def poly_prime_gcd(f,g,p):
"""
Algorithm 2.2.1 (gcd for polynomials), p. 90
from Crandall and Pomerance,
Prime Numbers, A Computational Perspective.
For given polynomials f(x),
g(x) in F[x], not both zero, this algorithm
returns d(x) = gcd(f(x),g(x)
"""
# 1. [initialize]
zero = poly1d([0])
if f.order < g.order or f == zero:
f,g = g,f
# 2. [Euclid loop]
while g != zero:
f,g = g, poly_prime_mod(f,g,p)
# 3. [Make Monic]
# take the leading coefficient.
flc = f.c[0]
# invert it.
c = inverse(flc,p)
# and multiply with f.
return reduce_prime_poly(f*c,p)
def poly_prime_divmod(f,g,p):
"""
f and g are polynomials in F_p[x]
return q,r such that
f = q*g + r
and r.order < g.order
"""
zero = poly1d([0])
x = poly1d([1,0])
d = f.order - g.order
if d < 0:
return zero, f
if g == zero:
raise ZeroDivisionError ("divmod(f,0) is undefined for all f")
# this is a case which requires some thought.
if f.order == 0:
flc,glc = f.c[0],g.c[0] # get leading terms
c = poly1d([inverse(glc,p)*flc % p])
c = reduce_prime_poly(c,p)
return c,zero
count = 0
while d >= 0 and f != zero:
flc,glc = f.c[0],g.c[0] # get leading terms
c = poly1d([inverse(glc,p)*flc % p])
term = c*(x**d)
# term*g has same degree and same leading coefficient as f.
# The following is calculated to reduce degree of f.
f = f - term*g
f = reduce_prime_poly(f,p)
count += term
d = f.order - g.order
count = reduce_prime_poly(count,p)
return count, f
def poly_prime_divmod(f, g, p):
"""
f and g are polynomials in F_p[x]
return q,r such that
f = q*g + r
and r.order < g.order
"""
zero = poly1d([0])
x = poly1d([1, 0])
d = f.order - g.order
if d < 0:
return zero, f
if g == zero:
raise ZeroDivisionError("divmod(f,0) is undefined for all f")
# this is a case which requires some thought.
if f.order == 0:
flc, glc = f.c[0], g.c[0] # get leading terms
c = poly1d([inverse(glc, p) * flc % p])
c = reduce_prime_poly(c, p)
return c, zero
count = 0
while d >= 0 and f != zero:
flc, glc = f.c[0], g.c[0] # get leading terms
c = poly1d([inverse(glc, p) * flc % p])
term = c * (x**d)
# term*g has same degree and same leading coefficient as f.
# The following is calculated to reduce degree of f.
f = f - term * g
f = reduce_prime_poly(f, p)
count += term
d = f.order - g.order
count = reduce_prime_poly(count, p)
return count, f
def poly_gcd_extended(f,g,p):
"""
Algorithm 2.2.2 (Extended gcd for polynomials)
p. 91 from Crandall and Pomerance,
Prime Numbers, a Computational Perspective.
Let F be a field. For given polynomials,
f(x), g(x) in F[x], not both zero, with
either deg f(x) >= deg g(x) or g(x) == 0,
this algorithm returns (s(x),t(x),d(x)) in
F[x] such that d = gcd (f,g) and sf + tg =
d. (For ease of notation, we may drop the
x argument).
"""
def rpp(f):
return reduce_prime_poly(f,p)
# 1. [initialize]
zero = poly1d([0])
one = poly1d([1])
f = reduce_prime_poly(f,p)
g = reduce_prime_poly(g,p)
swap = False
if f.order < g.order or f == zero:
f,g = g,f
swap = True
s,t,d,u,v,w = one,zero,f,zero,one,g
# 2. [Extended Euclid loop]
while w != zero:
q,r = poly_prime_divmod(d,w,p)
s,t,d,u,v,w = u,v,w,rpp(s-q*u), rpp(t-q*v), r
# 3. [Make Monic]
# take the leading coefficient of d.
# invert it.
c = poly1d([inverse(d.c[0],p)])
if swap:
s,t = t,s
s,t,d = rpp(c*s), rpp(c*t), rpp(c*d)
return s,t,d
def poly_gcd_extended(f, g, p):
"""
Algorithm 2.2.2 (Extended gcd for polynomials)
p. 91 from Crandall and Pomerance,
Prime Numbers, a Computational Perspective.
Let F be a field. For given polynomials,
f(x), g(x) in F[x], not both zero, with
either deg f(x) >= deg g(x) or g(x) == 0,
this algorithm returns (s(x),t(x),d(x)) in
F[x] such that d = gcd (f,g) and sf + tg =
d. (For ease of notation, we may drop the
x argument).
"""
def rpp(f):
return reduce_prime_poly(f, p)
# 1. [initialize]
zero = poly1d([0])
one = poly1d([1])
f = reduce_prime_poly(f, p)
g = reduce_prime_poly(g, p)
swap = False
if f.order < g.order or f == zero:
f, g = g, f
swap = True
s, t, d, u, v, w = one, zero, f, zero, one, g
# 2. [Extended Euclid loop]
while w != zero:
q, r = poly_prime_divmod(d, w, p)
s, t, d, u, v, w = u, v, w, rpp(s - q * u), rpp(t - q * v), r
# 3. [Make Monic]
# take the leading coefficient of d.
# invert it.
c = poly1d([inverse(d.c[0], p)])
if swap:
s, t = t, s
s, t, d = rpp(c * s), rpp(c * t), rpp(c * d)
return s, t, d
def make_monic(f,p):
if type(f) == int:
f = poly1d([f])
f = reduce_prime_poly(f,p)
r = deque(f.c.tolist())
# multiply each term by inverse of leading coefficient
factor = inverse(f.c[0],p)
l = []
while r:
# take c off of the right
c = r.popleft()
# reduce it mod p and append it to the left.
l.append(int(factor * c)%p)
# return a monic poly
return poly1d(l)