Interplay between resiliency and polynomial degree — Recursive amplification, higher order sensitivity and beyond

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well kn...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 365; pp. 138 - 159
Main Authors: Maitra, Subhamoy, Mukherjee, Chandra Sekhar, Stănică, Pantelimon, Tang, Deng
Format: Journal Article
Language:English
Published: Elsevier B.V 15.04.2025
Subjects:
Higher order sensitivity
Maiorana-McFarland construction
Polynomial degree
Resiliency
Sensitivity
Separation
06E30
Sensitivity
94D10
Separation
Maiorana-McFarland construction
94A60
Resiliency
Polynomial degree
Higher order sensitivity
ISSN:0166-218X
Online Access:Get full text
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Summary:Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.
ISSN:0166-218X
DOI:10.1016/j.dam.2025.01.003