To identify and specify trace bent functions of the form Tr(P(x)), where P(x) ∈ F(2n)[x], has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form F(x) = Trkn(Σi=0(2k) aixi((2k)-1)), where n = 2k, in terms of finding an explicit expression for the coefficients aiso that F is vectorial hyperbent. These coefficients only depend on the choice of the interpolating polynomial used in the Lagrange interpolation of the elements of U and some prespecified outputs, where U is the cyclic group of (2n/2+ 1)th roots of unity in F(2n). We show that these interpolation polynomials can be chosen in exactly (2k+ 1)!2k-1ways and this is the exact number of vectorial hyperbent functions of the form Trkn(Σi=02kaixi((2k)-1)). Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.

The authors of this paper recently researched the possibility of developing programming language implementation, that is neither compiler, nor interpreter. The concept is based on keeping the complete program in native machine code, but the specialized editor can 'on the fly' decompile the machine code and display it as high level language. The displayed code can be edited and saved again as pure machine code. This paper reviews the possibility of optimizing generated code, while still retaining the possibility of decompilation. We found many important code sequences which can be replaced with shorter ones while keeping the code in decompilable executable format.