[-1], [], 0, p, 1, K)
T.append(t)
S.append(s)
result, S = [], S + [T[-1]]
for s, f in zip(S, F):
s = gf_from_int_poly(s, p)
f = gf_from_int_poly(f, p)
r = gf_rem(gf_lshift(s, m, K), f, p, K)
s = gf_to_int_poly(r, p)
result.append(s)
return result
def test_gf_euclidean():
assert gf_gcd([], [], 11, ZZ) == []
assert gf_gcd([2], [], 11, ZZ) == [1]
assert gf_gcd([], [2], 11, ZZ) == [1]
assert gf_gcd([2], [2], 11, ZZ) == [1]
assert gf_gcd([], [1, 0], 11, ZZ) == [1, 0]
assert gf_gcd([1, 0], [], 11, ZZ) == [1, 0]
assert gf_gcd([3, 0], [3, 0], 11, ZZ) == [1, 0]
assert gf_gcd([1, 8, 7], [1, 7, 1, 7], 11, ZZ) == [1, 7]
assert gf_gcdex([], [], 11, ZZ) == ([1], [], [])
assert gf_gcdex([2], [], 11, ZZ) == ([6], [], [1])
assert gf_gcdex([], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([2], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([], [3, 0], 11, ZZ) == ([], [4], [1, 0])
assert gf_gcdex([3, 0], [], 11, ZZ) == ([4], [], [1, 0])
assert gf_gcdex([3, 0], [3, 0], 11, ZZ) == ([], [4], [1, 0])
assert gf_gcdex([1, 8, 7], [1, 7, 1, 7], 11, ZZ) == ([5, 6], [6], [1, 7])
def test_gf_euclidean():
assert gf_gcd([], [], 11, ZZ) == []
assert gf_gcd([2], [], 11, ZZ) == [1]
assert gf_gcd([], [2], 11, ZZ) == [1]
assert gf_gcd([2], [2], 11, ZZ) == [1]
assert gf_gcd([], [1,0], 11, ZZ) == [1,0]
assert gf_gcd([1,0], [], 11, ZZ) == [1,0]
assert gf_gcd([3,0], [3,0], 11, ZZ) == [1,0]
assert gf_gcd([1,8,7], [1,7,1,7], 11, ZZ) == [1,7]
assert gf_gcdex([], [], 11, ZZ) == ([1], [], [])
assert gf_gcdex([2], [], 11, ZZ) == ([6], [], [1])
assert gf_gcdex([], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([2], [2], 11, ZZ) == ([], [6], [1])
assert gf_gcdex([], [3,0], 11, ZZ) == ([], [4], [1,0])
assert gf_gcdex([3,0], [], 11, ZZ) == ([4], [], [1,0])
assert gf_gcdex([3,0], [3,0], 11, ZZ) == ([], [4], [1,0])
assert gf_gcdex([1,8,7], [1,7,1,7], 11, ZZ) == ([5,6], [6], [1,7])
def inv(self, x):
s, t, h = gf_gcdex(x, self.reducing, self.p, ZZ)
return s
def gf_inv(p1,p2,m):
res = gf_gcdex(gf_strip(p2), p1, m, ZZ)
if res[2] != [1]:
print("Nie ma odwrotnego")
return res[0]
def gf_inv(a): # irriducible polynomial
# mod = 0x18f57 => x^16 + x^15 + x^11 + x^10 + x^9 + x^8 + x^6 + x^4 + x^2 + x^1 + 1 Polynome irreductible
mod = [1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1]
a = hextolist(a)
s, t, g = gf_gcdex(ZZ.map(gf_strip(a)), ZZ.map(mod), 2, ZZ)
return listtohex(s)
def inv(self, x: list) -> list:
s, t, h = gf_gcdex(x, self.reducing, self.p, ZZ)
return s
