Abstract
For multi-output Boolean functions (also called S-boxes), various measures of nonlinearity have been widely discussed in the literature but many problems are left open in this topic. The purpose of this paper is to present a new approach to estimating the nonlinearity of S-boxes. A more fine-grained view on the notion of nonlinearity of S-boxes is presented and new connections to some linear codes are established. More precisely, we mainly study the nonlinearity indicator (denoted by \(\mathcal {N}_{\mathrm {v}}\)) for S-boxes from a coding theory point of view. Such a cryptographic parameter \(\mathcal {N}_{\mathrm {v}}\) is more related to best affine approximation attacks on stream ciphers. We establish a direct link between \(\mathcal {N}_{\mathrm {v}}\) and the minimum distance of the corresponding linear code. We exploit that connection to derive the first general lower bounds on \(\mathcal {N}_{\mathrm {v}}\) of non-affine functions from \(\mathbb {F}_{2^{n}}\) to \(\mathbb {F}_{2^{m}}\) for m dividing n. Furthermore, we show that \(\mathcal {N}_{\mathrm {v}}\) can be determined directly by the weight distribution of the corresponding linear code.
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References
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean models and methods in mathematics, computer science, and engineering, pp 257–397. University Press, Cambridge (2010)
Carlet, C.: Vectorial boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean models and methods in mathematics, computer science, and engineering, pp 398–469. University Press, Cambridge (2010)
Carlet, C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Crypt. 59(1–3), 89–109 (2011)
Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20(2–3), 205–244 (2004)
Carlet, C., Ding, C.: Nonlinearities of S-boxes. Finite Fields Appl. 13(1), 121–135 (2007)
Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005)
Carlet, C., Prouff, E.: On a new notion of nonlinearity relevant to multi-output pseudo-random generators. In: 10th Annual International Workshop, Selected Areas in Cryptography 2003, vol. 3006, pp 291–305. Springer-Verlag, Berlin (2004)
Chen, L., Fu, F.: On the nonlinearity of multi-output Boolean functions. Acta Sci. Nat. Univ. Nankai. 34(4), 28–33 (2001). (in Chinese)
Cohen, G., Honkala, I., Litsyn, S., Lobstein, A.: Covering Codes. Amsterdam, North-Holland (1997)
Delsarte, P.: On subfield sub-codes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)
Ding, C., Xiao, G., Shan, W.: The stability theory of stream ciphers. Lect. Notes Comput. Sci, vol. 561. Springer-Verlag, Berlin (1991)
Gold, R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968)
Lidl, R., Niederreiter, H.: Encyclopedia of mathematics and its applications. Finite Fields, vol. 20. Addison-Wesley Publishing Company, Massachusetts (1983)
Liu, J., Chen, L.: On nonlinearity of the second type of multi-output Boolean functions. Chinese Journal of Engineering Mathematics 31(1), 9–22 (2014). (in Chinese)
MacWilliams, F. J., Sloane, N. J. A.: The theory of error-correcting codes. North-Holland Publishing Company, Amsterdam (1977)
Matsui, M.: Linear cryptanalysis method for DES cipher. In: Advances in Cryptology—EUROCRYPT’93, vol. 765, pp 386–397. Springer-Verlag, Berlin (1993)
Menezes, A., VanOorschot, P., Vanstone, S.: Handbook of applied cryptography. CRC Press, Boca Raton (1996)
Nyberg, K.: Perfect nonlinear S-boxes. In: Advances in Cryptology—EUROCRYPT’91, vol. 547, pp 378–386. Springer-Verlag, Berlin (1992)
Nyberg, K.: On the construction of highly nonlinear permutations. In: Advances in Cryptology—EUROCRYPT’92, vol. 658, pp 92–98. Springer-Verlag, Berlin (1993)
Acknowledgments
The authors are grateful to the anonymous referees for their insightful comments and help in improving the technical quality of this paper. This work is supported by the National Key Basic Research Program of China under Grant 2013CB834204.
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Liu, J., Mesnager, S. & Chen, L. On the nonlinearity of S-boxes and linear codes. Cryptogr. Commun. 9, 345–361 (2017). https://doi.org/10.1007/s12095-015-0176-z
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DOI: https://doi.org/10.1007/s12095-015-0176-z
Keywords
- Symmetric cryptography
- Multi-output Boolean functions
- S-boxes
- Affine approximation attack
- Nonlinearity
- Linear codes