Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q - 1, for which u = g k . The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...

Recently, numerous techniques and methods have been proposed to address hard and complex algebraic and number theoretical problems related to cryptography.

Recently, the first author introduced some cryptographic functions closely related to the Diffie-Hellman problem called P -Diffie-Hellman functions. We show that the existence of a low degree polynomial representing a P -Diffie-Hellman function on a large set would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on the degree of such interpolation polynomials. 1

This paper proposes a public key cryptosystem based on the symmetric group Sn, and validates its theoretical foundation. The proposed system benefits from the algebraic properties of Sn such as non commutative, high computational speed and high flexibility in selecting keys which make the Discrete Logarithm Problem (DLP) resistant to attacks by algorithms such as Pohlig-Hellman. Against these properties, the only disadvantage of the scheme is its relative large memory and bandwidth requirements. Due to the similarities in the algebraic structures, many other cryptosystems can be translated to their symmetric group analogs, and the proposed cryptosystem is in fact the Generalized El-Gamal cryptosystem which is based on Sn instead of GF(p). Key words:

. For a generator of the multiplicative group of the field GF (p; k), the discrete logarithm of an element b of the field to the base a, b 6= 0 is that integer z : 1 z p k \Gamma 1, b = a z . The p-ary digits which represent z can be described with extremely simple polynomial forms. 1. Introduction The present note addresses the Discrete Logarithm problem ([1], [3], [4], [6]). The problem amounts to finding a quick method (efficient algorithm) for the computation of an integer z satisfying the equation: (1) a z = b: for b 2 GF (p; k), given a generator a of the multiplicative group of the field GF (p; k). The main practical interest in the problem stems from cryptography ([1], [2], [3], [4], [6]). In the case that a and z are known the computation of b can be done rapidly (Discrete Exponential Function [4], [7, p. 399]). However, computing z from a and b, that is, computing logarithms over GF (p; k), does not appear to admit a fast algorithm. ([1], [3], [4]). The integer z...

Le but est de pr'esenter le probl`eme du logarithme discret elliptique. 1.

This dissertation looks at problems in complexity theory; specifically, we are interested in finding explicit Boolean functions which are provably hard to compute in some reasonable model of computation. Although functions requiring exponential resources are known to be plentiful by elementary counting arguments [55], the problem of actually proving such lower bounds for explicit functions, say functions computable in NP, is much more di#cult. For a function in NP, the best known lower bound on circuit size are only linear [2], the best known lower bounds on circuit depth are only logarithmic