A general secondary construction of Boolean functions including the indirect sum and its generalizations

2025-05-17

AI Summary

This work addresses the theoretical unification and spectral analysis of secondary constructions of Boolean functions. Method: We propose a general quadratic construction framework encompassing direct sums, indirect sums, and their various generalizations—unifying classical constructions and their known extensions for the first time. Under the key condition $(g oplus g')(g oplus g'') equiv 0$, we derive a novel Walsh spectral identity relating $W_g$, $W_{g'}$, $W_{g''}$, and $W_{g oplus g' oplus g''}$. Using algebraic normal form (ANF) derivation, Walsh spectral analysis, and cryptographic parameter modeling, we rigorously characterize the Walsh transform of the constructed functions. Contribution/Results: Our analysis systematically reveals propagation rules for nonlinearity, algebraic degree, and balancedness. The framework strengthens the theoretical foundation for high-dimensional Boolean function design and provides new tools and perspectives for cryptographic function construction.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms $W_g,W_{g'},W_{g''}$, and $W_{goplus g'oplus g''}$ when $g,g',g''$ are Boolean functions such that $(goplus g')(goplus g'')$ equals the zero function.

Problem

• Generalizes secondary constructions of Boolean functions
• Unifies direct and indirect sum methods
• Analyzes Walsh transforms of constructed functions

Innovation

• Generalizes direct and indirect sum constructions
• Unifies known secondary Boolean function constructions
• Analyzes Walsh transforms of constructed functions