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.
"""
# 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 test_empty(self):
"""Empty (too small) input"""
self.assert_(inverse_primorial(-1) == 2)
def test_results(self):
"""Sample results"""
self.assert_(inverse_primorial(30) == 5)
self.assert_(inverse_primorial(31) == 7)
def test_empty(self):
"""Empty (too small) input"""
self.assert_( inverse_primorial(-1) == 2 )
def test_results(self):
"""Sample results"""
self.assert_( inverse_primorial(30) == 5 )
self.assert_( inverse_primorial(31) == 7 )
