This paper was motivated by the problem of determining what isogeny classes of abelian surfaces over a finite field k contain jacobians. In [MN] we performed a numerical exploration of this problem, that led to several conjectures. We present in this paper a complete answer for supersingular surfaces in characteristic 2 (section 5). We deal with this problem in a direct way by computing explici...

We study generalisations to totally real fields of the methods originating with Wiles and Taylor-Wiles ([32], [31]). In view of the results of Skinner-Wiles [26] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some app...

One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus g? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order 2, ...

The Edwards curves of the form x2 + y2 = 1 dx2y2 are investigated in this article. An exact formula for quantity points on over a field Fp is obtained odd prime numbers p. special attention paid to with exactly p+1 Fp. These called supersingular. They not recommended usage cryptography, because their structure relatively simple. supersingularity curve proved any p 4m+3. Also, some other values ...

We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to k-isomorphism and we exhibit equations and the zeta function of all their k/k-twists. As an application we obtain a direct...

Scott uses an efficiently computable isomorphism in order to optimize pairing computation on a particular class of curves with embedding degree 2. He points out that pairing implementation becomes thus faster on these curves than on their supersingular equivalent, originally recommended by Boneh and Franklin for Identity Based Encryption. We extend Scott’s method to other classes of curves with...

In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over nite elds. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms i...

In this paper, we present the results of computing the Tate pairing using a supersingular elliptic curve defined over a prime field. The aim of this work is to demonstrate the feasibility of the primitives of identity based cryptosystem for application in embedded processors such as a smartcard. The most computationally intensive operation in an Identity Based Protocol is the calculation of a p...

Recently, the supersingular elliptic curves over ternary fields are widely used in pairing based crypto-applications since they achieve the best possible ratio between security level and space requirement. We propose new algorithms for projective arithmetic on the curves, where the point tripling is field multiplication free, and point addition and point doubling requires one field multiplicati...

The recent introduction of Edwards curves has significantly reduced the cost of addition on elliptic curves. This paper presents new explicit formulae for pairing implementation in Edwards coordinates. We prove our method gives performances similar to those of Miller’s algorithm in Jacobian coordinates and is thus of cryptographic interest when one chooses Edwards curve implementations of proto...