CRYPTOGRAPHICALLY STABLE BOOLEAN FUNCTIONS: ANALYSIS OF NONLINEARITY, STABILITY, AND CORRELATIONAL IMMUNITY

Abstract

In the modern digital world, where the volume of information transmitted and stored electronically is rapidly increasing, the issue of ensuring the confidentiality, integrity, and authenticity of data has become critically important. Among the numerous areas of cryptography that provide information protection, symmetric encryption systems—specifically stream and block ciphers—hold a special place. One of the fundamental components of such systems is Boolean functions, which serve as the core for constructing cryptographic primitives. Boolean functions, as mappings from binary space to binary values, may seem like simple mathematical objects at first glance. However, in the context of cryptography, they must meet complex and often conflicting requirements. An ideal Boolean function for cryptographic purposes must be nonlinear to prevent linear attacks; stable, meaning balanced and insensitive to fixing a certain number of variables; and correlation immune to withstand correlation attacks based on the analysis of statistical relationships between input and output bits.

Unfortunately, achieving all these properties simultaneously is limited by a number of theoretical results, particularly the Zaytsev–Siegel–Chang theorem, which establishes the boundaries of possible coexistence of stability, nonlinearity, and immunity in Boolean functions. This presents a real challenge for cryptographers: to find functions that, while not ideal, provide an acceptable compromise between security and efficiency.

The aim of this article is a deep analysis of the main cryptographic properties of Boolean functions — nonlinearity, stability, and correlation immunity. Formal definitions of these characteristics, algorithms for their computation, and examples of functions used in practice are considered. Special attention is paid to the interrelationship of properties and the theoretical limitations that define the boundaries of constructing cryptographically strong Boolean functions.