×

zbMATH — the first resource for mathematics

G-perfect nonlinear functions. (English) Zbl 1179.94060
Summary: Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with \(G\)-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of \(G\)-perfect nonlinear functions, both \({\mathbb{Z}}_2\) -valued and \({\mathbb{Z}}_2^a\) -valued. Our final constructions demonstrate \(G\)-perfect nonlinear planar permutations (from \({\mathbb{Z}}_2^a\) to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.

MSC:
94A60 Cryptography
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arasu K.T., Ding C., Helleseth T., Kumar P.V., Martinsen H. (2001). Almost difference sets and their sequences with optimal autocorrelations. IEEE Trans. Inform. Theory 47(7): 2934–2943 · Zbl 1008.05027 · doi:10.1109/18.959271
[2] Beth T., Ding C.: On Almost Perfect Nonlinear Permutations, Advances in Cryptology - Eurocrypt ’93. Lecture Notes in Computer Science vol. 765, 65–76 Springer (1994). · Zbl 0951.94524
[3] Beth T., Jungnickel D., Lenz H. (1999). Design Theory, 2nd edn. Cambridge University Press, Cambridge · Zbl 0945.05004
[4] Biham E., Shamir A. (1991). Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1): 3–72 · Zbl 0729.68017 · doi:10.1007/BF00630563
[5] Budaghyan L., Carlet C., Felke P., Leander G. (2006). An infinite class of quadratic APN functions which are not equivalent to power mappings, IEEE Trans. Inform. Theory 52(2): 744–747 · Zbl 1246.11185 · doi:10.1109/TIT.2005.862128
[6] Carlet C., Charpin P., Zinoviev V. (1998). Codes, bent functions and permutations suitable for DES-like Cryptosystems. Des. Codes Cryptog. 15(2): 125–146 · Zbl 0938.94011 · doi:10.1023/A:1008344232130
[7] Carlet C., Ding C. (2004). Highly nonlinear mappings. J. Complexity 20(2): 205–244 · Zbl 1053.94011 · doi:10.1016/j.jco.2003.08.008
[8] Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis, Advances in Cryptology–Eurocrypt ’94. Lecture Notes in Computer Science Vol. 950, 356–365 Springer (1995). · Zbl 0879.94023
[9] Daemen J., Rijmen V.: The design of Rijndael: AES - The Advanced Encryption Standard. Springer-Verlag (2002). · Zbl 1065.94005
[10] Davis J.A. (1992). Construction of relative difference sets in p-groups. Discrete Math. 103: 7–15 · Zbl 0777.05023 · doi:10.1016/0012-365X(92)90034-D
[11] Dillon J.F.: Elementary Hadamard difference sets, PhD thesis, University of Maryland (1974). · Zbl 0346.05003
[12] Ding C., Yuan J. (2006). A family of skew Hadamard difference sets. J. Comb. Theory A. 113(7): 1526–1535 · Zbl 1106.05016 · doi:10.1016/j.jcta.2005.10.006
[13] Dobbertin H. (1999). Almost perfect nonlinear power functions on GF(2 n ): The Niho case. Inform. Comut. 151, 57–72 · Zbl 1072.94513 · doi:10.1006/inco.1998.2764
[14] Dobbertin H.: Almost perfect nonlinear power functions on GF(2 n ): a new case for n divisible by 5. In: Jugnickel, D., Niederreiter, H. (eds.). Proceedings of finite fields and applications Fq5, Springer 113–121 (2000). · Zbl 1010.94550
[15] Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping, preprint, http://arxiv.org/abs/math.CO/0506420 (2005), accessed November 2005. · Zbl 1246.11185
[16] Feistel H. (1973). Cryptography and computer privacy. Sci. Am. 228(5): 15–23 · doi:10.1038/scientificamerican0573-15
[17] FIPS 46–3, Data encryption standard, Federal Information Processing Standards Publication 46–3, U.S. Department of Commerce/N.I.S.T (1999).
[18] FIPS 197, Advanced encryption standard, Federal Information Processing Standards Publication 197, U.S. Department of Commerce/N.I.S.T (2001).
[19] Gold R. (1968). Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory 14, 154–156 · Zbl 0228.62040 · doi:10.1109/TIT.1968.1054106
[20] Hou X.-D.: Affinity of permutations of \({\mathbb{F}}_2^n\) . In: Augot, D., Charpin, P., Kabatiansky, G. (eds.) Workshop on Coding and Cryptography 2003 pp. 273–280 (2003).
[21] Lai X., Massey J.L.: A proposal for a new block encryption standard, Advances in Cryptology - Eurocrypt ’90. Lecture Notes in Computer Science 473, pp. 389–404 Springer (1991). · Zbl 0764.94017
[22] Meier W., Staffelbach O.: Nonlinearity criteria for cryptographic functions, Advances in Cryptology–Eurocrypt ’89. Lecture Notes in Computer Science, Vol 434, 549–562 Springer (1990). · Zbl 0724.94009
[23] Nyberg K.: Perfect nonlinear S-boxes, Advances in Cryptology–Eurocrypt ’91. Lecture Notes in Computer Sci. Springer, Vol. 547, pp. 378–386 (1992). · Zbl 0766.94012
[24] Nyberg K.: On the construction of highly nonlinear permutations, Advances in Cryptology–Eurocrypt ’92, Lecture Notes in Computer Science, Springer, Vol. 658, pp. 92–98 Springer (1993). · Zbl 0794.94008
[25] Nyberg K.: Differentially uniform mappings for cryptography, Advances in Cryptology–Eurocrypt ’93, Lecture Notes in Computer Science, Springer, Vol. 765, 55–64 (1994). · Zbl 0951.94510
[26] Paley R.E.A.C. (1933). On orthogonal matrices. J. Math. Phys. MIT. 12, 311–320 · Zbl 0007.10004
[27] Poinsot L.: Non linéarité parfaite généralisée au sens des actions de groupes, contribution aux fondements de la solidité cryptographique (available at http://poinsot.univ-tln.fr/These.pdf ), PhD thesis, University of South Toulon-Var (2005).
[28] Poinsot L., Harari S.: Generalized Boolean bent functions, Progress in Cryptology–Indocrypt 2004. Lecture Notes in Computer Science, 3348, pp. 107–119 Springer (2004). · Zbl 1115.94010
[29] Poinsot L., Harari S. (2005). Group actions based perfect nonlinearity, GESTS Int. Tr. Comput. Sci. Eng. 12(1): 1–14
[30] Pott A. (2004). Nonlinear functions in abelian groups and relative difference sets. Discrete Appl. Math. 138, 177–193 · Zbl 1035.05023 · doi:10.1016/S0166-218X(03)00293-2
[31] Rothaus O.S. (1976). On bent functions. J. Comb. Theory A 20, 300–305 · Zbl 0336.12012 · doi:10.1016/0097-3165(76)90024-8
[32] Shorin V.V., Jelezniakov V.V., Gabidulin E.M.: Linear and differential cryptanalysis of Russian GOST. In: Augot, D., Carlet, C. (eds.) Workshop on Coding and Cryptography 467–476 (2001). · Zbl 0985.94035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.