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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 basedprotocols have emerged. Towards improving the efficiency of such primitives, anumber of new block ciphers and hash functions over $\mathbb{F}_p$ wereproposed. These new primitives also showed that following alternative designstrategies to the classical Substitution-Permutation Network (SPN) and FeistelNetworks 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 designstrategies for constructing symmetric-key (iterative) permutations over$\mathbb{F}_p$. We first identify iterative polynomial dynamical systems overfinite fields as the central building block of almost all block cipher designstrategies. 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 mostwell-known design strategies, namely SPNs, Feistel networks and Lai--Massey.Consequently, the block ciphers that are constructed following these designstrategies can also be instantiated from our generic definition. Moreover, wefind that the recently proposed \texttt{Griffin} design, which neither followsthe Feistel nor the SPN design, can be described using the generic GTDS-baseddefinition. We also show that a new generalized Lai--Massey construction can beinstantiated from the GTDS-based definition.  We further provide generic analysis of the GTDS including an upper bound onthe differential uniformity and the correlation.

Generalized Triangular Dynamical System: An Algebraic System for Constructing Cryptographic Permutations over Finite Fields