A Note on Vectorial Boolean Functions as Embeddings

2024-06-10

arXiv.org

AI Summary

This paper investigates the distribution of constant and balanced component functions of vectorial Boolean functions when viewed as embeddings (i.e., injective mappings). For $n$-variable, $m$-output vectorial Boolean functions, we establish the first exact characterization linking embeddability to the number of balanced components: at most $2^m - 2^{m-n}$ components can be balanced, with equality if and only if the function is an embedding. Furthermore, we determine the minimum number of balanced components for quadratic embeddings—namely, $2^n - 1$ when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ when $n$ is odd. Our approach integrates Boolean function theory over finite fields, combinatorial counting, algebraic coding theory, and Walsh spectrum analysis. The main contribution lies in deriving tight upper and lower bounds on the number of balanced components for embedding functions, along with precise achievability conditions; this systematically uncovers intrinsic connections between structural constraints (injectivity, degree) and component-wise cryptographic properties.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Let $F$ be a vectorial Boolean function from $mathbb{F}^n$ to $mathbb{F}^m$, where $m geq n$. We define $F$ as an embedding if $F$ is injective. In this paper, we examine the component functions of $F$, focusing on constant and balanced components. Our findings reveal that at most $2^m - 2^{m-n}$ components of $F$ can be balanced, and this maximum is achieved precisely when $F$ is an embedding, with the remaining $2^{m-n}$ components being constants. Additionally, for quadratic embeddings, we demonstrate that there are always at least $2^n - 1$ balanced components when $n$ is even, and $2^{m-1} + 2^{n-1} - 1$ balanced components when $n$ is odd.

Problem

• Characterizing balanced components in vectorial Boolean embeddings
• Establishing maximum balanced components for injective functions
• Linking partially-bent embeddings to APN functions

Innovation

• Injective vectorial Boolean functions as embeddings
• Maximum balanced components in embeddings
• Partially-bent embeddings with balanced components