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

摘要

We summarize the state of the classification of almost perfect nonlinear (APN) power functions x/sup d/ on GF(2/sup 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 power function x/sup 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/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/sup m+1/.