2017/08/03 by Kangquan Li, Longjiang Qu, Li, Kangquan +3
Computer Science · Engineering · #06E30 #11T06 #94A60 #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1708.01165
openalex publication_date 2017/08/03 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form xrh(xq-1) over \mathbbFq2, where q=2k, h(x)=1+xs+xt and r, s, t, k>0 are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \mathbbFq2 into that of showing the corresponding fractional polynomials permute a smaller set μq+1, where μq+1:=\x∈\mathbbFq2 : xq+1=1\. Motivated by these results, we characterize the permutation polynomials of the form xrh(xq-1) over \mathbbFq2 such that h(x)∈\mathbbFq[x] is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \mathbbFq2 into that of showing permutations over a small subset S of a proper subfield \mathbbFq, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form xrh(xq-1) over \mathbbFq2. Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic.