Design of hash functions and pseudo-random permutations over Galois extensions of \(\mathbb {Z}_{\varvec{q}}\) for prime powers \(\varvec{q}\) has recently gained some interest in relation to recent directions in advanced cryptography, such as multiparty computation and zero-knowledge protocol design. Thus investigating optimality of cryptographic properties of S-boxes defined by polynomials over Galois rings is of interest. Of particular interest is the differential uniformity of such functions. To our knowledge, there are very few results on the differential uniformity for polynomials over Galois rings GR \(\varvec{(p^k,m)}\) when \(\varvec{k,m\ge 2}\) . Motivated by designing secure hash functions and block ciphers over Galois rings, a main contribution of this paper is an investigation into the differential properties of polynomials over Galois rings. Finally, we provide a classification of APN permutations in GR \(\varvec{(4,2)}\) up to affine and CCZ-equivalence.
