Almost perfect nonlinear trinomials and hexanomials

作者：Goeloglu Faruk

来源：Finite Fields and Their Applications, 2015, 33: 258-282.

DOI：10.1016/j.ffa.2014.12.006

• APN functions
• Crooked functions

In this paper we give a new family of almost perfect nonlinear (APN) trinomials of the form X2k+1 (tr(m)(n)(X))2(k+1) on F-2n where gcd(k,n) = 1 and n = 2m = 4t, and prove its important properties. The family satisfies for all n = 4f an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on F-22m. As another contribution of the paper, we consider a family of hexanomials g(C,k) which was shown to be differentially 2(gcd(m,k))-uniform by Budaghyan and Car let (2008) when a quadrinomial P-C,P-k has no roots in a specific subgroup. In this paper, for all (m, k) pairs, we characterize, construct and count all C is an element of F-2n satisfying the condition. Bracken, Tan and Tan (2014) and Qu, Tan and Li (2014) constructed some elements C satisfying the condition when m equivalent to 2 or 4 (mod 6) and m equivalent to 0 (mod 6) respectively, both requiring gcd(m, k) = 1. Bluher (2013) proved that such C exists if and only if k not equal m without characterizing, constructing or counting those C. To prove the results, we effectively use a Trace-0/Trace-1 (relative to the subfield F-2m) decomposition of F-2n.

• 出版日期2015-5