In this paper, we use a recent method given by Rudnev et al. (2020) to improve results on sum–product type problems due Pham and Mojarrad (2018).

The elliptic curve discrete logarithm problem is one of the most important problems in cryptography. In recent years, several index calculus algorithms have been introduced for elliptic curves defined over extension fields, but the most important curves in practice, defined over prime fields, have so far appeared immune to these attacks. In this paper we formally generalize previous attacks fro...

A variation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed. We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the class number one setting. Our heuristics are based upon a recent conjecture by R. Gross and J. Smith on n...

This paper describes a method of construction of a composite field representation from a given binary field representation. We derive the conversion (change of basis) matrix. The special case of when the degree of the ground field is relatively prime to the extension degree, where the irreducible polynomial generating the composite field has its coefficients from the binary prime field rather t...

This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by Gaudry and Lubicz in 2009 had suggested the use of the associated Kummer line to speed up scalar multiplication. In this work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necess...