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Nonlinearities of S-boxes. (English) Zbl 1122.94026
Authors’ abstract: We introduce an indicator of the non-balancedness of functions defined over Abelian groups, and deduce a new indicator, denoted by $$NB$$, of the nonlinearity of such functions. We prove an inequality relating $$NB$$ and the classical indicator $$NL$$, introduced by Nyberg and studied by Chabaud and Vaudenay, of the nonlinearity of S-boxes. This inequality results in an upper bound on $$NL$$ which unifies Sidelnikov-Chabaud-Vaudenay’s bound and the covering radius bound. We also deduce from bounds on linear codes three new bounds on $$NL$$ that improve upon Sidelnikov-Chabaud-Vaudenay’s bound and the covering radius bound in many cases.

##### MSC:
 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94B75 Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) to coding theory 94B05 Linear codes, general
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##### References:
 [1] Biham, E.; Shamir, A., Differential cryptanalysis of DES-like cryptosystems, J. cryptol., 4, 1, 3-72, (1991) · Zbl 0729.68017 [2] Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Designs codes cryptography, 15, 125-156, (1998) · Zbl 0938.94011 [3] Carlet, C.; Ding, C., Highly nonlinear mappings, J. complexity, 20, 205-244, (2004) · Zbl 1053.94011 [4] C. Carlet, S. Dubuc, On generalized bent and q-ary perfect nonlinear functions, in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Proceedings of Fq5, Springer, Berlin, 2000, pp. 81-94. · Zbl 1010.94549 [5] F. Chabaud, S. Vaudenay, Links between differential and linear cryptanalysis, in: Advances in Cryptology—EUROCRYPT’94, Lecture Notes in Computer Science, vol. 950, Springer, Berlin, 1995, pp. 356-365. · Zbl 0879.94023 [6] J.F. Dillon, Elementary Hadamard Difference sets, Ph.D. Thesis, University of Maryland, 1974. · Zbl 0346.05003 [7] Ling, S.; Xing, C., Coding theory, (2004), Cambridge University Press Cambridge [8] MacWilliams, F.J.; Sloane, N.J., The theory of error-correcting codes, (1977), North Holland Amsterdam · Zbl 0369.94008 [9] M. Matsui, Linear cryptanalysis method for DES cipher, in: Advances in Cryptography—EUROCRYPT’93, Lecture Notes in Computer Science, vol. 765, Springer, Heidelberg, 1994, pp. 386-397. · Zbl 0951.94519 [10] K. Nyberg, Perfect non-linear S-boxes, in: Advances in Cryptology—EUROCRYPT’ 91, Lecture Notes in Computer Science, vol. 547, Springer, Heidelberg, 1992, pp. 378-386. · Zbl 0766.94012 [11] K. Nyberg, On the construction of highly nonlinear permutations, in: Advances in Cryptology—EUROCRYPT’ 92, Lecture Notes in Computer Science, vol. 658, Springer, Heidelberg, 1993, pp. 92-98. · Zbl 0794.94008 [12] K. Nyberg, Differentially uniform mappings for cryptography, in: Advances in Cryptography—EUROCRYPT’93, Lecture Notes in Computer Science, vol. 765, Springer, Heidelberg, 1994, pp. 55-64. · Zbl 0951.94510 [13] Patterson, N.J.; Wiedemann, D.H., The covering radius of the $$[2^{15}, 16]$$ reed – muller code is at least 16276, IEEE trans. inform. theory, IT-36, 2, 443, (1983) · Zbl 0505.94021 [14] Rothaus, O.S., On bent functions, J. combin. theory, 20A, 300-305, (1976) · Zbl 0336.12012 [15] Shannon, C.E., Communication theory of secrecy systems, Bell system tech. J., 28, 656-715, (1949) · Zbl 1200.94005 [16] Sidel’nikov, V.M., On the mutual correlation of sequences, Soviet math. dokl., 12, 197-201, (1971) · Zbl 0241.94008 [17] Wadayama, T.; Hada, T.; Wakasugi, K.; Kasahara, M., Upper and lower bounds on maximum nonlinearity of n-input m-output Boolean function, Designs, codes cryptography, 23, 23-33, (2001) · Zbl 1020.94016 [18] Welch, L., Lower bounds on the maximum cross correlation of signals, IEEE trans. inform. theory, IT-20, 3, 397-399, (1974) · Zbl 0298.94006
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