1,589 research outputs found
Computing cardinalities of Q-curve reductions over finite fields
We present a specialized point-counting algorithm for a class of elliptic
curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo
inert primes and, more generally, any elliptic curve over F\_{p^2} with a
low-degree isogeny to its Galois conjugate curve. These curves have interesting
cryptographic applications. Our algorithm is a variant of the
Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree
endomorphism in place of Frobenius. While it has the same asymptotic asymptotic
complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of
Drew Sutherlan
Computing the cardinality of CM elliptic curves using torsion points
Let E be an elliptic curve having complex multiplication by a given quadratic
order of an imaginary quadratic field K. The field of definition of E is the
ring class field Omega of the order. If the prime p splits completely in Omega,
then we can reduce E modulo one the factors of p and get a curve Ep defined
over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a
fast way to find this sign. For this, we propose to use the action of the
Frobenius on torsion points of small order built with class invariants a la
Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for
computing the cardinality of a given elliptic curve modulo p. We apply our
results to the Elliptic Curve Primality Proving algorithm (ECPP).Comment: Revised and shortened version, including more material using
discriminants of curves and division polynomial
Four primality testing algorithms
In this expository paper we describe four primality tests. The first test is
very efficient, but is only capable of proving that a given number is either
composite or 'very probably' prime. The second test is a deterministic
polynomial time algorithm to prove that a given numer is either prime or
composite. The third and fourth primality tests are at present most widely used
in practice. Both tests are capable of proving that a given number is prime or
composite, but neither algorithm is deterministic. The third algorithm exploits
the arithmetic of cyclotomic fields. Its running time is almost, but not quite
polynomial time. The fourth algorithm exploits elliptic curves. Its running
time is difficult to estimate, but it behaves well in practice.Comment: 21 page
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
Linearizing torsion classes in the Picard group of algebraic curves over finite fields
We address the problem of computing in the group of -torsion rational
points of the jacobian variety of algebraic curves over finite fields, with a
view toward computing modular representations.Comment: To appear in Journal of Algebr
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