23 research outputs found
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
Constructing elliptic curves of prime order
We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.Comment: 13 page
A faster pseudo-primality test
We propose a pseudo-primality test using cyclic extensions of . For every positive integer , this test achieves the
security of Miller-Rabin tests at the cost of Miller-Rabin
tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal,
Springe
A note on Agrawal conjecture
We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Zp[X]/(Cr(X)))* and state the modified conjecture that the set {X-1, X+2} generate big enough subgroup of this group
Finding suitable paths for the elliptic curve primality proving algorithm
An important part of the Elliptic Curve Primality Proving algorithm consists of finding a sequence of elliptic curves with appropriate properties. In this paper we consider a strategy to search for an improved sequence, as part of an implementation (implemented in Magma 2.19) to obtain improved heuristics and compare it to an implementation which does not use such heuristics, namely to a built-in Magma function