Almost perfect nonlinear power functions on GF(2n): the Welch case

We summarize the state of the classification of almost perfect nonlinear (APN) power functions x^d on GF(2ⁿ) and contribute two new cases. To prove these cases we derive new permutation polynomials. The first case supports a well-known conjecture of Welch stating that for odd n=2m+1, the power function x^2m+3 is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2^m+3 takes on precisely the three values -1, -1±2^m+1