def _find_pow_of_frobenius(p, n, x, y):
"""
Find the power of Frobenius which yields `x` when applied to `y`.
INPUT:
- ``p`` -- prime number
- ``n`` -- positive integer
- ``x`` -- an element of a field `K` of `p^n` elements so that
the multiplicative order of `x` is `p^n - 1`.
- ``y`` -- an element of `K` with the same minimal polynomial as
`x`.
OUTPUT:
- an element `i` of the integers modulo `n` such that `x = y^{p^i}`.
EXAMPLES::
sage: from sage.rings.finite_rings.conway_polynomials import _find_pow_of_frobenius
sage: K.<a> = GF(3^14)
sage: x = K.multiplicative_generator()
sage: y = x^27
sage: _find_pow_of_frobenius(3, 14, x, y)
11
"""
from integer_mod import mod
for i in xrange(n):
if x == y: break
y = y**p
else:
raise RuntimeError("No appropriate power of Frobenius found")
return mod(i, n)
def _find_pow_of_frobenius(p, n, x, y):
"""
Find the power of Frobenius which yields `x` when applied to `y`.
INPUT:
- ``p`` -- prime number
- ``n`` -- positive integer
- ``x`` -- an element of a field `K` of `p^n` elements so that
the multiplicative order of `x` is `p^n - 1`.
- ``y`` -- an element of `K` with the same minimal polynomial as
`x`.
OUTPUT:
- an element `i` of the integers modulo `n` such that `x = y^{p^i}`.
EXAMPLES::
sage: from sage.rings.finite_rings.conway_polynomials import _find_pow_of_frobenius
sage: K.<a> = GF(3^14)
sage: x = K.multiplicative_generator()
sage: y = x^27
sage: _find_pow_of_frobenius(3, 14, x, y)
11
"""
from integer_mod import mod
for i in xrange(n):
if x == y: break
y = y**p
else:
raise RuntimeError("No appropriate power of Frobenius found")
return mod(i, n)
def _frobenius_shift(K, generators, check_only=False):
"""
Given a field `K` of degree `n` over ``GF(p)`` and a dictionary
holding, for each divisor `q` of `n`, an element with minimal
polynomial a pseudo-Conway polynomial of degree `n/q`, modify
these generators into a compatible system.
Such a system of generators is said to be compatible if for each
pair of prime divisors `q_1` and `q_2` and each common divisor `m`
of `n/q_1` and `n/q_2`, the equality
``generators[q1]^((p^(n/q1)-1)/(p^m-1)) == generators[q2]^((p^(n/q2)-1)/(p^m-1))``
holds.
INPUT:
- ``K`` -- a finite field of degree `n` over its prime field
- ``generators`` -- a dictionary, indexed by prime divisors `q` of
`n`, whose entries are elements of `K` satisfying the `n/q`
pseudo-Conway polynomial.
- ``check_only`` -- if ``True``, just check that the given
generators form a compatible system.
EXAMPLES::
sage: R.<x> = GF(2)[]
sage: f30 = x^30 + x^28 + x^27 + x^25 + x^24 + x^20 + x^19 + x^18 + x^16 + x^15 + x^12 + x^10 + x^7 + x^2 + 1
sage: f20 = x^20 + x^19 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + x^2 + x + 1
sage: f12 = x^12 + x^10 + x^9 + x^8 + x^4 + x^2 + 1
sage: K.<a> = GF(2^60, modulus='first_lexicographic')
sage: x30 = f30.any_root(K)
sage: x20 = f20.any_root(K)
sage: x12 = f12.any_root(K)
sage: generators = {2: x30, 3: x20, 5: x12}
sage: from sage.rings.finite_rings.conway_polynomials import _frobenius_shift, _find_pow_of_frobenius
sage: _frobenius_shift(K, generators)
sage: _find_pow_of_frobenius(2, 30, x30, generators[2])
0
sage: _find_pow_of_frobenius(2, 20, x20, generators[3])
13
sage: _find_pow_of_frobenius(2, 12, x12, generators[5])
8
"""
if len(generators) == 1:
return generators
p = K.characteristic()
n = K.degree()
compatible = {}
from integer_mod import mod
for m in n.divisors():
compatible[m] = {}
for q, x in generators.iteritems():
for m in (n//q).divisors():
compatible[m][q] = x**((p**(n//q)-1)//(p**m-1))
if check_only:
for m in n.divisors():
try:
q, x = compatible[m].popitem()
except KeyError:
break
for qq, xx in compatible[m].iteritems():
assert x == xx
return
crt = {}
qlist = sorted(generators.keys())
for j in range(1, len(qlist)):
for i in range(j):
crt[(i, j)] = []
for m in n.divisors():
mqlist = sorted(compatible[m].keys())
for k in range(1,len(mqlist)):
j = qlist.index(mqlist[k])
i = qlist.index(mqlist[k-1])
crt[(i,j)].append(_find_pow_of_frobenius(p, m, compatible[m][qlist[j]], compatible[m][qlist[i]]))
from integer_mod import mod
pairs = crt.keys()
for i, j in pairs:
L = crt[(i,j)]
running = mod(0,1)
for a in L:
running = _crt_non_coprime(running, a)
crt[(i,j)] = [(mod(running, q**(running.modulus().valuation(q))), running.modulus().valuation(q)) for q in qlist]
crt[(j,i)] = [(-a, level) for a, level in crt[(i,j)]]
# Let x_j be the power of Frobenius we apply to generators[qlist[j]], for 0 < j < len(qlist)
# We have some direct conditions on the x_j: x_j reduces to each entry in crt[(0,j)].
# But we also have the equations x_j - x_i reduces to each entry in crt[(i,j)].
# We solve for x_j one prime at a time. For each prime, we have an equations of the form
# x_j - x_i = c_ij. The modulus of the currently known value of x_j, x_i and c_ij will all be powers
# (possibly 0, possibly different) of the same prime.
# We can set x_0=0 everywhere, can get an initial setting of x_j from the c_0j.
# We go through prime by prime.
import bisect
frob_powers=[mod(0,1) for q in qlist]
def find_leveller(qindex, level, x, xleveled, searched, i):
searched[i] = True
crt_possibles = []
for j in range(1,len(qlist)):
if i==j: continue
if crt[(i,j)][qindex][1] >= level:
if xleveled[j]:
return [j]
elif j not in searched:
crt_possibles.append(j)
for j in crt_possibles:
path = find_leveller(qindex, level, x, xleveled, searched, j)
if path is not None:
path.append(j)
return path
return None
def propagate_levelling(qindex, level, x, xleveled, i):
for j in range(1, len(qlist)):
if i==j: continue
if not xleveled[j] and crt[(i,j)][qindex][1] >= level:
newxj = x[i][0] + crt[(i,j)][qindex][0]
x[j] = (newxj, min(x[i][1], crt[(i,j)][qindex][1]))
xleveled[j] = True
propagate_levelling(qindex, level, x, xleveled, j)
for qindex in range(len(qlist)):
q = qlist[qindex]
# We include the initial 0 to match up our indexing with crt.
x = [0] + [crt[(0,j)][qindex] for j in range(1,len(qlist))]
# We first check that our equations are consistent and
# determine which powers of q occur as moduli.
levels = []
for j in range(2, len(qlist)):
for i in range(j):
# we need crt[(0,j)] = crt[(0,i)] + crt[(i,j)]
if i != 0:
assert x[j][0] == x[i][0] + crt[(i,j)][qindex][0]
level = crt[(i,j)][qindex][1]
if level > 0:
ins = bisect.bisect_left(levels,level)
if ins == len(levels):
levels.append(level)
elif levels[ins] != level:
levels.insert(ins, level)
for level in levels:
xleveled = [0] + [x[i][1] >= level for i in range(1,len(qlist))]
while True:
try:
i = xleveled.index(False, 1)
searched = {}
levelling_path = find_leveller(qindex, level, x, xleveled, searched, i)
if levelling_path is None:
# Any lift will work, since there are no constraints.
x[i] = (mod(x[i][0].lift(), q**level), level)
xleveled[i] = True
propagate_levelling(qindex, level, x, xleveled, i)
else:
levelling_path.append(i)
for m in range(1,len(path)):
# This point on the path may have already
# been leveled in a previous propagation.
if not xleveled[path[m]]:
newx = x[path[m-1]][0] + crt[(path[m-1],path[m])][qindex][0]
x[path[m]] = (newx, min(x[path[m-1]][1], crt[(path[m-1],path[m])][qindex][1]))
xleveled[path[m]] = True
propagate_levelling(qindex, level, x, xleveled, path[m])
except ValueError:
break
for j in range(1,len(qlist)):
frob_powers[j] = frob_powers[j].crt(x[j][0])
for j in range(1, len(qlist)):
generators[qlist[j]] = generators[qlist[j]]**(p**(-frob_powers[j]).lift())
_frobenius_shift(K, generators, check_only=True)
def _frobenius_shift(K, generators, check_only=False):
"""
Given a field `K` of degree `n` over ``GF(p)`` and a dictionary
holding, for each divisor `q` of `n`, an element with minimal
polynomial a pseudo-Conway polynomial of degree `n/q`, modify
these generators into a compatible system.
Such a system of generators is said to be compatible if for each
pair of prime divisors `q_1` and `q_2` and each common divisor `m`
of `n/q_1` and `n/q_2`, the equality
``generators[q1]^((p^(n/q1)-1)/(p^m-1)) == generators[q2]^((p^(n/q2)-1)/(p^m-1))``
holds.
INPUT:
- ``K`` -- a finite field of degree `n` over its prime field
- ``generators`` -- a dictionary, indexed by prime divisors `q` of
`n`, whose entries are elements of `K` satisfying the `n/q`
pseudo-Conway polynomial.
- ``check_only`` -- if ``True``, just check that the given
generators form a compatible system.
EXAMPLES::
sage: R.<x> = GF(2)[]
sage: f30 = x^30 + x^28 + x^27 + x^25 + x^24 + x^20 + x^19 + x^18 + x^16 + x^15 + x^12 + x^10 + x^7 + x^2 + 1
sage: f20 = x^20 + x^19 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + x^2 + x + 1
sage: f12 = x^12 + x^10 + x^9 + x^8 + x^4 + x^2 + 1
sage: K.<a> = GF(2^60, modulus='first_lexicographic')
sage: x30 = f30.any_root(K)
sage: x20 = f20.any_root(K)
sage: x12 = f12.any_root(K)
sage: generators = {2: x30, 3: x20, 5: x12}
sage: from sage.rings.finite_rings.conway_polynomials import _frobenius_shift, _find_pow_of_frobenius
sage: _frobenius_shift(K, generators)
sage: _find_pow_of_frobenius(2, 30, x30, generators[2])
0
sage: _find_pow_of_frobenius(2, 20, x20, generators[3])
13
sage: _find_pow_of_frobenius(2, 12, x12, generators[5])
8
"""
if len(generators) == 1:
return generators
p = K.characteristic()
n = K.degree()
compatible = {}
from integer_mod import mod
for m in n.divisors():
compatible[m] = {}
for q, x in generators.iteritems():
for m in (n//q).divisors():
compatible[m][q] = x**((p**(n//q)-1)//(p**m-1))
if check_only:
for m in n.divisors():
try:
q, x = compatible[m].popitem()
except KeyError:
break
for qq, xx in compatible[m].iteritems():
assert x == xx
return
crt = {}
qlist = sorted(generators.keys())
for j in range(1, len(qlist)):
for i in range(j):
crt[(i, j)] = []
for m in n.divisors():
mqlist = sorted(compatible[m].keys())
for k in range(1,len(mqlist)):
j = qlist.index(mqlist[k])
i = qlist.index(mqlist[k-1])
crt[(i,j)].append(_find_pow_of_frobenius(p, m, compatible[m][qlist[j]], compatible[m][qlist[i]]))
from integer_mod import mod
pairs = crt.keys()
for i, j in pairs:
L = crt[(i,j)]
running = mod(0,1)
for a in L:
running = _crt_non_coprime(running, a)
crt[(i,j)] = [(mod(running, q**(running.modulus().valuation(q))), running.modulus().valuation(q)) for q in qlist]
crt[(j,i)] = [(-a, level) for a, level in crt[(i,j)]]
# Let x_j be the power of Frobenius we apply to generators[qlist[j]], for 0 < j < len(qlist)
# We have some direct conditions on the x_j: x_j reduces to each entry in crt[(0,j)].
# But we also have the equations x_j - x_i reduces to each entry in crt[(i,j)].
# We solve for x_j one prime at a time. For each prime, we have an equations of the form
# x_j - x_i = c_ij. The modulus of the currently known value of x_j, x_i and c_ij will all be powers
# (possibly 0, possibly different) of the same prime.
# We can set x_0=0 everywhere, can get an initial setting of x_j from the c_0j.
# We go through prime by prime.
import bisect
frob_powers=[mod(0,1) for q in qlist]
def find_leveller(qindex, level, x, xleveled, searched, i):
searched[i] = True
crt_possibles = []
for j in range(1,len(qlist)):
if i==j: continue
if crt[(i,j)][qindex][1] >= level:
if xleveled[j]:
return [j]
elif j not in searched:
crt_possibles.append(j)
for j in crt_possibles:
path = find_leveller(qindex, level, x, xleveled, searched, j)
if path is not None:
path.append(j)
return path
return None
def propagate_levelling(qindex, level, x, xleveled, i):
for j in range(1, len(qlist)):
if i==j: continue
if not xleveled[j] and crt[(i,j)][qindex][1] >= level:
newxj = x[i][0] + crt[(i,j)][qindex][0]
x[j] = (newxj, min(x[i][1], crt[(i,j)][qindex][1]))
xleveled[j] = True
propagate_levelling(qindex, level, x, xleveled, j)
for qindex in range(len(qlist)):
q = qlist[qindex]
# We include the initial 0 to match up our indexing with crt.
x = [0] + [crt[(0,j)][qindex] for j in range(1,len(qlist))]
# We first check that our equations are consistent and
# determine which powers of q occur as moduli.
levels = []
for j in range(2, len(qlist)):
for i in range(j):
# we need crt[(0,j)] = crt[(0,i)] + crt[(i,j)]
if i != 0:
assert x[j][0] == x[i][0] + crt[(i,j)][qindex][0]
level = crt[(i,j)][qindex][1]
if level > 0:
ins = bisect.bisect_left(levels,level)
if ins == len(levels):
levels.append(level)
elif levels[ins] != level:
levels.insert(ins, level)
for level in levels:
xleveled = [0] + [x[i][1] >= level for i in range(1,len(qlist))]
while True:
try:
i = xleveled.index(False, 1)
searched = {}
levelling_path = find_leveller(qindex, level, x, xleveled, searched, i)
if levelling_path is None:
# Any lift will work, since there are no constraints.
x[i] = (mod(x[i][0].lift(), q**level), level)
xleveled[i] = True
propagate_levelling(qindex, level, x, xleveled, i)
else:
levelling_path.append(i)
for m in range(1,len(path)):
# This point on the path may have already
# been leveled in a previous propagation.
if not xleveled[path[m]]:
newx = x[path[m-1]][0] + crt[(path[m-1],path[m])][qindex][0]
x[path[m]] = (newx, min(x[path[m-1]][1], crt[(path[m-1],path[m])][qindex][1]))
xleveled[path[m]] = True
propagate_levelling(qindex, level, x, xleveled, path[m])
except ValueError:
break
for j in range(1,len(qlist)):
frob_powers[j] = frob_powers[j].crt(x[j][0])
for j in range(1, len(qlist)):
generators[qlist[j]] = generators[qlist[j]]**(p**(-frob_powers[j]).lift())
_frobenius_shift(K, generators, check_only=True)
