Paper 2025/955

Towards Better Integral Distinguishers over $\mathbb{F}_{p}$ Based on Exact Coefficients of Monomials

Abstract

Symmetric primitives used in multi-party computation, fully homomorphic encryption, and zero-knowledge proofs are often defined over Finite Field $\mathbb{F}_{q}$ with $q=2^t$ or an odd prime $p$. Integral attack is one of the most effective methods against such primitives due to the common use of low-degree non-linear layers. This in turn highlights the importance of a deeper understanding of degree growth. For ciphers defined over $\mathbb{F}_{2^t}$, numerous works have explored the growth of the algebraic degree. However, these methods cannot be directly applied to $\mathbb{F}_{p}$. At CRYPTO 2020, Beyne et al. extended the integral cryptanalysis to $\mathbb{F}_{p}$ by comparing degree with $s(p-1)$ when using $p^s$ data. However, given that the precise degree evaluation remains fundamentally challenging and often computationally infeasible, one may lose better integral distinguishers. In this paper, we present the first automatic search model over $\mathbb{F}_{p}$ based on the exact coefficient $\mathcal{A}$ of the monomial $\prod_{w=1}^{s}x_w^{p-1}$ contained in the algebraic representation. This model is constructed following the Computation-Traceback-Determine framework, where $\mathcal{A}$ is represented by several sums of multinomial coefficients under specific conditions. The existence of integral properties is then transformed into a determination of whether these sums can consistently equal $0\bmod{p}$. This determination is facilitated by four newly developed propositions based on Lucas Theorem. To demonstrate the effectiveness of our framework, we apply it to all variants of GMiMC. As a result, we achieve the best integral distinguishers for GMiMC-erf/-crf using large primes when they are used as block ciphers. For GMiMC-nyb/-mrf using 32/64-bit primes, our integral distinguishers cover more rounds than all other attacks. Meanwhile, all distinguishers we identified are no worse than those trivial ones predicted only considering the maximal degree. This shows the necessity of considering exact coefficients when searching for integral distinguishers over $\mathbb{F}_p$. Our framework is further employed to assess the security of two HADES designs: HadesMiMC and Poseidon2$^\pi$. The results reveal that the full rounds at the beginning and end of HADES provide sufficient resistance against integral cryptanalysis.