Simple vs. vectorial: exploiting structural symmetry to beat the ZeroSum distinguisher: Applications to SHA3, Xoodyak and Bash

Higher order differential properties constitute a very insightful tool at the hands of a cryptanalyst allowing for probing a cryptographic primitive from an algebraic perspective. In FSE 2017, Saha et al. reported SymSum (referred to as SymSum_Vec in this paper), a new distinguisher based on higher order vectorial Boolean derivatives of SHA-3, constituting one of the best distinguishers on the latest cryptographic hash standard. SymSum_Vec exploits the difference in the algebraic degree of highest degree monomials in the algebraic normal form of SHA-3 with regards to their dependence on round constants. Later in AFRICACRYPT 2020, Suryawanshi et al. extended SymSum_Vec using linearization techniques and in SSS 2023 also applied it to NIST-LWC finalist Xoodyak. However, a major limitation of SymSum_Vec is the maximum attainable derivative (MAD) of the polynomial representation, which is less than half of the widely studied ZeroSum distinguisher. This is attributed to SymSum_Vec being dependent on k-fold vectorial derivatives while ZeroSum relies on k-fold simple derivatives. In this work we overcome this limitation of SymSum_Vec by developing and validating the theory of computing SymSum_Vec with simple derivatives. This gives us a close to 100% improvement in the MAD that can be computed. The new distinguisher reported in this work can also be combined with 1/2-round linearization to penetrate more rounds. Moreover, we identify an issue with the 2-round linearization claim made by Suryawanshi et al. which renders it invalid and also furnishes an algebraic fix at the cost of some additional constraints. Combining all the results we report SymSum_Sim, a new variant of the SymSum_Vec distinguisher based on k-fold simple derivatives that outperforms ZeroSum by a factor of 2²⁵⁷,2¹²⁹ for 10-round SHA3-384 and 9-round SHA3-512 respectively while enjoying the same MAD as ZeroSum. For every other SHA-3 variant, SymSum_Sim maintains an advantage of factor 2 over the ZeroSum. Combined with 1/2-round linearization, SymSum_Sim improves upon all existing ZeroSum and SymSum_Vec distinguishers on both SHA-3 and Xoodyak. As regards Keccak-p, the internal permutation of SHA-3, we report the best 15-round distinguisher with a complexity of 2²⁵⁶ and the first better than birthday-bound 16-round distinguisher with a complexity of 2⁵¹² (improving upon the 15/16-round results by Guo et al. in ASIACRYPT 2016). We also devise the best full-round distinguisher on the Xoodoo internal permutation of Xoodyak with a practically verifiable complexity of 2³² and furnish the first third-party distinguishers on the Belarushian-standard hash function Bash. All distinguishers presented in this work have been verified through implementations whenever practically viable. Overall, with the MAD barrier broken, SymSum_Sim emerges as a better distinguisher than ZeroSum on all fronts and adds to the state-of-the-art of cryptanalytic tools investigating non-randomness of crypto primitives.