Paper 2023/1680

On the cryptographic properties of weightwise affine and weightwise quadratic functions

Pierrick Méaux, Luxembourg University
Yassine Ozaim

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 lower 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.

Available format(s)
Secret-key cryptography
Publication info
Boolean functionscryptographysymmetric functionsHWBF
Contact author(s)
pierrick meaux @ uni lu
yassine ozaim @ gmail com
2023-11-03: approved
2023-10-30: received
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      author = {Pierrick Méaux and Yassine Ozaim},
      title = {On the cryptographic properties of weightwise affine and weightwise quadratic functions},
      howpublished = {Cryptology ePrint Archive, Paper 2023/1680},
      year = {2023},
      note = {\url{}},
      url = {}
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