Exploration of Balanced Boolean Functions: Advancements in Cryptography and Coding Theory
Boolean functions have become integral components of various fields such as cryptography, coding theory, and sequence design due to their unique mathematical properties. Recent research has focused on enhancing these functions, particularly those characterized by few-valued spectra. The latest advancements in the parametric construction approach have yielded promising results for generating balanced Boolean functions through specialized mappings, including the innovative $-to-$ transformation expressed as $P(x^2+x)$, where $P$ represents carefully selected permutation polynomials.
This study introduces a significant contribution to the field with the establishment of a new family of four-valued spectrum Boolean functions. These functions are noteworthy not only for their theoretical underpinnings but also for their practical applications. Among their attributes are enhanced cryptographic features, including the same nonlinearity as semi-bent functions, which are pivotal for ensuring security in cryptographic systems. Furthermore, these new functions boast a maximal algebraic degree alongside optimal algebraic immunity for dimensions $n leq 14$. Algebraic immunity, a critical measure in assessing the resilience of cryptographic functions against algebraic attacks, underscores the practical relevance of these findings in the design of secure cryptographic protocols.
In addition to the development of four-valued spectra, the research further identifies seven distinct classes of plateaued functions. This includes four infinite families of semi-bent functions and a suite of near-bent functions, presenting a diverse landscape of Boolean functions with varying degrees of complexity and applicability. The identification of these functions is particularly relevant for cryptographic applications, where the ability to design functions with controlled nonlinearity and resistance to specific types of attacks can significantly enhance security measures.
Overall, the implications of this research extend beyond theoretical curiosity; they provide a foundation for future innovations in cryptography and related fields. By refining the construction of Boolean functions through systematic mappings and studying their algebraic properties, researchers bolster the robustness of cryptographic systems. As demand for secure digital communication continues to grow, the exploration of Boolean functions remains a crucial area of study, promising future advancements in not only security but also the efficiency of data transmission technologies. This work thus marks a significant step forward in the quest for more effective cryptographic techniques, paving the way for further exploration and development in mathematical function theory.
