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/.
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机译：我们总结了GF（2 / sup n /）上的近乎完美的非线性（APN）幂函数x / sup d /的分类状态，并贡献了两个新案例。为了证明这些情况，我们导出了新的置换多项式。第一种情况支持韦尔奇（Welch）的一个众所周知的猜想，即对于奇数n = 2m + 1，幂函数x / sup 2m + 3 /甚至是最大非线性的，或者换句话说，二进制最大值与整数之间的互相关函数长度为n的线性移位寄存器序列和该序列的2 / sup m / + 3的抽取精确地取三个值-1，-1 / spl plusmn / 2 / sup m + 1 /。
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