Elliptic curves over finite fields have applications in public key cryptography. A Koblitz curve is an elliptic curve E over F2; the group E(F2n ) has convenient features for efficient implementation of elliptic curve cryptography. Wiener and Zuccherato and Gallant, Lambert and Vanstone showed that one can accelerate the Pollard rho algorithm for the discrete logarithm problem on Koblitz curves...

Our purpose is to describe elliptic curves with complex multiplication which in characteristic 2 have the following useful properties for constructing Diffie-HeUman type cryptosystems: (1) they are nonsupersingular (so that one cannot use the Menezes-Okamoto-Vanstone reduction of discrete log from elliptic curves to finite fields); (2) the order of the group h a s a large prime factor (so that ...

This position paper summarizes our perspectives on the selection of next-generation elliptic curves for standardization. It also contains generation algorithms intended as a foundation for choosing elliptic curves for cryptography in a simple, consistent and rigid way.

This book is about elliptic curves and modular functions, two topics that are intimately related in both accidental and essential ways. As emphasized by André Weil in his magisterial historical introduction to contemporary number theory [W], the arithmetic study of elliptic curves is, in spite of the clear reference to the integral calculus in the adjective elliptic, in many respects antecedent...

Some notes on the analogy between number theory over Z and Fq[t] and an attempt to translate a paper of Gouvêa and Mazur on ranks of quadratic twists of elliptic curves over Q to elliptic curves over Fq(t).

The security of most elliptic curve cryptosystems is based on the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Such a problem turns out to be computationally unfeasible when elliptic curves are suitably chosen. This paper provides an algorithm to obtain cryptographically good elliptic curves from a given one. The core of such a procedure lies on the usage of successi...

In this article we will generalize some known constructions to produce pseudorandom sequences with the aid of elliptic curves. We will make use of both additive and multiplicative characters on elliptic curves.

We give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces. The Fano surface of the Fermat cubic of P is the only one to contain 30 elliptic curves...

Stange, generalizingWard’s elliptic divisibility sequences, introduced elliptic nets, and showed an equivalence between elliptic nets and elliptic curves. This note relates Stange’s recursion for elliptic nets and the Coxeter group F4.

Many interesting questions can be asked about the decomposition of Jacobians of curves. For instance, we may want to know which curves have completely decomposable Jacobians (Jacobians which are the product of g elliptic curves) [4]. We may ask about number theoretic properties of the elliptic curves that show up in the decomposition of Jacobians of curves [2]. We would also like to know how ma...