Abstract
Since the introduction of the notions of nonlinearity in the mid-70’s (the term has been in fact introduced later), of correlation immunity and resiliency in the mid-80’s, and of algebraic immunity recently, the problem of efficiently constructing Boolean functions satisfying, at high levels, one or several of these criteria has received much attention. Only few primary constructions are known, and secondary constructions are also necessary to obtain functions achieving or approaching the best possible cryptographic characteristics. After recalling the background on cryptographic criteria and making some general observations, we try to give a survey of all these constructions and their properties. We then show that a nice and simple property of Boolean functions leads to a general secondary construction building an n-variable function from three known n-variable functions. This construction generalizes secondary constructions recently obtained for Boolean bent functions and also leads to secondary constructions of highly nonlinear balanced or resilient functions, with potentially better algebraic immunities than the three functions used as building blocks.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Assmus, E.F., Key, J.D.: Designs and their Codes. Cambridge Univ. Press, Cambridge
Botev, A., Tarannikov, Y.: Lower bounds on algebraic immunity for recursive constructions of nonlinear filters (2004) (Preprint)
Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 86–100. Springer, Heidelberg (1992)
Canteaut, A., Trabbia, M.: Improved fast correlation attacks using parity-check equations of weight 4 and 5. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 573–588. Springer, Heidelberg (2000)
Carlet, C.: Partially-bent functions. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 280–291. Springer, Heidelberg (1993)
Carlet, C.: Two new classes of bent functions. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 77–101. Springer, Heidelberg (1994)
Carlet, C.: Generalized Partial Spreads. IEEE Transactions on Information Theory 41(5), 1482–1487 (1995)
Carlet, C.: A construction of bent functions. In: Finite Fields and Applications, London Mathematical Society. Lecture Series, vol. 233, pp. 47–58. Cambridge University Press, Cambridge (1996)
Carlet, C.: More correlation-immune and resilient functions over Galois fields and Galois rings. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 422–433. Springer, Heidelberg (1997)
Carlet, C.: Recent results on binary bent functions. In: International Conference on Combinatorics, Information Theory and Statistics; Journal of Combinatorics, Information and System Sciences 24(3-4), 275–291 (1999)
Carlet, C.: On the coset weight divisibility and nonlinearity of resilient and correlation-immune functions. In: Proceedings of SETA 2001 (Sequences and their Applications 2001). Discrete Mathematics and Theoretical Computer Science, pp. 131–144. Springer, Heidelberg (2001)
Carlet, C.: A larger Class of Cryptographic Boolean Functions via a Study of the Maiorana-McFarland Construction. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 549–564. Springer, Heidelberg (2002)
Carlet, C.: On the confusion and diffusion properties of Maiorana-McFarland’s and extended Maiorana-McFarland’s functions. Special Issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday. Journal of Complexity 20, 182–204 (2004)
Carlet, C.: On the secondary constructions of resilient and bent functions. In: Feng, K., Niederreiter, H., Xing, C. (eds.) Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003, pp. 3–28. Birkhäuser Verlag, Basel (2004)
Carlet, C.: On the degree, nonlinearity, algebraic thickness and non-normality of Boolean functions, with developments on symmetric functions. IEEE Transactions on Information Theory 50, 2178–2185 (2004)
Carlet, C.: Concatenating indicators of flats for designing cryptographic functions. Design, Codes and Cryptography 36(2), Ê189–202 (2005)
Carlet, C.: Designing bent functions and resilient functions from known ones, without extending their number of variables. In: Proceedings of International Symposium on Information Theory (2005)
Carlet, C.: Boolean Functions for Cryptography and Error Correcting Codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models. Cambridge University Press, Cambridge, Chapter of the monography (winter 2005-2006) (to appear)
Carlet, C., Dalai, D.K., Gupta, K.C., Maitra, S.: Algebraic Immunity for Cryptographically Significant Boolean Functions: Analysis and Construction (Preprint)
Carlet, C., Dobbertin, H., Leander, G.: Normal extensions of bent functions. IEEE Transactions on Information Theory 50(11), 2880–2885 (2004)
Carlet, C., Gaborit, P.: Hyper-bent functions and cyclic codes. To appear in the Journal of Combinatorial Theory, Series A (2005)
Carlet, C., Gaborit, P.: On the construction of balanced Boolean functions with a good algebraic immunity. In: Proceedings of International Symposium on Information Theory (2005) (To appear)
Carlet, C., Sarkar, P.: Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions. Finite fields and Applications 8, 120–130 (2002)
Carlet, C., Yucas, J.L.: Piecewise Constructions of Bent and Almost Optimal Boolean Functions. To appear in Designs, Codes and Cryptography (2005)
Courtois, N.: Fast Algebraic Attacks on Stream Ciphers with Linear Feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 177–194. Springer, Heidelberg (2003)
Courtois, N., 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)
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)
Dalai, D.K., Gupta, K.C., Maitra, S.: Results on Algebraic Immunity for Cryptographically Significant Boolean Functions. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 92–106. Springer, Heidelberg (2004)
Dalai, D.K., Gupta, K.C., Maitra, S.: Cryptographically Significant Boolean functions: Construction and Analysis in terms of Algebraic Immunity. In: Gilbert, H., Handschuh, H. (eds.) FSE 2005. LNCS, vol. 3557, pp. 98–111. Springer, Heidelberg (2005)
Dillon, J.F.: Elementary Hadamard Difference sets. Ph. D. Thesis, Univ. of Maryland (1974)
Dillon, J.F.: Elementary Hadamard Difference sets. In: Hoffman, F., et al. (eds.) Proc. Sixth S-E Conf. Comb. Graph Theory and Comp. Winnipeg Utilitas Math., pp. 237–249 (1975)
Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields and Their Applications 10, 342–389 (2004)
Dobbertin, H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 61–74. Springer, Heidelberg (1995)
Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of Bent Functions via Niho Power Functions. To appear in the Journal of Combinatorial Theory, Series A (2005)
Gong, G.: On Existence and Invariant of Algebraic Attacks. Technical report (2004), http://www.cacr.math.uwaterloo.ca/techreports/2004/corr2004-17.pdf
Hawkes, P., Rose, G.G.: Rewriting variables: the complexity of fast algebraic attacks on stream ciphers. Cryptology ePrint Archive, Report 2004/081 (2004), http://eprint.iacr.org/2004/081
Hou, X.-D.: New constructions of bent functions. In: International Conference on Combinatorics, Information Theory and Statistics; Journal of Combinatorics, Information and System Sciences 24(3-4), 275–291 (1999)
Hou, X.-D.: Group actions on binary resilient functions. Appl. Algebra Eng. Commun. Comput. 14(2), 97–115 (2003)
Hou, X.-D., Langevin, P.: Results on bent functions. Journal of Combinatorial Theory, Series A 80, 232–246 (1997)
Johansson, T., Jönsson, F.: Improved fast correlation attack on stream ciphers via convolutional codes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 347–362. Springer, Heidelberg (1999)
Johansson, T., Jönsson, F.: Fast correlation attacks based on turbo code techniques. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 181–197. Springer, Heidelberg (1999)
Khoo, K., Tan, G.-E., Lee, H.-K., Gong, G.: Comparision of Boolean function design. In: Proceedings of International Symposium on Information Theory (2005)
Leander, G.: Bent functions with 2r Niho exponents. In: Proceedings of the Workshop on Coding and Cryptography 2005, pp. 454–461 (2005)
Leander, G.: Monomial bent functions. In: Proceedings of the Workshop on Coding and Cryptography 2005, Bergen, pp. 462–470 (2005)
Mac Williams, F.J., Sloane, N.J.: The theory of error-correcting codes. North Holland, Amsterdam (1977)
Maity, S., Maitra, S.: Minimum distance between bent and 1-resilient Boolean functions. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 143–160. Springer, Heidelberg (2004)
Meier, W., Staffelbach, O.: Nonlinearity Criteria for Cryptographic Functions. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 549–562. Springer, Heidelberg (1990)
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)
Olejár, D., Stanek, M.: On cryptographic properties of random Boolean functions. Journal of Universal Computer Science 4(8), 705–717 (1998)
Pasalic, E., Maitra, S.: A Maiorana-McFarland type construction for resilient Boolean functions on n variables (n even) with nonlinearity > 2n − 1 − 2n/2 + 2n/2 − 2. In: Proceedings of the Workshop on Coding and Cryptography 2003, pp. 365–374 (2003)
Olsen, J.D., Scholtz, R.A.: Bent function sequences. IEEE Trans. on Inf. Theory IT- 28(6) (1982)
Pasalic, E.: Degree optimized resilient Boolean functions from Maiorana-McFarland class. In: 9th IMA Conference on Cryptography and Coding (2003)
Rothaus, O.S.: On “bent” functions. J. Comb. Theory 20A, 300–305 (1976)
Pasalic, E., Johansson, T., Maitra, S., Sarkar, P.: New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity. In: Proceedings of the Workshop on Coding and Cryptography. Electronic Notes in Discrete Mathematics, vol. 6, pp. 425–434. Elsevier, Amsterdam (2001)
Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 485–506. Springer, Heidelberg (2000)
Sarkar, P., Maitra, S.: Nonlinearity Bounds and Constructions of Resilient Boolean Functions. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 515–532. Springer, Heidelberg (2000)
Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information theory IT-30(5), 776–780 (1984)
Siegenthaler, T.: Decrypting a Class of Stream Ciphers Using Ciphertext Only. IEEE Transactions on Computer C-34(1), 81–85 (1985)
Simpson, L., Dawson, E., Golic, J., Millan, W.: LILI Keystream generator. In: Proceedings of SAC 2000. LNCS, vol. 1807, pp. 248–261. Springer, Heidelberg (2001), www.isrc.qut.edu.au/lili/
Tarannikov, Y.V.: On resilient Boolean functions with maximum possible nonlinearity. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 19–30. Springer, Heidelberg (2000)
Guo-Zhen, X., Massey, J.L.: A Spectral Characterization of Correlation-Immune Combining Functions. IEEE Trans. Inf. Theory IT 34(3), 569–571 (1988)
Zheng, Y., Zhang, X.M.: Plateaued functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carlet, C. (2006). On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_1
Download citation
DOI: https://doi.org/10.1007/11617983_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31423-3
Online ISBN: 978-3-540-31424-0
eBook Packages: Computer ScienceComputer Science (R0)