A compact QUBO encoding of computational logic formulae demonstrated on cryptography constructions

TL;DR

This paper presents a compact QUBO encoding method for cryptography algorithms, significantly reducing problem size and potentially increasing their vulnerability to quantum annealing attacks.

Contribution

The authors introduce a novel QUBO encoding approach that minimizes variables for cryptography functions, outperforming previous methods in size reduction.

Findings

Achieved over 8x reduction in QUBO variables for AES-256.

Reduced logical variables by thousands compared to prior work.

Maintained sparse QUBO matrices with low coefficient magnitudes.

Abstract

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.