In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the the group of rational points of an elliptic curve and is thus not a generic algorithm. The algorithm that we describe has some similarities with the most powerful indexcalculus algorithm for the discrete logarithm problem over a finite field.
The elliptic curve discrete logarithm problem is considered a secure
cryptographic primitive. The purpose of this paper is to propose a paradigm
shift in attacking the elliptic curve discrete logarithm problem. In this
paper, we will argue that initial minors are a viable way to solve this
problem. This paper will present necessary algorithms for this attack. We have
written a code to verify the conjecture of initial minors using Schur
complements. We were able to solve the problem for groups of order up to
$2^{50}$.
Comment: 13 pages; revised for publication
It is currently known from the work of Shoup and Nechaev that a generic algorithm to solve the discrete logarithm problem in a group of prime order must have complexity at least k √ N where N is the order of the group. In many collision search algorithms this complexity is achieved. So with generic algorithms one can only hope to make the k smaller. This k depends on the complexity of the iterative step in the generic algorithms. The √ N comes from the fact there is about √ N iterations before a collision. So if we can find ways that can reduce the amount of work in one iteration then that is of great interest and probably the only possible modification of a generic algorithm. The modified r-adding walk allegedly does just that. It claims to reduce the amount of work done in one iteration of the original r-adding walk. In this paper we study this modified r-adding walk, we critically analyze it and we compare it with the original r-adding walk.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.