Permutation polynomials over Zpn are useful in the design of cryptographic algorithms. In this paper, we obtain an equivalent condition for polynomial functions over Zpn to be permutations, and this equivalent condition can help us to analysis the randomness of such functions. Our results provide a method to distinguish permutation polynomials from random functions. We also introduce how to improve the randomness of permutation polynomials over Zpn.

Feedback with carry shift registers (FCSRs) implemented using Galois representation have been found to have a weakness called LFSRization. It leads to powerful attacks against the stream ciphers based on them. A new representation called ring representation has been proposed to avoid the attacks. It was considered to circumvent the weaknesses of Galois FCSRs. This correspondence presents a class of ring FCSRs, which meet the implementation criteria, but are still possible to maintain linear behavior for several clock cycles. Their LFSRization probability and how to improve their security are also mentioned.