def gfq_gap_to_sage(x, F):
"""
INPUT:
- ``x`` - gap finite field element
- ``F`` - Sage finite field
OUTPUT: element of F
EXAMPLES::
sage: x = gap('Z(13)')
sage: F = GF(13, 'a')
sage: F(x)
2
sage: F(gap('0*Z(13)'))
0
sage: F = GF(13^2, 'a')
sage: x = gap('Z(13)')
sage: F(x)
2
sage: x = gap('Z(13^2)^3')
sage: F(x)
12*a + 11
sage: F.multiplicative_generator()^3
12*a + 11
AUTHOR:
- David Joyner and William Stein
"""
from sage.rings.finite_rings.constructor import FiniteField
s = str(x)
if s[:2] == '0*':
return F(0)
i1 = s.index("(")
i2 = s.index(")")
q = eval(s[i1+1:i2].replace('^','**'))
if q == F.order():
K = F
else:
K = FiniteField(q, F.variable_name())
if s.find(')^') == -1:
e = 1
else:
e = int(s[i2+2:])
if F.degree() == 1:
g = int(gap.eval('Int(Z(%s))'%q))
else:
g = K.multiplicative_generator()
return F(K(g**e))
def gfq_gap_to_sage(x, F):
"""
INPUT:
- ``x`` - gap finite field element
- ``F`` - Sage finite field
OUTPUT: element of F
EXAMPLES::
sage: x = gap('Z(13)')
sage: F = GF(13, 'a')
sage: F(x)
2
sage: F(gap('0*Z(13)'))
0
sage: F = GF(13^2, 'a')
sage: x = gap('Z(13)')
sage: F(x)
2
sage: x = gap('Z(13^2)^3')
sage: F(x)
12*a + 11
sage: F.multiplicative_generator()^3
12*a + 11
AUTHOR:
- David Joyner and William Stein
"""
from sage.rings.finite_rings.constructor import FiniteField
s = str(x)
if s[:2] == '0*':
return F(0)
i1 = s.index("(")
i2 = s.index(")")
q = eval(s[i1 + 1:i2].replace('^', '**'))
if q == F.order():
K = F
else:
K = FiniteField(q, F.variable_name())
if s.find(')^') == -1:
e = 1
else:
e = int(s[i2 + 2:])
if F.degree() == 1:
g = int(gap.eval('Int(Z(%s))' % q))
else:
g = K.multiplicative_generator()
return F(K(g**e))
def polynomial(self, n):
r"""
Return the pseudo-Conway polynomial of degree `n` in this
lattice.
INPUT:
- ``n`` -- positive integer
OUTPUT:
- a pseudo-Conway polynomial of degree `n` for the prime `p`.
ALGORITHM:
Uses an algorithm described in [HL99]_, modified to find
pseudo-Conway polynomials rather than Conway polynomials. The
major difference is that we stop as soon as we find a
primitive polynomial.
REFERENCE:
.. [HL99] L. Heath and N. Loehr (1999). New algorithms for
generating Conway polynomials over finite fields.
Proceedings of the tenth annual ACM-SIAM symposium on
discrete algorithms, pp. 429-437.
EXAMPLES::
sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
sage: PCL = PseudoConwayLattice(2, use_database=False)
sage: PCL.polynomial(3)
x^3 + x + 1
sage: PCL.polynomial(4)
x^4 + x^3 + 1
sage: PCL.polynomial(60)
x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
"""
if n in self.nodes:
return self.nodes[n]
p = self.p
if n == 1:
f = self.ring.gen() - FiniteField(p).multiplicative_generator()
self.nodes[1] = f
return f
# Work in an arbitrary field K of order p**n.
K = FiniteField(p**n, names='a')
# TODO: something like the following
# gcds = [n.gcd(d) for d in self.nodes.keys()]
# xi = { m: (...) for m in gcds }
xi = {q: self.polynomial(n//q).any_root(K, -n//q, assume_squarefree=True)
for q in n.prime_divisors()}
# The following is needed to ensure that in the concrete instantiation
# of the "new" extension all previous choices are compatible.
_frobenius_shift(K, xi)
# Construct a compatible element having order the lcm of orders
q, x = xi.popitem()
v = p**(n//q) - 1
for q, xitem in xi.iteritems():
w = p**(n//q) - 1
g, alpha, beta = v.xgcd(w)
x = x**beta * xitem**alpha
v = v.lcm(w)
r = p**n - 1
# Get the missing part of the order to be primitive
g = r // v
# Iterate through g-th roots of x until a primitive one is found
z = x.nth_root(g)
root = K.multiplicative_generator()**v
while z.multiplicative_order() != r:
z *= root
# The following should work but tries to create a huge list
# whose length overflows Python's ints for large parameters
#Z = x.nth_root(g, all=True)
#for z in Z:
# if z.multiplicative_order() == r:
# break
f = z.minimal_polynomial()
self.nodes[n] = f
return f
def polynomial(self, n):
r"""
Return the pseudo-Conway polynomial of degree `n` in this
lattice.
INPUT:
- ``n`` -- positive integer
OUTPUT:
- a pseudo-Conway polynomial of degree `n` for the prime `p`.
ALGORITHM:
Uses an algorithm described in [HL99]_, modified to find
pseudo-Conway polynomials rather than Conway polynomials. The
major difference is that we stop as soon as we find a
primitive polynomial.
REFERENCE:
.. [HL99] L. Heath and N. Loehr (1999). New algorithms for
generating Conway polynomials over finite fields.
Proceedings of the tenth annual ACM-SIAM symposium on
discrete algorithms, pp. 429-437.
EXAMPLES::
sage: from sage.rings.finite_rings.conway_polynomials import PseudoConwayLattice
sage: PCL = PseudoConwayLattice(2, use_database=False)
sage: PCL.polynomial(3)
x^3 + x + 1
sage: PCL.polynomial(4)
x^4 + x^3 + 1
sage: PCL.polynomial(60)
x^60 + x^59 + x^58 + x^55 + x^54 + x^53 + x^52 + x^51 + x^48 + x^46 + x^45 + x^42 + x^41 + x^39 + x^38 + x^37 + x^35 + x^32 + x^31 + x^30 + x^28 + x^24 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^10 + x^8 + x^7 + x^5 + x^3 + x^2 + x + 1
"""
if n in self.nodes:
return self.nodes[n]
p = self.p
if n == 1:
f = self.ring.gen() - FiniteField(p).multiplicative_generator()
self.nodes[1] = f
return f
# Work in an arbitrary field K of order p**n.
K = FiniteField(p**n, names='a')
# TODO: something like the following
# gcds = [n.gcd(d) for d in self.nodes.keys()]
# xi = { m: (...) for m in gcds }
xi = {q: self.polynomial(n//q).any_root(K, -n//q, assume_squarefree=True)
for q in n.prime_divisors()}
# The following is needed to ensure that in the concrete instantiation
# of the "new" extension all previous choices are compatible.
_frobenius_shift(K, xi)
# Construct a compatible element having order the lcm of orders
q, x = xi.popitem()
v = p**(n//q) - 1
for q, xitem in xi.iteritems():
w = p**(n//q) - 1
g, alpha, beta = v.xgcd(w)
x = x**beta * xitem**alpha
v = v.lcm(w)
r = p**n - 1
# Get the missing part of the order to be primitive
g = r // v
# Iterate through g-th roots of x until a primitive one is found
z = x.nth_root(g)
root = K.multiplicative_generator()**v
while z.multiplicative_order() != r:
z *= root
# The following should work but tries to create a huge list
# whose length overflows Python's ints for large parameters
#Z = x.nth_root(g, all=True)
#for z in Z:
# if z.multiplicative_order() == r:
# break
f = z.minimal_polynomial()
self.nodes[n] = f
return f
