Permutation binomials over finite fields

By Ariane M. Masuda, Michael E. Zieve. 2007

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We prove that if x m + ax n permutes the prime field F p , where m > n > 0 and a ∈ F * p , then gcd(m - n,p ― 1) > √/p ― 1. Conversely, we prove that if q > 4 and m > n > 0 are fixed and satisfy gcd(m - n, q ― 1) > 2q(log log q)/ log q, then there exist permutation binomials over F q of the form x m + ax n if and only if gcd(m, n, q - 1) = 1.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/327741d858e3a565b4ad2ea23c5e7526b421b069