Transformation of the discrete logarithm problem over $\mathbb F_{2^n}$ to the QUBO problem using normal bases

TL;DR

This paper introduces a polynomial transformation of the discrete logarithm problem over binary fields into a QUBO formulation, enabling quantum annealing solutions and addressing a previously unexplored area in quantum cryptanalysis.

Contribution

It presents the first known transformation of DLP over binary fields into QUBO form, utilizing normal bases and approximately 3n^2 variables, expanding quantum cryptanalysis capabilities.

Findings

Transformation enables quantum annealing for DLP over binary fields

Uses approximately 3n^2 variables with normal bases

Bridges a gap in quantum cryptanalysis research

Abstract

Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately $3n^2$ logical variables for the binary field $\mathbb{F}_{2^n}$. In our estimations, we assume the existence of an optimal normal base of II type in the given fields. Such a QUBO instance can then be solved using quantum annealing.