  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.
