It has been suggested that a major obstacle in finding an index calculus attack on the elliptic curve discrete logarithm problem lies in the difficulty of lifting points from elliptic curves over finite fields to global fields. We explore the possibility of circumventing the problem of explicitly lifting points by investigating whether partial information about the lifting would be sufficient f...

The paper gives a formula for the probability that a randomly chosen elliptic curve over a nite eld has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a nite eld has kq points where k is a small number and where q is a prime. 1. I...

This paper merges between elliptic curve addition presents a modified processor architecture for Elliptic Curve Cryptography computations in Galois Fields GF(p). The architecture incorporates the methodology of pipelining to utilize the benefit of both parallel and serial implementations. It allows the exploitation of the inherited independency that exists in elliptic curve point addition and d...

Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. Public –key encryption schemes are secure only if the authenticity of the public-key is assured. Elliptic curve arithmetic can be used to develop a variety of elliptic curve cryptographic (ECC) schemes including key exchange, encryption and digital signature. The principal attraction of ellip...

The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rh...

The most popular public-key cryptography systems nowadays are RSA and Elliptic Curve Cryptography (ECC). ECC is a type of public-key cryptosystem which uses the additive group of points on a nonsingular elliptic curve as a cryptographic medium. The basic operation in most elliptic curve cryptosystems is a scalar multiplication. Scalar Multiplication is the costliest operation among all in ECC w...

that is both analytic (as a mapping of complex manifolds) and algebraic: addition of points in E(C) corresponds to addition in C modulo the lattice L. This correspondence between lattices and elliptic curves over C is known as the Uniformization Theorem; we will spend this lecture and part of the next proving it. To make the correspondence explicit, we need to specify the map Φ from C/L and an ...

In this paper, we develop the idea in [16] to obtain finer results on the structure of Selmer modules for p-adic representations than the usual main conjecture in Iwasawa theory. We determine the higher Fitting ideals of the Selmer modules under several assumptions. Especially, we describe the structure of the classical Selmer group of an elliptic curve over Q, using the ideals defined from mod...

We extend the notion of an invalid-curve attack from elliptic curves to genus 2 hyperelliptic curves. We also show that invalid singular (hyper)elliptic curves can be used in mounting invalid-curve attacks on (hyper)elliptic curve cryptosystems, and make quantitative estimates of the practicality of these attacks. We thereby show that proper key validation is necessary even in cryptosystems bas...