def frobenius_trace(curve):
"""
Compute the trace of the Frobenius endomorphism for the given EllpiticCurve
@p curve.
This is an implementation of Schoof's original algorithm for counting the
points of an elliptic curve over a finite field.
@return The trace @f$ t @f$ of the Frobenius endomorphism. The number of
points on the curve then is @f$ q + 1 - t @f$, where @f$ q @f$
is the size of the finite field over which the curve was defined.
"""
# Initialize variables and parameters
trace_congruences = []
search_range = possible_frobenius_trace_range(curve.field())
upper_prime_bound = inverse_primorial(
len(search_range), shunned=curve.field().characteristic())
# Collect the congruence equations (avoid multivariate
# polynomial arithmetic by handling 2-torsion separately)
trace_congruences.append(frobenius_trace_mod_2(curve))
torsion_group = LTorsionGroup(curve)
for prime in primes_range(3, upper_prime_bound + 1):
if prime != curve.field().characteristic():
trace_congruences.append(
frobenius_trace_mod_l(torsion_group(prime)))
# Recover the unique valid trace representative
trace_congruence = solve_congruence_equations(trace_congruences)
return representative_in_range(trace_congruence, search_range)
def frobenius_trace(curve):
"""
Compute the trace of the Frobenius endomorphism for the given EllpiticCurve
@p curve.
This is an implementation of Schoof's original algorithm for counting the
points of an elliptic curve over a finite field.
@return The trace @f$ t @f$ of the Frobenius endomorphism. The number of
points on the curve then is @f$ q + 1 - t @f$, where @f$ q @f$
is the size of the finite field over which the curve was defined.
"""
trace_congruences = []
search_range = hasse_frobenius_trace_range(curve.field())
torsion_primes = greedy_prime_factors(len(search_range),
curve.field().characteristic())
# To avoid multivariate polynomial arithmetic, make l=2 a special case.
if 2 in torsion_primes:
trace_congruences.append(frobenius_trace_mod_2(curve))
torsion_primes.remove(2)
torsion_group = LTorsionGroup(curve)
for prime in torsion_primes:
trace_congruences.append(frobenius_trace_mod_l(torsion_group(prime)))
trace_congruence = solve_congruence_equations(trace_congruences)
return representative_in_range(trace_congruence, search_range)
def frobenius_trace(curve):
"""
Compute the trace of the Frobenius endomorphism for the given EllpiticCurve
@p curve.
This is an implementation of Schoof's original algorithm for counting the
points of an elliptic curve over a finite field.
@return The trace @f$ t @f$ of the Frobenius endomorphism. The number of
points on the curve then is @f$ q + 1 - t @f$, where @f$ q @f$
is the size of the finite field over which the curve was defined.
"""
trace_congruences = []
search_range = hasse_frobenius_trace_range( curve.field() )
torsion_primes = greedy_prime_factors(
len(search_range),
curve.field().characteristic()
)
# To avoid multivariate polynomial arithmetic, make l=2 a special case.
if 2 in torsion_primes:
trace_congruences.append( frobenius_trace_mod_2( curve ) )
torsion_primes.remove( 2 )
torsion_group = LTorsionGroup( curve )
for prime in torsion_primes:
trace_congruences.append(
frobenius_trace_mod_l( torsion_group( prime ) )
)
trace_congruence = solve_congruence_equations( trace_congruences )
return representative_in_range( trace_congruence, search_range )
def frobenius_trace(curve):
"""
Compute the trace of the Frobenius endomorphism for the given EllpiticCurve
@p curve.
This is an implementation of Schoof's original algorithm for counting the
points of an elliptic curve over a finite field.
@return The trace @f$ t @f$ of the Frobenius endomorphism. The number of
points on the curve then is @f$ q + 1 - t @f$, where @f$ q @f$
is the size of the finite field over which the curve was defined.
"""
# Initialize variables and parameters
trace_congruences = []
search_range = possible_frobenius_trace_range( curve.field() )
upper_prime_bound = inverse_primorial(
len(search_range),
shunned = curve.field().characteristic()
)
# Collect the congruence equations (avoid multivariate
# polynomial arithmetic by handling 2-torsion separately)
trace_congruences.append( frobenius_trace_mod_2( curve ) )
torsion_group = LTorsionGroup( curve )
for prime in primes_range( 3, upper_prime_bound+1 ):
if prime != curve.field().characteristic():
trace_congruences.append(
frobenius_trace_mod_l( torsion_group( prime ) )
)
# Recover the unique valid trace representative
trace_congruence = solve_congruence_equations(
trace_congruences
)
return representative_in_range( trace_congruence, search_range )
