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.

翻译：量子计算是现代密码学的一个重要分支。根据通用量子计算机和量子退火器中可用的工作物理量子比特数量，当前量子退火器能够求解更大规模的问题并非偶然。本文聚焦于利用量子退火求解二元域上的离散对数问题。值得注意的是，尽管此前已有研究考虑利用量子退火求解素数域上的离散对数问题，但迄今为止尚未有作者探讨利用量子退火求解二元域上的该问题。因此，本文旨在填补这一空白。我们提出了一种将二元域上的离散对数问题转化为二次无约束二进制优化问题的多项式变换方法，对于二元域 $\mathbb{F}_{2^n}$，该方法使用约 $3n^2$ 个逻辑变量。在我们的估算中，假设给定域存在 II 型最优正规基。此类 QUBO 实例随后可通过量子退火进行求解。