## Cryptology ePrint Archive: Report 2008/196

A New Family of Perfect Nonlinear Binomials

Zhengbang Zha and Gohar M. Kyureghyan and Xueli Wang

Abstract: We prove that the binomials $x^{p^s+1}-\alpha x^{p^k+p^{2k+s}}$ define perfect nonlinear mappings in $GF(p^{3k})$ for an appropriate choice of the integer $s$ and $\alpha \in GF(p^{3k})$. We show that these binomials are inequivalent to known perfect nonlinear monomials. As a consequence we obtain new commutative semifields for $p\geq 5$ and odd $k$.

Category / Keywords: foundations / perfect nonlinear functions, almost perfect nonlinear functions