On the cryptographic properties of weightwise affine and weightwise quadratic functions

MEAUX, Pierrick; OZAIM, Yassine

doi:10.1016/J.DAM.2024.04.008

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On the cryptographic properties of weightwise affine and weightwise quadratic functions

MEAUX, Pierrick; OZAIM, Yassine

2024 • In Discrete Applied Mathematics

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https://hdl.handle.net/10993/57255

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10.1016/J.DAM.2024.04.008

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Keywords :

Boolean functions; Cryptography; symmetric functions; HWBF

Abstract :

[en] Weightwise degree-d functions are Boolean functions that take the values of a function of degree at most d on each set of fixed Hamming weight. The class of weightwise affine functions encompasses both the symmetric functions and the Hidden Weight Bit Function (HWBF). The good cryptographic properties of the HWBF, except for the nonlinearity, motivates to investigate a larger class with functions that share the good properties and have a better nonlinearity. Additionally, the homomorphic friendliness of symmetric functions exhibited in the context of hybrid homomorphic encryption and the recent results on homomorphic evaluation of Boolean functions make this class of functions appealing for efficient privacy-preserving protocols. In this article we realize the first study on weightwise degree-d functions, focusing on weightwise affine and weightwise quadratic functions. We show some properties on these new classes of functions, in particular on the subclass of cyclic weightwise functions. We provide balanced constructions and prove nonlinearity upper bounds for all cyclic weightwise affine functions and for a family of weightwise quadratic functions. We complement our work with experimental results, they show that other cyclic weightwise linear functions than the HWBF have better cryptographic parameters, and considering weightwise quadratic functions allows to reach higher algebraic immunity and substantially better nonlinearity.

Disciplines :

Mathematics

Author, co-author :

MEAUX, Pierrick ^✱; University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT) > PI Coron

OZAIM, Yassine

^✱ These authors have contributed equally to this work.

External co-authors :

yes

Language :

English

Title :

On the cryptographic properties of weightwise affine and weightwise quadratic functions

Original title :

[en] On the cryptographic properties of weightwise affine and weightwise quadratic functions

Publication date :

2024

Journal title :

Discrete Applied Mathematics

ISSN :

0166-218X

eISSN :

1872-6771

Publisher :

Elsevier, Amsterdam, Netherlands

Peer reviewed :

Peer Reviewed verified by ORBi

Focus Area :

Security, Reliability and Trust

Available on ORBilu :

since 27 October 2023

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