Some power mappings with low differential uniformity (Q1360984)

For a self-map \(f\) of the finite field \(\mathrm{GF}(q)\) and for \(a,b\in \mathrm{GF}(q)\) with \(a\neq 0\), let \(N(a,b)\) be the number of solutions \(x\in \mathrm{GF}(q)\) of \(f(x+ a)-f(x)= b\). Let \(\Delta_f\) denote the maximum of the numbers \(N(a,b)\). The map \(f\) is said to be differentially \(k\)-uniform if \(\Delta_f= k\). Such maps are of interest in the cryptanalysis of block ciphers. The authors construct three infinite families of maps of the form \(f(x)= x^d\) with low differential uniformity. In the first family, constructed in Theorem 1, the value of \(\Delta_f\) is the smallest possible. Note that in this theorem the condition that \(\mathrm{GF}(q)\) be of characteristic 2 has to be added. It is also shown that any differentially 1-uniform map \(f(x)= x^d\) yields a family of sequences with good correlation properties.