Bijective S-boxes of different sizes obtained from quasi-cyclic codes

Dusan Bikov, Iliya Bouyukliev, Stefka Bouyuklieva


The aim of this paper is to construct S-boxes of different sizes with good cryptographic properties. An algebraic construction for bijective S-boxes is described. It uses quasi-cyclic representations of the binary simplex code. Good S-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18 are obtained.

Full Text:



D. Bikov, I. Bouyukliev, BoolSPLG: A library with parallel algorithms for Boolean functions and S-boxes for GPU.

D. Bikov, I. Bouyukliev, Parallel Fast Walsh Transform Algorithm and its implementation with CUDA on GPUs, Cybernetics and Information Technologies, Cybernetics and Information Technologies 18(5) (2018) 21–43.

I. Bouyukliev, D. Bikov, S. Bouyuklieva, S-boxes from binary quasi-cyclic codes, Electronic Notes in Discrete Mathematics 57 (2017) 67–72.

C. Carlet, Boolean Functions for Cryptography and Error Correcting Codes, In: Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Crama, Hammer, Cambridge University Press, 2010.

C. Carlet, Vectorial Boolean Functions for Cryptography, In: Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Crama, Hammer, (Eds.), Cambridge University Press, 2010.

E. Z. Chen, New quasi-cyclic codes from simplex codes, IEEE Trans. Inform. Theory 53(3) (2007) 1193–1196.

CUDA Zone.

J.Daeman, V.Rijmen, The Design of Rijndael, AES–the advanced encryption standard, Springer-Verlag Berlin Heidelberg, 2002.

I. Hussain, T. Shah, M. A. Gondal, W. A. Khan, Construction of Cryptographically Strong $8times 8$ S-boxes, World Applied Sciences Journal 13 (2011) 2389–2395.

G. Ivanov, N. Nikolov, S. Nikova, Reversed genetic algorithms for generation of bijective S-boxes with good cryptographic properties, Cryptogr. Commun. 8(2) (2016) 247–276.

K. Lally, P. Fitzpatrick, Algebraic structure of quasi-cyclic codes, Discrete Applied Mathematics 111(1–2) (2001) 157–175.

G. Leander, A. Poschmann, On the Classification of 4 Bit S-Boxes, In: Carlet C., Sunar B. (eds) Arithmetic of Finite Fields. WAIFI 2007. Lecture Notes in Computer Science, vol 4547. Springer, Berlin, Heidelberg (2007) 159–176.

S. Ling, P. Solé, On the algebraic structure of quasi-cyclic codes I: finite fields, IEEE Trans. Inform. Theory 47(7) (2001) 2751–2760.

F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam 1977.

NVIDIA Data Center.

M. J. O. Saarinen, Cryptographic Analysis of all $ 4times4 $–bit S–boxes, In: Proceedings of the 18th International Conference on Selected Areas in Cryptography, ser. SAC 11. Springer-Verlag (2012) 118–133.

W. Zhang, Z. Bao, V. Rijmen, M. Liu, A New Classification of 4-bit Optimal S-boxes and Its Application to PRESENT, RECTANGLE and SPONGENT. In: Leander G. (eds) Fast Software Encryption. Lecture Notes in Computer Science, vol 9054. Springer, Berlin, Heidelberg (2015) 494–515.


  • There are currently no refbacks.

ISSN: 2148-838X