Abstract
This paper investigates the correlation of n-bit to m-bit vectorial Boolean functions denoted by F. At Crypto 2000, Zhang and Chan showed that the maximum of linear approximations for F with Boolean functions g have a higher bias than those based on the usual correlation attack. The correlation for this linear approximation has been named the maximum correlation and has been shown to be a useful tool for correlation attack resistance. In this work, we deal with two issues. Firstly, we show that combining F with any g does not always increase the bias as stated by several works. To justify such results, we demonstrate the exact correlation link between F, g and the combination of F by g. Secondly, we provide the exact condition in which the correlation coefficients for this approximation are maximum.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Braeken, A.: Cryptographic properties of boolean functions and S-boxes. Ph.D. thesis, phd thesis-2006 (2006)
Canteaut, A., Naya-Plasencia, M.: Correlation attacks on combination generators. Crypt. Commun. 4(3–4), 147–171 (2012)
Carlet, C.: Boolean methods and models, ch. boolean functions for cryptography and error correcting codes (2009)
Carlet, Claude, Khoo, Khoongming, Lim, Chu-Wee, Loe, Chuan-Wen: Generalized correlation analysis of vectorial boolean functions. In: Biryukov, Alex (ed.) FSE 2007. LNCS, vol. 4593, pp. 382–398. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74619-5_24
Carlet, C., Khoo, K., Lim, C.W., Loe, C.W.: On an improved correlation analysis of stream ciphers using multi-output boolean functions and the related generalized notion of nonlinearity. Adv. Math. Commun. 2(2), 201 (2008)
Carlet, Claude, Prouff, Emmanuel: On a new notion of nonlinearity relevant to multi-output pseudo-random generators. In: Matsui, Mitsuru, Zuccherato, Robert J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 291–305. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24654-1_21
Daemen, Joan, Govaerts, René, Vandewalle, Joos: Correlation matrices. In: Preneel, Bart (ed.) FSE 1994. LNCS, vol. 1008, pp. 275–285. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60590-8_21
Fuller, J., Millan, W., Dawson, E.: Multi-objective optimisation of bijective s-boxes. New Gener. Comput. 23(3), 201–218 (2005)
Ivanov, G., Nikolov, N., Nikova, S.: Reversed genetic algorithms for generation of bijective s-boxes with good cryptographic properties. Crypt. Commun. 8(2), 247–276 (2016)
Kazymyrov, O., Kazymyrova, V., Oliynykov, R.: A method for generation of high-nonlinear s-boxes based on gradient descent. IACR Cryptology ePrint Arch. 2013, 578 (2013)
Khoo, K., Lim, C.W., Gong, G.: Highly nonlinear balanced s-boxes with improved bound on unrestricted and generalized nonlinearity. Appl. Algebra Eng., Commun. Comput. 19(4), 323–338 (2008)
Nyberg, Kaisa: S-boxes and round functions with controllable linearity and differential uniformity. In: Preneel, Bart (ed.) FSE 1994. LNCS, vol. 1008, pp. 111–130. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60590-8_9
Picek, S., Carlet, C., Jakobovic, D., Miller, J.F., Batina, L.: Correlation immunity of boolean functions: an evolutionary algorithms perspective. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation. pp. 1095–1102. ACM (2015)
Rose, Gregory G., Hawkes, Philip: Turing: a fast stream cipher. In: Johansson, Thomas (ed.) FSE 2003. LNCS, vol. 2887, pp. 290–306. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-39887-5_22
Rueppel, R.A.: Stream ciphers, in\(\backslash \)contemporary cryptology: the science of information integrity. Simmons, G.J. (ed.) (1991)
Tarannikov, Yuriy, Korolev, Peter, Botev, Anton: Autocorrelation coefficients and correlation immunity of boolean functions. In: Boyd, Colin (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 460–479. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45682-1_27
Zhang, Muxiang, Chan, Agnes: Maximum correlation analysis of nonlinear s-boxes in stream ciphers. In: Bellare, Mihir (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 501–514. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44598-6_31
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendices
A Appendix-A
-
\({\varGamma _1}\)
-
\(\varGamma _2\)
B Appendix-B
As Theorem 2 is linked to \(\max \limits _b(c_F(a,b)\pm c_g(b))^2\), we fix \(\max \limits _bc_F(a,b)\) and we vary \(c_g(b)\). The y-axis indicates \(c_{g\circ F}(a)\) and the x-axis indicates \(\max \limits _bc_g(b)\). By computing the white area surface (\(|\varepsilon _{F}|\le |\varepsilon _{g\circ F}|\)), the probability \(Pr_g\) is determined as the ratio of white area surface over the rectangle area surface (Fig. 2).
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Harmouch, Y., El Kouch, R., Ben-Azza, H. (2019). An Improvement of Correlation Analysis for Vectorial Boolean Functions. In: Buchmann, J., Nitaj, A., Rachidi, T. (eds) Progress in Cryptology – AFRICACRYPT 2019. AFRICACRYPT 2019. Lecture Notes in Computer Science(), vol 11627. Springer, Cham. https://doi.org/10.1007/978-3-030-23696-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-23696-0_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-23695-3
Online ISBN: 978-3-030-23696-0
eBook Packages: Computer ScienceComputer Science (R0)