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

We summarize the state of the classification of almost perfect nonlinear (A PN) power functions x(d) on GF(2(n)) 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 powe r function x(2m)+3 is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift reg ister sequences of degree n and a decimation of that sequence by 2(m) + 3 t akes on precisely the three values -1, -1 +/-2(m+1).