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，一类基于有限域$\mathbb{F}_p$的新型对称密钥原语在多方计算和零知识证明协议中发挥着关键作用。为提升此类原语的效率，研究者提出了多个基于$\mathbb{F}_p$的新型分组密码和哈希函数。这些新原语表明，相较于经典的替换-置换网络（SPN）和Feistel网络，采用替代设计策略能为大奇素数$p$上的$\mathbb{F}_p$密码与哈希函数设计带来更高效率。基于上述研究进展，本文构建了一个**代数框架**，可系统探索在$\mathbb{F}_p$上构建对称密钥（迭代）置换的可行且高效的设计策略。我们首先将有限域上的迭代多项式动力系统识别为几乎所有分组密码设计策略的核心构建模块。随后提出广义三角多项式动力系统（GTDS），并基于该框架为$\mathbb{F}_p^n$上的迭代（带密钥）置换给出通用定义。基于GTDS的通用定义能够描述三种最著名的设计策略：SPN、Feistel网络和Lai-Massey结构。遵循这些设计策略构建的分组密码均可通过我们的通用定义实例化。此外，我们发现近期提出的\texttt{Griffin}设计（既非Feistel也非SPN结构）同样可通过基于GTDS的通用定义描述。我们还展示了基于GTDS定义可实例化一种新型广义Lai-Massey结构。最后，我们提供了GTDS的通用分析，包括差分均匀性和相关性的上界。