On multiple output bent functions

Abstract

In this article we investigate the possibilities of obtaining multiple output bent functions from certain power polynomials over finite fields. So far multiple output bent functions F:GF(2)n→GF(2)m (where n is even and m⩽n/2), for any particular class of Boolean bent functions, has been generated using a suitable collection of m Boolean bent functions so that any nonzero linear combination of these functions is again bent. Here, we take a different approach by deriving these functions directly from the known classes of so-called monomial trace bent functions. We derive a sufficient condition for a bent Boolean function of the form f(x)=Tr1n(λxd) so that the associated mapping F(x)=Trmn(λxd), where F:GF(2)n→GF(2)m, is a multiple output bent function. We consider all the main cases of monomial trace bent functions and specify the restrictions on λ and m that yield multiple output bent functions F(x)=Trmn(λxd). Interestingly enough, in one particular case when n=4r, d=(2r+1)2, a multiple bent function F(x)=Tr2rn(axd) could not be obtained by considering a collection of 2r Boolean bent functions of the form fi(x)=Tr1n(λixd) for some suitable coefficients λi∈GF(2n).