Differential spectrum of some power functions in odd prime characteristic (Q1946681)

The main result of the paper is a formula of differential spectrum for two power functions. First, for \(f(x)=x^{\frac{p^k+1}{2}}\) defined over the finite field \(\mathbb F_{p^n}\) where \(p>2\) is a prime number. So, the result from [\textit{T. Helleseth, C. Rong} and \textit{D. Sandberg}, IEEE Trans. Inf. Theory 45, No. 2, 475--485 (1999; Zbl 0960.11051)] -- where only an upper bound is computed -- is improved. Second, for \(f(x)=x^{\frac{p^n+1}{p^m+1}+\frac{p^n-1}{2}}\) where \(p\equiv 3\pmod 4\) is a prime number, and \(n\) is an odd integer with \(m|n\). The authors consider that this is the first paper where the differential spectrum of power functions with an odd prime characteristic is determined exactly. As to remember, the characteristic of a function \(f:\mathbb F_{p^n}\longrightarrow \mathbb F_{p^n}\) is \[ \Delta_f=\max_{a\in \mathbb F_{p^n}^*,\, b\in\mathbb F_{p^n}}N_f(a,b) \] where \(N_f(a,b)\) is the number of solutions of the equation \(f(x+a)-f(x)=b\). The differential spectrum of the function \(f\) with \(\Delta_f=k\) is \((\omega_0,\omega_1,\dots,\omega_k)\) where \[ \omega_i=\text{card}\left(\{b\in \mathbb F_{p^n}\mid N_f(1,b)=i\}\right). \] In cryptography the mappings \(f\) with \(\Delta_f=1\) (perfect nonlinear) and \(\Delta_f=2\) (almost perfect nonlinear) are very important and widely studied. The paper offers a general method to obtain such type of functions.