The discrete logarithm problem for exponents of bounded height

Problem 3 : The Discrete Logarithm

Generic algorithms for the discrete logarithm problem

This can be a significantly harder problem. For example, say we are using a randomized (Las Vegas) algorithm. If β lies in 〈α〉 then we are guaranteed to eventually find logα β, but if not, we will never find it and it may be impossible to tell whether we are just very unlucky or β 6∈ 〈α〉. On the other hand, with a deterministic algorithm such as the baby-steps giant-steps method, we can unequiv...

Generalized Jacobian and Discrete Logarithm Problem on Elliptic Curves

Let E be an elliptic curve over the finite field F_{q}, P a point in E(F_{q}) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*_{q} , the multiplicative group of nonzero elements of Fq, in the case where n | q...

Attacking the Elliptic Curve Discrete Logarithm Problem

Definition 1.1. Given a finite abelian group G written multiplicatively and elements b and g in G, the discrete logarithm problem (DLP) consists of finding an integer n such that bn = g, if such an n exists. The difficulty involved in computing the discrete logarithm varies with the choice of G. For example, in the additive group of integers modulo n, (Z/nZ)+, the problem can be solved efficien...

On the Bounded Sum-of-digits Discrete Logarithm Problem in Kummer and Artin-Schreier Extensions

In this paper, we study the discrete logarithm problem in the finite fields Fqn where n|q−1. The field is called a Kummer field or a Kummer extension of Fq. It plays an important role in improving the AKS primality proving algorithm. It is known that we can efficiently construct an element g with order greater than 2 in the fields. Let Sq(•) be the function from integers to the sum of digits in...