Transformation of the discrete logarithm problem over $\mathbb F_{2^n}$ to the QUBO problem using normal bases
Michał Wroński, Mateusz Leśniak
TL;DR
The paper tackles solving the discrete logarithm problem over binary fields using quantum annealing by introducing a polynomial transformation from $ ext{F}_{2^n}$ DLP to a QUBO formulation. It relies on the existence of an optimal normal basis of II type to bound the variable count at approximately $3n^2$, and demonstrates the approach with a concrete $ ext{F}_{2^3}$ example solved on a D-Wave system. A worked transformation yields a QUBO with 11 logical variables, achieving a non-random success rate of about 75% in 10,000 trials for the small-field case. The work highlights both the potential of quantum annealing for binary-field DLPs and the current limitations posed by the existence of optimal normal bases, pointing to future experiments on larger fields.
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.