The role of Boolean functions in hiding sums as trapdoors for some block ciphers

Most modern block ciphers are built using components whose cryptographic strength is evaluated in terms of their resistance to attacks on the whole cipher. In particular, differential properties of vectorial Boolean functions are studied for the S-Boxes to thwart differential cryptanalysis. Little is known on similar properties to avoid trapdoors in the design of the block cipher. In this paper we present a form of trapdoors coming from alternative vector space structures, which we call hidden sums, and give a characterization on the Boolean function S-Box to avoid any such hidden sum. We also study some properties of this new class of vectorial Boolean functions, which we call anti-crooked, and provide a toy cipher with a hidden sum trapdoor.