Abstract
Algebraic attack studies ciphers from the point of view of solving equations. It is important to measure the security of block ciphers how many linearly independent bi-affine or quadratic equations they satisfy. As the S-box is the main nonlinear part of block ciphers, it really makes sense to get the number of linearly independent bi-affine and quadratic equations that an S-box satisfies to analyse the security of block ciphers. The article answers this question for two S-boxes based on APN power functions, and shows how to find out the equations by two toy examples. The techniques can be generalized to other S-boxes constructed by power functions. According to these conclusions, we can estimate the safety of such kind of block ciphers.
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References
Courtois, N.T., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 346–359. Springer, Heidelberg (2003)
Mihaljevie, M., Imai, H.: Cryptanalysis of Toyocrypt-HIS stream cipher. IEICE Transactions on Fundamentals E85-A, 66-73 (2002), http://www.csl.esat.sony.co.jp/atl/papers/IEICEjan02.pdf
Babbage, S.: Cryptanalysis of LILI-128. Technical report (January 2001), http://www.cosic.esat.kuleuven.ac.be/nessie/reports/
Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of boolean functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)
Shannon, C.E.: Communication Theory of Secrecy System. Bell System Technical Journal 28, 656–715 (1949), http://netlab.cs.ucla.edu/wiki/files/shannon1949.pdf
Youssef, A.M., Gong, G.: Hyper-bent functions. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, p. 406. Springer, Heidelberg (2001)
Cheon, J., Lee, D.: Resistance of S-boxes against algebraic attacks. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 83–94. Springer, Heidelberg (2004)
Cheon, J., Lee, D.H.: Quadratic equations from APN power functions. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E89-A(1), 19–27 (2006)
Nawaz, Y., Gong, G., Gupta, K.C.: Upper bounds on algebraic immunity of Boolean power functions, http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-09.pdf
Courtois, N.T., Debraize, B., Garrido, E.: On exact algebraic [non-]immunity of S-boxes based on power functions, http://eprint.iacr.org/2005/203.ps
Courtois, N., Pieprzyk, J.: Cryptanalysis of block ciphers with overdefined systems of equations. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 267–287. Springer, Heidelberg (2002)
Gong, G.: On existence and invariant of algebraic attack, http://www.cacr.math.uwaterloo.ca/techreports/2004/corr2004-17.pdf
Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge University Press, Cambridge (ISBN 0-521-30706-6)
Dobbertin, H.: Almost perfect nonlinear power functions on GF(2n): The Welch Case. IEEE Trans. Infrom. Theory 45(4), 1271–1275 (1999)
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Xie, J., Cao, W., Wang, T. (2009). Quadratic Equations from a Kind of S-boxes. In: Youm, H.Y., Yung, M. (eds) Information Security Applications. WISA 2009. Lecture Notes in Computer Science, vol 5932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10838-9_18
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DOI: https://doi.org/10.1007/978-3-642-10838-9_18
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