Shift-invariant transformations and almost liftings

TL;DR

This paper studies shift-invariant transformations derived from Boolean functions, introducing the concept of almost liftings, and analyzes their cryptographic properties and collision bounds, with implications for hash function design.

Contribution

It defines almost liftings for Boolean functions, establishes collision bounds, and explores their cryptographic potential, generalizing known hash function components.

Findings

Maximum collisions for almost liftings are 2^{k-1} for diameter k functions.

Almost liftings can have good cryptographic properties despite non-bijectivity.

Generalization of the Keccak map hi for cryptographic applications.

Abstract

We investigate shift-invariant transformations, also known as rotation-symmetric vectorial Boolean functions, on $n$ bits that are induced from Boolean functions on $k$ bits, for $k\leq n$. We consider such transformations that are not necessarily permutations, but are, in some sense, almost bijective, and study their cryptographic properties. In this context, we define an almost lifting as a Boolean function for which there is an upper bound on the number of collisions of its induced transformation that does not depend on $n$. We show that if a Boolean function with diameter $k$ is an almost lifting, then the maximum number of collisions of its induced transformation is $2^{k-1}$ for any $n$. Moreover, we search for functions in the class of almost liftings that have good cryptographic properties and for which the non-bijectivity does not cause major security weaknesses. These functions generalize the well-known map $\chi$ used in the Keccak hash function.