The differential uniformity of the power functions xpn+52 over Fpn

Abstract

Cryptographic functions with low differential uniformity have important applications in designing S-box in the block ciphers. In this paper, we mainly investigate the differential uniformity ${\Delta }_F$ on a new class of power mappings $F\left(x\right)=x^{\frac{p^n+5}{2}}$ over $F_{p^n}$ with p being an odd prime and n being a positive integer. More precisely, for $p=3$, the differential uniformity and the differential spectrum of F have been determined explicitly. The results indicate that F is a locally-PN function with differentially $\frac{3^n+1}{4}$-uniform when n is odd and a locally-APN function with differentially $\frac{3^n+3}{4}$-uniform when n is even. Then, for $p=5$, we prove that F is APN for even n and ${\Delta }_F=6$ for odd n through specific differential equations and quadratic character over $F_{5^n}$. The method is different from the existing one. Moreover, for primes $p>5$, we show that ${\Delta }_F\le 5$ when $p^n\equiv 3\left(\mathrm{mod}4\right)$ and ${\Delta }_F\le 8$ when $p^n\equiv 1\left(\mathrm{mod}4\right)$.