This paper generalises the results of Miyaji, Nakabayashi and Takano by giving families corresponding to non-prime group orders with embedding degree suitable for pairing applications and considers the case of ordinary abelian varieties of dimension 2.Expand

An analysis of the security of keys generated by the proposed key generation method for RSA moduli, and a new birthday attack on low Hamming-weight private exponents are given.Expand

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as… Expand

A factoring method is presented which, heuristically, splits composite n in O(n 1/4+ ∈) steps, well-suited for use with small computers: the storage required is negligible, and one never needs to work with numbers larger than n itself.Expand

A new bound for the Schur-Siegel-Smyth trace problem is produced and many new examples of such polynomials of minimal absolute trace (for given degree) are found.Expand

for some constant A (depending on a, b and c). Apparently this is due to Bellman and Shapiro (unpublished), and Bellman describes the proof as 'not elementary, although not difficult' [1]. The first… Expand

Recurrence relations for the coefficients of the nth division polynomial for elliptic curves are presented. These provide an algorithm for computing the general division polynomial without using… Expand

It is shown that f_k(n) ~ n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists, which holds for infinitely many k, including all k < 195 and all k such that k+1 or 2k+1 is prime.Expand

We show that there are Salem numbers of every trace. The nontrivial part of this result is for Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction,… Expand