def pinch_method(p, n, m, f = None, g = None):
'''
Function that given three integers p, n and m such as :
- p is prime,
- m is dividing n and there exists a primitive mth root that spans F_{p^n}
and none of its subfields,
returns the image, in a finite field G, of the polynomial generator of a
finite field F with the same cardinality as G.
'''
c, w = cputime(), walltime()
R = PolynomialRing(GF(p), 'X')
# If no polynomials are given, we compute both of them randomly.
if f is None:
f = R.irreducible_element(n, algorithm='random')
if g is None:
g = R.irreducible_element(n, algorithm='random')
while f == g:
g = R.irreductible_element(n, algorithm='random')
# We compute two fields of cardinality p^n and two primitive m-rooth
rootmf, rootmg, F, G = find_mroots_and_fields(p, n, m, f, g)
# The matrixes will contain the coefficients of rootmf and rootmg in the
# basis x^i and y^i respectively
A = matrix(GF(p), n, n)
B = matrix(GF(p), n, n)
for i in range(n):
A[i,:] = (rootmf**i).vector()
# Failsafe, but probably outdated
try:
Ainv = A.inverse()
except ZeroDivisionError:
print 'erreur'
return A
# We will try to find the power s such as phi(rootmf) = rootmg^s, since
# rootmg and rootmf are both primitive mrooth it is bound to happen
# since the multiplicative group is cyclic.
s = 1
while s <= m :
for i in range(n):
B[i,:] = ((rootmg**s)**i).vector()
# This will be the isomorphism's matrix
C = Ainv*B
v = C[1,:] # The second line correponds to the image of x
res = G(v[0])
# I realized that you could try to find if the image of
# rootmf is also a zero of the minimal polynomial of rootmf
# but it would force us to compute yet another minimal polynomials.
# Instead, if you find that the image of x is a root of f,
# then you win!
if f(res) == 0:
print 'CPU %s, Wall %s' % (cputime(c), walltime(w))
# Some of what is returned is probably useless and only here
# for testing purposes.
return (res, C, rootmf, rootmg, s, f, F, G)
s = s + 1
print 'No isomorphism found, check your m.'
return 1
