2014/10/27 by Gologlu, Faruk
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1411.2981
In this paper we give a new family of APN trinomials of the form X2k+1 + (trnm(X))2k+1 on \mathbbF2n where gcd(k,n)=1 and n = 2m = 4t, and prove its important properties. The family satisfies for all n = 4t an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on \mathbbF22m. As another contribution of the paper, we consider a family of hexanomials gC,k which was shown to be differentially 2gcd(m,k)-uniform by Budaghyan and Carlet (2008) when a quadrinomial PC,k has no roots in a specific subgroup. In this paper, for all (m,k) pairs, we characterize, construct and count all C ∈ \mathbbF2n satisfying the condition. Bracken, Tan and Tan (2014) and Qu, Tan and Li (2014) constructed some elements C satisfying the condition when m ≡ 2 \textrmor 4 \pmod6 and m ≡ 0 \pmod6 respectively, both requiring gcd(m,k) = 1. Bluher (2013) proved that such C exists if and only if k ≠ m without characterizing, constructing or counting those C. To prove the results, we effectively use a Trace-0/Trace-1 (relative to the subfield \mathbbF2m) decomposition of \mathbbF2n.