On multiple output bent functions

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)mF:GF(2)n→GF(2)m (where n   is even and m⩽n/2m⩽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)mF: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=4rn=4r, d=(2r+1)2d=(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)λi∈GF(2n).

► Turning Boolean trace bent functions into multiple output bent functions is considered. ► A sufficient condition for the derivation of multiple output bent functions is given. ► Several cases of monomial Boolean trace bent functions are analyzed. ► Multiple output bent functions are derived from these classes.