Construction of balanced vectorial Boolean functions with almost optimal nonlinearity and very low differential-linear uniformity

Abstract

The differential-linear connectivity table (DLCT) of a vectorial Boolean function was recently introduced by Bar-On et al. at EUROCRYPT'19, whose value at a point is related to the autocorrelation value of its component functions. Further, in INDOCRYPT'19, we proposed a new construction method for vectorial Boolean functions with very low differential-linear uniformity using Maiorana–McFarland bent functions. The difficulty of that construction method was to identify the permutations and the sub-functions that satisfy the conditions to attain good cryptographic properties. In this paper we discover novel techniques to construct such sub-functions to generate vectorial Boolean functions with substantially improved cryptographic properties. Our proposed methods are based on ideas from combinatorics as well as finite fields. In particular, we construct the sub-functions to generate (4t,t−1)-function, t≥5, in a different manner than our Indocrypt'19 paper. Further our new methods help in obtaining sub-functions to generate balanced (4t+2,t−1)-function and (2k,k)-function with very good nonlinearity and very low differential-linear uniformity, that were never demonstrated earlier.