A New Representation of Binary Sequences by means of Boolean Functions

TL;DR

This paper introduces a novel bijection between Boolean functions and binary sequences with periods as powers of two, enabling new analysis methods in cryptography.

Contribution

It presents a new representation called reverse-ANF based on Boolean functions, linking different sequence representations and properties.

Findings

Established a bijection between Boolean functions and binary sequences.

Defined the reverse-ANF sequence representation based on algebraic normal form.

Analyzed generalized self-shrinking sequences using the new representations.

Abstract

Boolean functions and binary sequences are main tools used in cryptography. In this work, we introduce a new bijection between the set of Boolean functions and the set of binary sequences with period a power of two. We establish a connection between them which allows us to study some properties of Boolean functions through binary sequences and vice versa. Then, we define a new representation of sequences, based on Boolean functions and derived from the algebraic normal form, named reverse-ANF. Next, we study the relation between such a representation and other representations of Boolean functions as well as between such a representation and the binary sequences. Finally, we analyse the generalized self-shrinking sequences in terms of Boolean functions and some of their properties using the different representations.