On generalized Redei functions (Q1115920)

Let L be a finite extension of the field K and let \(\theta =(\theta_ 1,...,\theta_ n)\) be a basis of L over a field K. The discriminant matrix of L over K with respect to this basis is denoted by D. If \(f\in K[x]\) a vector function can be obtained by \(f((x_ 1,...,x_ n))=(f(x_ 1),...,f(x_ n))\). Then by \(\bar f^{\theta}((x_ 1,...,x_ n)^ T)=D^{-1}f(D(x_ 1,...,x_ n)^ T)\) a polynomial vector is defined. - For \(f(x)=x^ k\) this definition yields the so- called generalized Rédei functions denoted by \(\bar f_ k^{\theta}\). A necessary and sufficient condition to induce a permutation of \((GF(q))^ n\) for Rédei function vectors \(\bar f_ k^{\theta}\) is given. In order to obtain also non-trivial results for residue class rings \({\mathbb{Z}}/(m)\) instead of \(x^ k\) the Dickson polynomial \(G_ k(x,a)\) of degree k and parameter \(a\in {\mathbb{Z}}\) is taken as f(x). Finally, applications of these function vectors in cryptography are given.