In this paper we identify polynomial dynamical systems over finite fields as
the central component of almost all iterative block cipher design strategies
over finite fields. We propose a generalized triangular polynomial dynamical
system (GTDS), and give a generic algebraic definition of iterative (keyed)
permutation using GTDS. Our GTDS-based generic definition is able to describe
widely used and well-known design strategies such as substitution permutation
network (SPN), Feistel network and their variants among others. We show that
the Lai-Massey design strategy for (keyed) permutations is also described by
the GTDS. Our generic algebraic definition of iterative permutation is
particularly useful for instantiating and systematically studying block ciphers
and hash functions over Fp​ aimed for multiparty computation and
zero-knowledge based cryptographic protocols. Finally, we provide the
discrepancy analysis a technique used to measure the (pseudo-)randomness of a
sequence, for analyzing the randomness of the sequence generated by the generic
permutation or block cipher described by GTDS