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Open Questions on Nonlinearity and on APN Functions

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Arithmetic of Finite Fields (WAIFI 2014)

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Abstract

In a first part of the paper, we recall some known open questions on the nonlinearity of Boolean and vectorial functions and on the APN-ness of vectorial functions. All of them have been extensively searched and seem quite difficult. We also indicate related less well-known open questions. In the second part of the paper, we introduce four new open problems (leading to several related sub-problems) and the results which lead to them. Addressing these problems may be less difficult since they have not been much worked on.

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Notes

  1. 1.

    By that, we mean a criterion related to those attacks on the cryptosystem viewed as a black box, by opposition to the attacks exploiting leaks, like side channel attacks.

  2. 2.

    The precise parameter allowing to quantify the complexity of this attack is the fast algebraic immunity, introduced in a preliminary version of the paper [30] and studied in [16], but a large algebraic degree is a necessary condition for the resistance to the fast algebraic attack.

  3. 3.

    Unless the nonlinearity of the CF function becomes much worse for values of \(n\) which are too large for allowing its computation.

References

  1. Browning, K., Dillon, J., McQuistan, M.: APN polynomials and related codes. Special volume of Journal of Combinatorics, Information and System Sciences, honoring the 75-th birthday of Prof. D.K.Ray-Chaudhuri 34, 135–159 (2009)

    Google Scholar 

  2. Browning, K., Dillon, J., McQuistan, M., Wolfe, A.: An APN permutation in dimension six. Contemp. Math. 58, 33–42 (2010)

    Article  MathSciNet  Google Scholar 

  3. Budaghyan, L., Carlet, C., Helleseth, T.: On bent functions associated to AB functions. In: Proceedings of IEEE Information Theory Workshop (2011)

    Google Scholar 

  4. Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear functions. IEEE Trans. Inform. Theory 52(3), 1141–1152 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carlet, C.: Generalized partial spreads. IEEE Trans. Inform. Theory 41(5), 1482–1487 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carlet, C., Feng, K.: An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)

    Chapter  Google Scholar 

  8. Carlet, C.: Vectorial Boolean Functions for Cryptography, Idem, pp. 398–469 (2010)

    Google Scholar 

  9. Carlet, C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent. Des. Codes Crypt. 59(1), 89–109 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carlet, C.: Open problems on binary bent functions. In: Proceedings of the Conference Open Problems in Mathematical and Computational Sciences, September 18–20, 2013, in Istanbul, Turkey, pp. 203–241. Springer (2014)

    Google Scholar 

  11. Carlet. C.: A survey on nonlinear boolean functions with optimal algebraic immunity suitable for stream ciphers. In: Proceedings of the SMF-VMS Conference, Hué, Vietnam, 20–24 August 2012. (Special issue of the Vietnam Journal of Mathematics, Volume 41, Issue 4, Page 527–541, 2013)

    Google Scholar 

  12. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Crypt. 15(2), 125–156 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Carlet, C., Guillot, P.: A new representation of boolean functions. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 94–103. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  14. Carlet, C., Guillot, P.: Bent, resilient functions and the numerical normal form. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56, 87–96 (2001)

    MathSciNet  Google Scholar 

  15. Carlet, C., Klapper, A.: Upper bounds on the numbers of resilient functions and of bent functions. This paper was meant to appear in an issue of Lecture Notes in Computer Sciences dedicated to Philippe Delsarte, Editor Jean-Jacques Quisquater. But this issue finally never appeared. A shorter version has appeared in the Proceedings of the 23rd Symposium on Information Theory in the Benelux, Louvain-La-Neuve, Belgium (2002)

    Google Scholar 

  16. Carlet, C., Tang, D.: Enhanced Boolean functions suitable for the filter model of pseudo-random generator. Designs, Codes and Cryptography (to appear)

    Google Scholar 

  17. Courtois, N.T.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  18. Chepyzhov, V., Smeets, B.J.M.: On a fast correlation attack on certain stream ciphers. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 176–185. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  19. Dillon, J.: A survey of bent functions. NSA Tech. J., 191–215 (1972). Special Issue

    Google Scholar 

  20. Dillon, J.F.: Elementary Hadamard Difference sets. Ph. D. Thesis, Univ. of Maryland (1974)

    Google Scholar 

  21. Dobbertin, H.: Another proof of Kasami’s Theorem. Des. Codes Crypt. 17, 177–180 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Dobbertin, H.: Kasami power functions, permutation polynomials and cyclic difference sets. In: Proceedings of the NATO-A.S.I. Workshop “Difference sets, sequences and their correlation properties”, Bad Windsheim, Kluwer Verlag, pp. 133–158 (1998)

    Google Scholar 

  23. Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields Appl. 10, 342–389 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gologlu, F.: Projective polynomials and their applications in cryptography. In: International Workshop on Boolean Functions and Their Applications, Bergen, September 2014. http://www.people.uib.no/lbu061/gologlu.pdf

  26. Guillot, P.: Completed GPS covers all bent functions. J. Comb. Theory Ser. A 93, 242–260 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ding, C., Shan, W., Xiao, G. (eds.): The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)

    MATH  Google Scholar 

  28. Helleseth, T.: Kholosha, Alexander: \(x^{2^{l}+1}+x+a\) and related affine polynomials over \(GF(2^k)\). Crypt. Commun. 2(1), 85–109 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Langevin, P., Leander, G.: Counting all bent functions in dimension eight 99270589265934370305785861242880. Des. Codes Crypt. 59(1–3), 193–205 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu, M., Lin, D., Pei, D.: Fast algebraic attacks and decomposition of symmetric boolean functions. IEEE Trans. Inform. Theory 57, 4817–4821 (2011). A preliminary version of this paper was presented in ArXiv: 0910.4632v1 [cs.CR]. http://arxiv.org/abs/0910.4632

    Article  MathSciNet  Google Scholar 

  31. Massey, J.L.: Shift-register analysis and BCH decoding. IEEE Trans. Inf. Theory 15, 122–127 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  32. Nyberg, K.: Perfect nonlinear S-boxes. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 378–386. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  33. Piret, G., Roche, T., Carlet, C.: PICARO – a block cipher allowing efficient higher-order side-channel resistance. In: Bao, F., Samarati, P., Zhou, J. (eds.) ACNS 2012. LNCS, vol. 7341, pp. 311–328. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  34. Rønjom, S., Helleseth, T.: A new attack on the filter generator. IEEE Trans. Inf. Theory 53(5), 1752–1758 (2007)

    Article  Google Scholar 

  35. Rothaus, O.S.: On “bent” functions. J. Comb. Theory 20A, 300–305 (1976)

    Article  MathSciNet  Google Scholar 

  36. Tang, D., Carlet, C., Tang, X.: Highly nonlinear boolean functions with optimal algebraic immunity and good behavior against fast algebraic attacks. IEEE Tran. Inf. Theory 59(1), 653–664 (2013)

    Article  MathSciNet  Google Scholar 

  37. Tokareva, N.: On the number of bent functions from iterative constructions: lower bounds and hypotheses. Adv. Math. Commun. (AMC) 5, 609–621 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wu, B.: \({\cal PS}\) bent functions constructed from finite pre-quasifield spreads. http://arxiv.org/abs/1308.3355

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Acknowledgement

We wish to thank Lilya Budaghyan, Faruk Gologlu and Sihem Mesnager for useful information.

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Authors and Affiliations

  1. LAGA, Universities of Paris 8 and Paris 13; CNRS, UMR 7539, Saint-Denis Cedex 02, France

    Claude Carlet

  2. Department of Mathematics, University of Paris 8, 2 rue de la Liberté, 93526, Saint-Denis Cedex 02, France

    Claude Carlet

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  1. Claude Carlet
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Correspondence to Claude Carlet .

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Editors and Affiliations

  1. Department of Computer Science, Univ of California, Santa Barbara, Santa Barbara, California, USA

    Çetin Kaya Koç

  2. University of Paris VIII, Paris, France

    Sihem Mesnager

  3. Sabancı University, Istanbul, Turkey

    Erkay Savaş

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Carlet, C. (2015). Open Questions on Nonlinearity and on APN Functions. In: Koç, Ç., Mesnager, S., Savaş, E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science(), vol 9061. Springer, Cham. https://doi.org/10.1007/978-3-319-16277-5_5

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  • DOI: https://doi.org/10.1007/978-3-319-16277-5_5

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