Fusion Discrete Logarithm Problems

TL;DR

This paper introduces a generalized exponential function for discrete logarithm problems, allowing cryptosystems to operate over a broader class of algebraic structures without compromising security.

Contribution

It presents a novel axiomatic framework for exponential functions that extends discrete logarithm applications to new algebraic settings, enhancing cryptographic versatility.

Findings

Generalized exponential functions can be constructed without altering security proofs.

Cryptosystems can be implemented using the new framework with minimal modifications.

The approach potentially strengthens the hardness of discrete logarithm problems.

Abstract

The Discrete Logarithm Problem is well-known among cryptographers, for its computational hardness that grants security to some of the most commonly used cryptosystems these days. Still, many of these are limited to a small number of candidate algebraic structures which permit implementing the algorithms. In order to extend the applicability of discrete-logarithm-based cryptosystems to a much richer class of algebraic structures, we present a generalized form of exponential function. Our extension relaxes some assumptions on the exponent, which is no longer required to be an integer. Using an axiomatic characterization of the exponential function, we show how to construct mappings that obey the same rules as exponentials, but can raise vectors to the power of other vectors in an algebraically sound manner. At the same time, computational hardness is not affected (in fact, the problem could possibly be strengthened). Setting up standard cryptosystems in terms of our generalized exponential function is simple and requires no change to the existing security proofs. This opens the field for building much more general schemes than the ones known so far.