We aim to advance the state-of-the-art in Quadratic Unconstrained Binary Optimization formulation with a focus on cryptography algorithms. As the minimal QUBO encoding of the linear constraints of optimization problems emerges as the solution of integer linear programming (ILP) problems, by solving special boolean logic formulas (like ANF and DNF) for their integer coefficients it is straightforward to handle any normal form, or any substitution for multi-input AND, OR or XOR operations in a QUBO form. To showcase the efficiency of the proposed approach we considered the most widespread cryptography algorithms including AES-128/192/256, MD5, SHA1 and SHA256. For each of these, we achieved QUBO instances reduced by thousands of logical variables compared to previously published results, while keeping the QUBO matrix sparse and the magnitude of the coefficients low. In the particular case of AES-256 cryptography function we obtained more than 8x reduction in variable count compared to previous results. The demonstrated reduction in QUBO sizes notably increases the vulnerability of cryptography algorithms against future quantum annealers, capable of embedding around $30$ thousands of logical variables.

翻译：我们旨在推进二次无约束二元优化（QUBO）公式化的前沿研究，重点关注密码学算法。由于优化问题线性约束的最小QUBO编码可转化为整数线性规划（ILP）问题的求解，通过求解特殊布尔逻辑公式（如ANF与DNF）的整数系数，即可直接处理QUBO形式中的任意范式或对多输入与门（AND）、或门（OR）及异或门（XOR）运算的任意替换。为展示所提方法的效率，我们选取了最广泛的密码学算法（涵盖AES-128/192/256、MD5、SHA1及SHA256）进行验证。针对每种算法，我们实现了QUBO实例的降维——相较于已有成果，逻辑变量数量减少数千个，同时保持QUBO矩阵的稀疏性及系数幅值的低水平。以AES-256密码函数为例，其变量数量较先前结果实现了超过8倍的缩减。这种QUBO规模的显著降低进一步增强了密码学算法在未来量子退火器（可嵌入约3万个逻辑变量）面前的脆弱性。