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On weakly APN functions and 4-bit S-boxes. (English) Zbl 1271.94019

Summary: S-boxes are important security components of block ciphers. We provide theoretical results on necessary or sufficient criteria for an (invertible) 4-bit S-box to be weakly APN. Thanks to a classification of 4-bit invertible S-boxes, achieved independently by De Cannière and Leander-Poschmann, we can strengthen our results with a computer-aided proof. We also propose a class of 4-bit S-boxes which are very strong from a security point of view.

MSC:

94A60 Cryptography
06E30 Boolean functions
20B40 Computational methods (permutation groups) (MSC2010)

Software:

Serpent; PRESENT
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References:

[1] Adams, C.; Tavares, S., The structured design of cryptographically good S-boxes, J. Cryptology, 3, 1, 27-41 (1990) · Zbl 0711.94016
[2] Anderson, R. J.; Biham, E.; Knudsen, L. R., Serpent: A new block cipher proposal, (Proc. of FSE 1998. Proc. of FSE 1998, Lecture Notes in Comput. Sci., vol. 1372 (1998)), 222-238 · Zbl 1385.94015
[3] Aubry, Y.; McGuire, G.; Rodier, F., A few more functions that are not APN infinitely often, (Finite Fields: Theory and Applications. Finite Fields: Theory and Applications, Contemp. Math., vol. 518 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 23-31 · Zbl 1206.94025
[4] Bogdanov, A.; Knudsen, L. R.; Leander, G.; Paar, C.; Poschmann, A.; Robshaw, M.; Seurin, Y.; Vikkelsoe, C., PRESENT: An ultra-lightweight block cipher, (Proc. of CHES 2007. Proc. of CHES 2007, Lecture Notes in Comput. Sci., vol. 7427 (2007)), 450-466 · Zbl 1142.94334
[5] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., Fourier spectra of binomial APN functions, SIAM J. Discrete Math., 23, 2, 596-608 (2009) · Zbl 1208.11133
[6] Canteaut, A.; Charpin, P.; Kyureghyan, G. M., A new class of monomial bent functions, Finite Fields Appl., 14, 1, 221-241 (2008) · Zbl 1162.94004
[7] Caranti, A.; Dalla Volta, F.; Sala, M., On some block ciphers and imprimitive groups, Appl. Algebra Engrg. Comm. Comput., 20, 5-6, 339-350 (2009) · Zbl 1178.94183
[8] C. De Cannière, Analysis and design of symmetric encryption algorithms, PhD thesis, Katholieke Universiteit Leuven, 2007.; C. De Cannière, Analysis and design of symmetric encryption algorithms, PhD thesis, Katholieke Universiteit Leuven, 2007.
[9] Leander, G.; Poschmann, A., On the classification of 4 bit S-boxes, (Proc. of WAIFI 2007. Proc. of WAIFI 2007, Lecture Notes in Comput. Sci., vol. 4547 (2007)), 159-176 · Zbl 1184.94239
[10] National Institute of Standards and Technology, The Advanced Encryption Standard, FIPS 197, 2001.; National Institute of Standards and Technology, The Advanced Encryption Standard, FIPS 197, 2001.
[11] Paterson, K. G., Imprimitive permutation groups and trapdoors in iterated block ciphers, (Proc. of FSE 1999. Proc. of FSE 1999, Lecture Notes in Comput. Sci., vol. 1636 (1999)), 201-214 · Zbl 0942.94008
[12] Preneel, B.; Van Leekwijck, W.; Van Linden, L.; Govaerts, R.; Vandewalle, J., Propagation characteristics of Boolean functions, (Proc. of EUROCRYPT 1990. Proc. of EUROCRYPT 1990, Lecture Notes in Comput. Sci., vol. 473 (1991)), 161-173 · Zbl 0764.94024
[13] V. Pulice, A security classification of Boolean functions, Master thesis, Univ. of Trento, 2011.; V. Pulice, A security classification of Boolean functions, Master thesis, Univ. of Trento, 2011.
[14] M.J. Saarinen, Cryptographic analysis of all \(4 \times 4\); M.J. Saarinen, Cryptographic analysis of all \(4 \times 4\) · Zbl 1292.94132
[15] Zhang, W.; Wu, C.-K.; Li, S., Construction of cryptographically important Boolean permutations, Appl. Algebra Engrg. Comm. Comput., 15, 3-4, 173-177 (2004) · Zbl 1062.94043
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