Poschmann, On the classicication of 4 bit S-boxes, WAIFI 2007 (2007), 159–176. Qu, Construction of highly nonlinear resilient S-boxes with given degree, Designs, Codes and Cryptography, 64 (3) (2012), 241–253. Shan, The stability Theory of Strream Ciphers, Lectrure Notes in Computer Science, Spring-Verlag, Berlin, Heidelberg, New York, 561, 1991.
De Canniere, Analysis and Design of Symmtric Encryption Algoithms, PhD thesis, KU Leuven, 2007. Sethumadhavan, Affine equivalence of monomial rotation symmetric Boolean functions: A P ólya’s theorem approach, Journal of Mathematical Cryptology, 10 (3-4) (2016), 145–156. Shamir, Differential cryptanalysis of DES-like cryptosystems, In International Cryptology Conference on Advances in Cryptology, CRYPTO 1990, Springer-Verlag, London, pp. Khonji, Linear and differential cryptanalysis of small-sized random ( n, m) S-boxes, 2016 11-th International Conference for Internet Technology and Secured Transactions (ICITST), pp. The authors wish to thank the anonymous referees for their valuable comments to improve the presentation of this paper. It is also interesting to extend the method for constructing new S-boxes. In the next step, it will be an important issue to study the difference properties, the algebraic immunity, and the global avalanche characteristics of cross-correlation function of the S-box in this paper. These properties can help us to evaluate whether the S-boxes can be used in Block cipher or Stream cipher or not. Meanwhile, these results of this paper are given for any n-bit S-box and any G ∈ ? n, if G is replaced with the rotation symmetric Boolean function, then more specific results can be obtained. Our results are a supplement to the result in. In this paper, we give the distributions of Walsh spectrum and the distributions of the autocorrelation functions for ( n + 1)-bit S-boxes in, and obtain the correlation immunity, propagation criterion, etc. Σ ( v, v n + 1 ) ⋅ S ′ = 2 σ v ⋅ S, v n + 1 = 0 2 σ v ⋅ S ⊕ G, v n + 1 = 1 f o r g i v e n v ∈ F 2 n. Every Boolean function f ∈ ? n admits a unique representation (called its algebraic normal form ( ANF)) as a polynomial over ? 2: Let ? n be the set of n-variable Boolean functions, and ⊕ be additions in ? 2, in F 2 n and in ? n. In Section 3, we give some cryptographic properties of this construction.
In Section 2, the basic concepts and notions are presented. The organization of this paper is as follows. These results, which are obtained in this paper, can help us to further understand the cryptographic properties of such construction method. Therefore, we give these cryptographic properties in this paper. But they did not give cryptographic properties of ( n + 1)-bit S-boxes, such as resilient, the propagation criterion, the distribution of the Walsh spectrums and the autocorrelation functions, etc. From, some results on the presence of self-equivalent S-boxes and involutions in affine equivalence classes were presented. In this paper, we focus on ( n + 1)-bit S-boxes constructed by n-bit S-boxes in. Meanwhile, the classification of all 3-bit S-boxes and 4-bit S-boxes according to affine equivalency were given for the first time in, respectively. constructed the ( n + 1)-bit S-boxes from n-bit S-boxes with known sharings, and investigated the self-equivalency of S-boxes.