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 greedy_prime_factors(n, shunned=0):
"""
Return a list of the first primes whose product is greater than, or equal
to @p n, but do not use @p shunned.
For example, if @p n is 14, then the returned list will consist of 3 and
5, but not 2, because 3 times 5 is greater than 14. The function behaves
like inverse_primorial() except that it removes unnecessary smaller primes.
@note Canceling of unnecessary primes follows a greedy algorithm.
Therefore the choice of primes might be suboptimal; perfect
choice, however, is an NP-complete problem (KNAPSACK).
@note This function uses primes_range() to obtain a list of primes.
See the notes there for use case limitations.
"""
primes = primes_range(2, n + 1)
# Find the smallest product of primes that is at least n, but don't use
# the shunned prime.
product = 1
for index, prime in enumerate(primes):
if prime != shunned:
product *= prime
if product >= n:
break
# Throw away excess primes
primes = primes[:index + 1]
if shunned in primes:
primes.remove(shunned)
# Try to cancel unnecessary primes, largest first.
# (This greedy search is not optimal; however, we did not set out to solve
# the KNAPSACK problem, did we?)
for index, prime in enumerate(reversed(primes)):
canceled_product = product / prime
if canceled_product >= n:
product = canceled_product
primes[-(index + 1)] = 0
return list(filter(None, primes))
def greedy_prime_factors(n, shunned=0):
"""
Return a list of the first primes whose product is greater than, or equal
to @p n, but do not use @p shunned.
For example, if @p n is 14, then the returned list will consist of 3 and
5, but not 2, because 3 times 5 is greater than 14. The function behaves
like inverse_primorial() except that it removes unnecessary smaller primes.
@note Canceling of unnecessary primes follows a greedy algorithm.
Therefore the choice of primes might be suboptimal; perfect
choice, however, is an NP-complete problem (KNAPSACK).
@note This function uses primes_range() to obtain a list of primes.
See the notes there for use case limitations.
"""
primes = primes_range( 2, n+1 )
# Find the smallest product of primes that is at least n, but don't use
# the shunned prime.
product = 1
for index, prime in enumerate( primes ):
if prime != shunned:
product *= prime
if product >= n:
break
# Throw away excess primes
primes = primes[ : index+1 ]
if shunned in primes:
primes.remove( shunned )
# Try to cancel unnecessary primes, largest first.
# (This greedy search is not optimal; however, we did not set out to solve
# the KNAPSACK problem, did we?)
for index, prime in enumerate( reversed( primes ) ):
canceled_product = product / prime
if canceled_product >= n:
product = canceled_product
primes[ -(index+1) ] = 0
return list( filter( None, primes ) )
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_iterable(self):
"""Iterable return type"""
self.assert_([2, 3, 5] == [p for p in primes_range(2, 6)])
def test_empty(self):
"""Empty primes interval"""
self.assert_(not primes_range(3, 2))
def test_upper_bound(self):
"""Exclusive upper bound"""
self.assert_(list(primes_range(1, 7)) == [2, 3, 5])
def test_lower_bound(self):
"""Inclusive lower bound"""
self.assert_(list(primes_range(3, 10)) == [3, 5, 7])
def test_results(self):
"""Sample results"""
self.assert_(list(primes_range(1, 10)) == [2, 3, 5, 7])
def test_iterable(self):
"""Iterable return type"""
self.assert_( [2, 3, 5] == [ p for p in primes_range(2, 6) ] )
def test_empty(self):
"""Empty primes interval"""
self.assert_( not primes_range(3, 2) )
def test_upper_bound(self):
"""Exclusive upper bound"""
self.assert_( list(primes_range(1, 7)) == [2, 3, 5] )
def test_lower_bound(self):
"""Inclusive lower bound"""
self.assert_( list(primes_range(3, 10)) == [3, 5, 7] )
def test_results(self):
"""Sample results"""
self.assert_( list(primes_range(1, 10)) == [2, 3, 5, 7] )
