def frobenius_trace_mod_l(torsion_group):
"""
Compute the trace of the Frobenius endomorphism modulo @f$ l @f$, where
@f$ l @f$ is the torsion of @p torsion_group.
The function guesses candidates and verifies the frobenius_equation() for
every point in the @p torsion_group.
@note A torsion of 2 requires multivariate polynomial arithmetic,
which is unavailable. Therefore @f$ l @f$ must be greater than 2.
Use frobenius_trace_mod_2() to get the trace modulo 2.
@return The congruence class of the trace of the Frobenius endomorphism.
This is an element of QuotientRing( Integers, l ).
"""
assert torsion_group.torsion() > 2, \
"torsion 2 requires multivariate polynomial arithmetic"
ints_mod_torsion = QuotientRing(Integers, torsion_group.torsion())
field_size = torsion_group.curve().field().size()
for trace_candidate in range(0, torsion_group.torsion()):
candidate_congruence = ints_mod_torsion(trace_candidate)
for point in torsion_group.elements():
if not frobenius_equation(candidate_congruence, field_size, point):
# Exit inner loop and skip the 'else' clause.
break
else:
# Execute after the iteration completed; skip upon break.
return candidate_congruence
message = "Frobenius equation held for no trace candidate"
raise ArithmeticError(message)
def __init_point(self):
"""
Create the point that implicitly represents the whole l-torsion group
and store it in the cache.
Calling the method has no effect if the point already exists.
"""
if self.__point:
return
# R = F[x] / (y**2 - x**3 - A*x - B)
R = self._division_polynomial_list().curve_polynomials()
# The n-th division polynomial
psi = self._division_polynomial_list()[self.__torsion]
# T = ( F[x] / (y**2 - x**3 - A*x - B) ) / psi(l)
S = QuotientRing(R, psi)
T = FractionField(S)
A, B = R.curve().parameters()
# Polynomials x and y on the curve interpreted
# as elements of the field of fractions
x = T(R((0, 1), 0))
y = T(R(0, 1))
self.__point = EllipticCurve(T, A, B)(x, y)
def frobenius_trace_mod_2(curve):
"""
Compute the trace of the Frobenius endomorphism modulo 2.
We cannot use the implicit torsion group representation of
frobenius_trace_mod_l() in the case @f$ l=2 @f$ because the second division
polynomial is @f$ y @f$, and multivariate polynomial arithmetic is
unavailable. Implementing it for this single case would be overkill.
Instead, we test whether there are any rational 2-torsion points. If so,
then @f$ E[2] @f$ is a subgroup of the @p curve and its order divides the
order of the curve (which is the number of rational points). Now, the
number of rational points in @f$ E[2] @f$ can only be 2 or 4 in this case, so
we know that the number of rational points is even. Hence, the Frobenius
trace must then be 0 modulo 2.
@return 0 if the number of points on the curve is even, 1 otherwise.
The result is a congruence class modulo 2, that is, an element
of @c QuotientRing( Integers, 2 ).
"""
R = Polynomials(curve.field())
x = R(0, 1)
A, B = curve.parameters()
defining_polynomial = x**3 + A * x + B
rational_characteristic = x**curve.field().size() - x
# gcd() has an arbitrary unit as leading coefficient;
# relatively prime polynomials have a constant gcd.
d = gcd(rational_characteristic, defining_polynomial)
if d.degree() == 0:
# The rational characteristic and the defining polynomial
# are relatively prime: no rational point of order 2 exists
# and the Frobenius trace must be odd.
return QuotientRing(Integers, 2)(1)
else:
return QuotientRing(Integers, 2)(0)
def solve_congruence_equations(congruences):
"""
Return a quotient class that simultaneously solves the list of
@p congruences; this is the Chinese Remainder Theorem.
All representatives of the returned congruence have the remainders of
@p congruences when taken modulo the respective moduli. The result is a
congruence modulo the product of all moduli. Thus the function returns a
number @f$ z \mod \prod_{i} m_i @f$ such that
@f{align*}{
z &\equiv a_1 \mod m_1 \\
\vdots \\
z &\equiv a_k \mod m_k
@f}
@note The moduli @f$ m_i @f$ of all congruences must be relatively prime.
@exception ValueError if @p congruences is empty.
@param congruences An iterable of objects of QuotientClass over the
Integers. Every pair of two different moduli
must have a greatest common divisor (gcd()) of 1.
@return An instance of rings.quotients.naive.QuotientRing over the
rings.integers.naive.Integers solving the @p congruences.
@see Robinson, D. J. S., "An Introduction to Abstract Algebra", p. 27
"""
# The Chinese remainder theorem
if not congruences:
raise ValueError("cannot solve empty equation system")
if __debug__:
# This test is expensive; remove it in optimized execution
pairs = [(c1, c2)
for c1 in congruences for c2 in congruences if c1 != c2]
pairwise_gcds = [gcd(c1.modulus(), c2.modulus()) for c1, c2 in pairs]
assert set(pairwise_gcds) == set([1]), \
"the Chinese Remainder Theorem requires relatively prime moduli"
common_modulus = reduce(mul, [c.modulus() for c in congruences])
common_representative = 0
for c in congruences:
neutralizer = common_modulus // c.modulus()
common_representative += c.remainder() * neutralizer \
* inverse_modulo(neutralizer, c.modulus())
quotient_ring = QuotientRing(Integers, common_modulus)
return quotient_ring(common_representative)
def frobenius_trace_mod_l(torsion_group):
"""
Compute the trace of the Frobenius endomorphism modulo @f$ l @f$, where
@f$ l @f$ is the torsion of @p torsion_group.
The function guesses candidates and verifies whether the function that
results from applying the characteristic polynomial @f$ \chi_\phi @f$
to @f$ \phi @f$ maps every point in the @p torsion_group onto the point
at infinity.
@note A torsion of 2 requires multivariate polynomial arithmetic,
which is unavailable. Therefore @f$ l @f$ must be greater than 2.
Use frobenius_trace_mod_2() to get the trace modulo 2.
@return The congruence class of the trace of the Frobenius endomorphism.
This is an element of @c QuotientRing( Integers, l ).
"""
assert torsion_group.torsion() > 2, \
"torsion 2 requires multivariate polynomial arithmetic"
torsion_quotient_ring = QuotientRing(Integers, torsion_group.torsion())
field_size = torsion_group.curve().field().size()
# Note: Technically, there could be several points so we would have to
# filter the one candidate that worked for all points in the end.
# Luckily, there is only one point.
for point in torsion_group.elements():
frobenius_point = frobenius(point, field_size)
frobenius2_point = frobenius(frobenius_point, field_size)
determinant_point = (field_size % torsion_group.torsion()) * point
point_sum = frobenius2_point + determinant_point
if point_sum.is_infinite():
return torsion_quotient_ring(0)
trace_point = frobenius_point
for trace_candidate in range(1, (torsion_group.torsion() + 1) // 2):
if point_sum.x() == trace_point.x():
if point_sum.y() == trace_point.y():
return torsion_quotient_ring(trace_candidate)
else:
return torsion_quotient_ring(-trace_candidate)
else:
trace_point += frobenius_point
message = "Frobenius equation held for no trace candidate"
raise ArithmeticError(message)
def setUp(self):
Z3 = QuotientRing(Integers, 3)
Z5 = QuotientRing(Integers, 5)
self._remainders = [Z3(2), Z5(1)]