A new algebraic approach to binary sequences and Boolean functions

Boolean functions and binary sequences are fundamental tools in cryptography. In this work, we introduce a new bijection between the set of Boolean functions and the set of binary sequences whose period is a power of two. This correspondence enables the study of properties of Boolean functions through binary sequences and vice versa. Building on this connection, we propose a novel algebraic description, derived from the algebraic normal form of Boolean functions, which we call the reverse-ANF. Then, we explore how this formulation relates both to existing representations of Boolean functions and to binary sequences. Moreover, several cryptographic properties are examined through this new approach. Finally, we analyse generalized self-shrunken sequences from the perspective of Boolean functions, highlighting several properties that emerge under these different frameworks.