On generalized bent and q-ary perfect nonlinear functions

On perfect nonlinear functions

“…where a Á x and b Á y are inner products. In fact, the additive character group [2] that a function from Z n q to Z q is perfect nonlinear if and only if, for every u 2 Z q nf0g, the function uf is a bent function.…”

“…The authors are indebted to the referees for their beneficial comments. The authors also express their profound gratitude to Prof. Q. Xiang for his continuous support and to Prof. C. Carlet for sending the paper [2].…”

“…Gupta and Sarkar [5], and Kurosawa and Satoh [7] constructed some perfect nonlinear binary S-boxes with some cryptographic properties (SAC(k), for example). Carlet [2] obtained some perfect nonlinear functions from Z n 4 to Z 4 where n > 1 and from Z 2n p k to Z p k where n; p; k 2 Z; p prime and n ! k. It seems that perfect nonlinear functions are quite rare.…”