Faster Quantum Algorithms for MQ2 and Applications

We study quantum algorithms for multivariate quadratic Boolean equation systems by focusing on their precise gate count. While better asymptotic algorithms are known, currently gate counts were only computed for exhaustive search (Schwabe and Westerbaan, SPACE 2016) and a variant of Grover's search using preprocessing (Pring, WAIFI 2018). This limits the applicability of Boolean equation solving to cryptanalysis, which considers relatively small numbers of variables (from 40 to 200) and is concerned with the exact complexity of the solver. In this paper, we introduce two new quantum algorithms. The first algorithm is an optimized quantum exhaustive search which amortizes the cost of polynomial evaluation at each quantum search iterate. The second algorithm adapts a method of Bouillaguet et al. (SOSA 2022) which proceeds by linearization of the system. In both cases, we implement the quantum circuits, study their complexity, and obtain significant improvements over previous results. Next, we apply these new algorithms to the cryptanalysis of the block ciphers LowMC and RAIN in the single-data setting. By adapting attacks from Liu et al. (ToSC 2022) and Liu et al. (ToSC 2023) we obtain the first quantum cryptanalysis results on these ciphers.