In their seminal paper, Miyaji, Nakabayashi and Takano [12] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of...

The hyperelliptic curve cryptosystem is one of the emerging cryptographic primitives of the last years. This system offers the same security as established public-key cryptosystems, such as those based on RSA or elliptic curves, with much shorter operand length. Consequently, this system allows highly efficient computation of the underlying field arithmetic. However, until recently the common b...

This paper introduces three new probabilistic encryption schemes using elliptic curves over rings. The cryptosystems are based on three specific trapdoor mechanisms allowing the recipient to recover discrete logarithms on different types of curves. The first scheme is an embodiment of Naccache and Stern’s cryptosystem and realizes a discrete log encryption as originally wanted in [23] by Vansto...

Elliptical curve cryptography (ECC) is a public key encryption technique based on elliptic curve theory that can be used to create faster, smaller, and more efficient cryptographic keys. ECC generates keys through the properties of the elliptic curve equation instead of the traditional method of generation as the product of very large prime numbers. Because ECC helps to establish equivalent sec...

Finding suitable non-supersingular elliptic curves becomes an important issue for the growing area of pairing-based cryptosystems. For this purpose, many methods have been proposed when embedding degree k and cofactor h are taken different values. In this paper we propose a new method to find pairing-friendly elliptic curves without restrictions on embedding degree k and cofactor h. We propose ...

This paper provides an overview of elliptic curves and their use in cryptography. The purpose of this paper is an in-depth examination of the Elliptic Curve Discrete Logarithm (ECDLP) including techniques in attacking cryptosystems dependent on the ECDLP. The paper includes properties of elliptic curve and methods for various attacks.

The Weil and Tate pairings are defined for elliptic curves over fields, including finite fields. These definitions extend naturally to elliptic curves over Z/NZ , for any positive integer N , or more generally to elliptic curves over any finite commutative ring, and even the reduced Tate pairing makes sense in this more general setting. This paper discusses a number of issues which arise if one...

Recently, and contrary to the common belief, Rivest and Silverman argued that the use of strong primes is unnecessary in the RSA cryptosystem. This paper analyzes how valid this assertion is for RSA-type cryptosystems over elliptic curves. The analysis is more diicult because the underlying groups are not always cyclic. Previous papers suggested the use of strong primes in order to prevent fact...

Elliptic Curve Cryptosystems (ECCs) are utilized as an alternative to traditional publickey cryptosystems, and are more suitable for resource limited environments due to smaller parameter size. In this dissertation we carry out a thorough investigation of side-channel attack aware ECC implementations over finite fields of prime characteristic including the recently introduced Edwards formulatio...