In recent years a new class of symmetric-key primitives over $\mathbb{F}_p$ that are essential to Multi-Party Computation and Zero-Knowledge Proofs based protocols have emerged. Towards improving the efficiency of such primitives, a number of new block ciphers and hash functions over $\mathbb{F}_p$ were proposed. These new primitives also showed that following alternative design strategies to the classical Substitution-Permutation Network (SPN) and Feistel Networks leads to more efficient cipher and hash function designs over $\mathbb{F}_p$ specifically for large odd primes $p$. In view of these efforts, in this work we build an \emph{algebraic framework} that allows the systematic exploration of viable and efficient design strategies for constructing symmetric-key (iterative) permutations over $\mathbb{F}_p$. We first identify iterative polynomial dynamical systems over finite fields as the central building block of almost all block cipher design strategies. We propose a generalized triangular polynomial dynamical system (GTDS), and based on the GTDS we provide a generic definition of an iterative (keyed) permutation over $\mathbb{F}_p^n$. Our GTDS-based generic definition is able to describe the three most well-known design strategies, namely SPNs, Feistel networks and Lai--Massey. Consequently, the block ciphers that are constructed following these design strategies can also be instantiated from our generic definition. Moreover, we find that the recently proposed \texttt{Griffin} design, which neither follows the Feistel nor the SPN design, can be described using the generic GTDS-based definition. We also show that a new generalized Lai--Massey construction can be instantiated from the GTDS-based definition. We further provide generic analysis of the GTDS including an upper bound on the differential uniformity and the correlation.

翻译：近年来，一类基于$\mathbb{F}_p$的对称密钥原语在多方计算和零知识证明协议中变得至关重要。为提升此类原语效率，学界提出了多种基于$\mathbb{F}_p$的新型分组密码与哈希函数。这些新型原语表明，相较于经典替换-置换网络(SPN)与Feistel网络，采用另类设计策略可构造出更高效的$\mathbb{F}_p$密码与哈希函数——尤其当$p$为大奇数素数时。基于此研究背景，本文构建了一个代数框架，可系统探索构造$\mathbb{F}_p$上对称密钥（迭代）置换的有效设计策略。我们首先识别出有限域上的迭代多项式动力系统是几乎所有分组密码设计策略的核心构建模块。进而提出广义三角多项式动力系统(GTDS)，并基于GTDS给出$\mathbb{F}_p^n$上迭代（密钥化）置换的泛化定义。该GTDS泛化定义能够完整描述三种最经典的设计策略——SPN、Feistel网络与Lai-Massey结构，因此遵循这些策略构造的分组密码均可通过我们的泛化定义实例化。值得注意的是，近期提出的\texttt{Griffin}设计方案虽不属于Feistel或SPN设计范畴，仍可用基于GTDS的泛化定义进行描述。我们还展示了一种新型广义Lai-Massey结构可从GTDS泛化定义实例化产生。最后提供GTDS的通用分析，包括其差分均匀性及相关性的上界。