In this paper, we discuss an isomorphism between elliptic curves defined over binary fields (curves defined over F2n). We introduce a simple public-key encryption scheme for binary elliptic curves. Here we demonstrate that this encryption scheme is as secure as the EC El Gamal cryptosystem. The basis of the encryption scheme is this isomorphism between binary elliptic curves. We use this same i...

This note explores the method of A. Néron [5] for constructing elliptic curves of (fairly) high rank over Q . Néron’s basic idea is very simple: although the moduli space of elliptic curves is only 1-dimensional, the vector space of homogeneous cubic polynomials in three variables is 10-dimensional. Therefore, one can construct elliptic curves which pass through any given 9 rational points. Wit...

We provide the first construction of a hash function into ordinary elliptic curves that is indifferentiable from a random oracle, based on Icart’s deterministic encoding from Crypto 2009. While almost as efficient as Icart’s encoding, this hash function can be plugged into any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle...

0.1. Endomorphisms of elliptic curves. Recall that a homomorphism of complex elliptic curves is just a holomorphic map E1 → E2 which preserves the origin. (It turns out that this condition is enough to force it to be a homomorphism of groups in the usual sense; why?) An isogeny of elliptic curves is a homomorphism whose kernel is a finite subgroup of E1. In fact the kernel of a homomorphism of ...

Elliptic curves as an area of mathematical study are initially simple to understand, but reveal startling complexity when considered over different fields. This paper discusses the general properties and characteristics of projective space, elliptic curves, and the group structure that arises with certain binary operations on the curve. We discuss elliptic curves over Q, including the topic of ...

The normal form x2+y2 = a2+a2x2y2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be

Finding suitable non-supersingular elliptic curves for pairing-based cryptosystems becomes an important issue for the modern public-key cryptography after the proposition of id-based encryption scheme and short signature scheme. In previous work different algorithms have been proposed for finding such elliptic curves when embedding degree k ∈ {3, 4, 6} and cofactor h ∈ {1, 2, 3, 4, 5}. In this ...

Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves ...

The purpose of this paper is to study the Hodge-Arakelov theory of elliptic curves (cf. [Mzk1-4]) in positive characteristic. The first two §’s (§1,2) are devoted to studying the relationship of the Frobenius and Verschiebung morphisms of an elliptic curve in positive characteristic to the Hodge-Arakelov theory of elliptic curves. We begin by deriving a “Verschiebung-Theoretic Analogue of the H...

We survey recent research on pairings on hyperelliptic curves and present a comparison of the performance characteristics of pairings on elliptic curves and hyperelliptic curves. Our analysis indicates that hyperelliptic curves are not more efficient than elliptic curves for general pairing applications.