100 research outputs found
On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
For an elliptic curve E/Q without complex multiplication we study the
distribution of Atkin and Elkies primes l, on average, over all good reductions
of E modulo primes p. We show that, under the Generalised Riemann Hypothesis,
for almost all primes p there are enough small Elkies primes l to ensure that
the Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1))
expected time.Comment: 20 pages, to appear in LMS J. Comput. Mat
On the Distribution of Atkin and Elkies Primes
Given an elliptic curve E over a finite field F_q of q elements, we say that
an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a
square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of
F_q-rational points on E; otherwise ell is called an Atkin prime. We show that
there are asymptotically the same number of Atkin and Elkies primes ell < L on
average over all curves E over F_q, provided that L >= (log q)^e for any fixed
e > 0 and a sufficiently large q. We use this result to design and analyse a
fast algorithm to generate random elliptic curves with #E(F_p) prime, where p
varies uniformly over primes in a given interval [x,2x].Comment: 17 pages, minor edit
The Counting function for Elkies primes
Let be an elliptic curve over a finite field where is
a prime power. The Schoof--Elkies--Atkin (SEA) algorithm is a standard method
for counting the number of -points on . The asymptotic
complexity of the SEA algorithm depends on the distribution of the so-called
Elkies primes.
Assuming GRH, we prove that the least Elkies prime is bounded by when . This is the first such explicit bound in the
literature. Previously, Satoh and Galbraith established an upper bound of
.
Let denote the number of Elkies primes less than . Assuming GRH,
we also show
N_E(X)=\frac{\pi(X)}{2}+O\left(\frac{\sqrt{X}(\log qX)^2}{\log X}\right)\,.
$
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
On the evaluation of modular polynomials
We present two algorithms that, given a prime ell and an elliptic curve E/Fq,
directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the
j-invariants of the elliptic curves that are ell-isogenous to E. We do not
assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be
adapted to handle other types of modular polynomials, and we consider
applications to point counting and the computation of endomorphism rings. We
demonstrate the practical efficiency of the algorithms by setting a new
point-counting record, modulo a prime q with more than 5,000 decimal digits,
and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9
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