The discrete logarithm problem in finite groups is one of the supposedly difficult problems at the foundation of asymmetric or public key cryptography. The first cryptosystems based on discrete logarithms were implemented in the multiplicative groups of finite fields, in which the discrete logarithm problem turned out to be easier than one would wish, just as the factorisation problem at the he...

The security of many cryptographic protocols depends on the diiculty of solving the so-called \discrete logarithm" problem, in the multiplica-tive group of a nite eld. Although, in the general case, there are no polynomial time algorithms for this problem, constant improvements are being made { with the result that the use of these protocols require much larger key sizes, for a given level of s...

Nowadays the generation of cryptosystems requires two main aspects. First the security, and then the size of the keys involved in the construction and comunication process. About the former one needs a difficult mathematical assumption which ensures your system will not be broken unless a well known difficult problem is solved. In this context one of the most famous assumption underlying a wide...

In 1986 Victor Miller described an algorithm for computing the Weil pairing in his unpublished manuscript. This algorithm has then become the core of all pairing-based cryptosystems. Many improvements of the algorithm have been presented. Most of them involve a choice of elliptic curves of a special forms to exploit a possible twist during Tate pairing computation. Other improvements involve a ...

Since Edwards curves were introduced to elliptic curve cryptography by Bernstein and Lange in 2007, they have received a lot of attention due to their very fast group law operation. Pairing computation on such curves is slightly slower than on Weierstrass curves. However, in some pairing-based cryptosystems, they might require a number of scalar multiplications which is time-consuming operation...

This thesis analyzes the security and e ciency of public key cryptosystems. New attacks for several cryptosystems are proposed and the e ectiveness of the attacks is evaluated. Furthermore, solutions are given to several unsolved problems in computational number theory and algebraic geometry theory that are closely related to the security of public key cryptosystems. Moreover, new calculation m...

In this paper we give an introduction to elliptic curve public key cryptosystems. We explain how the discrete logarithm in an elliptic curve group can be used to construct cryptosystems. We also focus on practical aspects such as implementation, standardization and intellectual property.

Elliptic curves have become widespread in cryptographic applications since they offer the same cryptographic functionality as public-key cryptosystems designed over integer rings while needing a much shorter bitlength. The resulting speedup in computation as well as the smaller storage needed for the keys, are reasons to favor elliptic curves. Nowadays, elliptic curves are employed in scenarios...

This paper proposes a new modified variant of Menezes and Vanstone elliptic curve cryptosystem. This new variant uses the same original Menezes and Vanstone elliptic curve cryptosystem but in an elegant way. The new variant uses a not only one elliptic curve but a number of curves. Each curve is chosen with its corresponding keys to constitute a separate cryptosystem. The message is then divide...

‎In a (t,n)-threshold secret sharing scheme‎, ‎a secret s is distributed among n participants such that any group of t or more participants can reconstruct the secret together‎, ‎but no group of fewer than t participants can do‎. In this paper, we propose a verifiable (t,n)-threshold multi-secret sharing scheme based on Shao and Cao‎, ‎and the intractability of the elliptic curve discrete logar...