Weightwise perfectly balanced functions with high weightwise nonlinearity profile

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher (Méaux et al.: in Lecture Notes in Computer Science, vol. 9665, pp. 311–343, Springer, Berlin, 2016). In this paper, we propose a large class of weightwise perfectly balanced (WPB) functions, which is 2-rotation symmetric. This new class of WPB functions is not extended affinely equivalent to the known constructions. We also discuss the weightwise nonlinearity profile of these functions, and present general lower bounds on k-weightwise nonlinearity, where k is a power of 2. Moreover, we exhibit a subclass of the family. By a recursive lower bound, we show that these subclass of WPB functions have very high weightwise nonlinearity profile.

[1]  Selçuk Kavut,et al.  Results on rotation-symmetric S-boxes , 2012, Inf. Sci..

[2]  Yuval Filmus,et al.  An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube , 2014, Electron. J. Comb..

[3]  Claude Carlet,et al.  Vectorial Boolean Functions for Cryptography , 2006 .

[4]  Guy Kindler,et al.  Invariance Principle on the Slice , 2015, CCC.

[5]  Antoine Joux,et al.  Galois LFSR, Embedded Devices and Side Channel Weaknesses , 2006, INDOCRYPT.

[6]  Yuval Filmus.  Friedgut-Kalai-Naor Theorem for Slices of the Boolean Cube , 2016, Chic. J. Theor. Comput. Sci..

[7]  Claude Carlet,et al.  Towards Stream Ciphers for Efficient FHE with Low-Noise Ciphertexts , 2016, EUROCRYPT.

[8]  Joan Daemen,et al.  Cipher and hash function design strategies based on linear and differential cryptanalysis , 1995 .

[9]  Martin Hell,et al.  A Stream Cipher Proposal: Grain-128 , 2006, 2006 IEEE International Symposium on Information Theory.

[10]  Cunsheng Ding,et al.  On the nonlinearity of Boolean functions with restricted input , 2018, Cryptography and Communications.

[11]  Claude Carlet,et al.  Boolean functions with restricted input and their robustness; application to the FLIP cipher , 2017, IACR Trans. Symmetric Cryptol..

[12]  Xiaohu Tang,et al.  Systematic Constructions of Rotation Symmetric Bent Functions, 2-Rotation Symmetric Bent Functions, and Bent Idempotent Functions , 2017, IEEE Transactions on Information Theory.

[13]  Selçuk Kavut,et al.  9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class , 2010, Inf. Comput..

[14]  Wenfen Liu,et al.  A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions , 2014, J. Comb. Theory, Ser. A.

[15]  Virginie Lallemand,et al.  Cryptanalysis of the FLIP Family of Stream Ciphers , 2016, CRYPTO.

[16]  Claude Carlet,et al.  Boolean Functions for Cryptography and Error-Correcting Codes , 2010, Boolean Models and Methods.

[17]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[18]  Elchanan Mossel,et al.  Harmonicity and Invariance on Slices of the Boolean Cube , 2016, Computational Complexity Conference.