International Association
for Cryptologic Research

The LowMC-based post-quantum signature scheme Picnic was selected as a third-round candidate for NIST PQC, attracting wide attention to the design of efficient and secure post-quantum signature schemes using Symmetric Techniques for Advanced Protocols (STAP). Symmetric primitives designed for advanced protocols such as secure multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge (ZK) proof systems, with the goal of reducing the number of multiplication operations, are referred to as arithmetic-oriented (AO) primitives. These cryptographic primitives are typically constructed over large finite fields, which makes classical statistical analysis methods like differential and linear cryptanalysis inefficient. Due to their inherent algebraic properties, the mainstream security evaluation approaches are based on algebraic attacks. In this paper, we analyze the security of the MPC-friendly primitives \textsc{Rain} (CCS 2022) and AIM (CCS 2023) used in the post-quantum signature schemes Rainier and AIMer. Existing algebraic attacks on \textsc{Rain} and AIM were conducted over $\mathbb{F}_2$. We propose a novel algebraic attack over $\mathbb{F}_{2^n}$ that uses the polynomial decomposition to reduce degrees of equations. By further combining with the guess-and-determine technique, meet-in-the-middle modeling, and resultant, we are able to attack \textsc{Rain} and the full AIM. Specifically, we successfully attacked 2-round \textsc{Rain} with $2^{73.7}/2^{107.0}/2^{138.9}$ primitive calls, 3-round \textsc{Rain} with $2^{160.6}/2^{236.0}/2^{311.1}$ primitive calls, for the $128/192/256$-bit key. For the full AIM, we successfully attacked it with $2^{114.0}/2^{163.2}/2^{228.3}$ primitive calls for the $128/192/256$-bit key. The attack complexities mainly lie in solving univariate polynomial equations and computing resultants, and hence the complexity evaluations are accurate. \keywords{Polynomial Decomposition\and MITM \and \textsc{Rain} \and AIM \and Guess-and-determine \and Resultant.}

The rapid development of advanced cryptographic applications like multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge (ZK) proofs have motivated the designs of the so-called arithmetic-oriented (AO) primitives. Efficient AO primitives typically build over large fields and use large S-boxes. Such design philosophy brings difficulties in the cryptanalysis of these primitives as classical cryptanalysis methods do not apply well. The generally recognized attacks against these primitives are algebraic attacks, especially Gr\"obner basis attacks. Thus, the numbers of security rounds are usually derived through the complexity of solving the system of algebraic equations using Gr\"obner bases. In this paper, we propose a novel framework for algebraic attacks against AO primitives. Instead of using Gr\"obner basis, we use {\it resultants} to solve a system of multivariate equations that can better exploit the algebraic structures of AO primitives. We employ several techniques to redu