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Weightwise Perfectly Balanced Functions and Nonlinearity

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Codes, Cryptology and Information Security (C2SI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13874))

Abstract

In this article we realize a general study on the nonlinearity of weightwise perfectly balanced (WPB) functions. First, we derive upper and lower bounds on the nonlinearity from this class of functions for all n. Then, we give a general construction that allows us to provably provide WPB functions with nonlinearity as low as \(2^{n/2-1}\) and WPB functions with high nonlinearity, at least \(2^{n-1}-2^{n/2}\). We provide concrete examples in 8 and 16 variables with high nonlinearity given by this construction. In 8 variables we experimentally obtain functions reaching a nonlinearity of 116 which corresponds to the upper bound of Dobbertin’s conjecture, and it improves upon the maximal nonlinearity of WPB functions recently obtained with genetic algorithms. Finally, we study the distribution of nonlinearity over the set of WPB functions. We examine the exact distribution for \(n=4\) and provide an algorithm to estimate the distributions for \(n=8\) and 16, together with the results of our experimental studies for \(n=8\) and 16.

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References

  1. Braeken, A., Preneel, B.: On the algebraic immunity of symmetric Boolean functions. In: Maitra, S., Veni Madhavan, C.E., Venkatesan, R. (eds.) INDOCRYPT 2005. LNCS, vol. 3797, pp. 35–48. Springer, Heidelberg (2005). https://doi.org/10.1007/11596219_4

    Chapter  Google Scholar 

  2. Carlet, C.: On the degree, nonlinearity, algebraic thickness, and nonnormality of Boolean functions, with developments on symmetric functions. IEEE Trans. Inf. Theory 50(9), 2178–2185 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2021)

    MATH  Google Scholar 

  4. Carlet, C., Méaux, P.: A complete study of two classes of Boolean functions: direct sums of monomials and threshold functions. IEEE Trans. Inf. Theory 68(5), 3404–3425 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carlet, C., Méaux, P., Rotella, Y.: Boolean functions with restricted input and their robustness; application to the FLIP cipher. IACR Trans. Symmetric Cryptol. 3, 2017 (2017)

    Google Scholar 

  6. Canteaut, A., Videau, M.: Symmetric Boolean functions. IEEE Trans. Inf. Theory 51(8), 2791–2811 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of Boolean functions with maximum possible annihilator immunity. Des. Codes Crypt. 40, 41–58 (2006). https://doi.org/10.1007/s10623-005-6300-x

    Article  MathSciNet  MATH  Google Scholar 

  8. 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). https://doi.org/10.1007/3-540-60590-8_5

    Chapter  Google Scholar 

  9. Gini, A., Méaux, P.: On the weightwise nonlinearity of weightwise perfectly balanced functions. Discret. Appl. Math. 322, 320–341 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gini, A., Méaux, P.: Weightwise almost perfectly balanced functions: secondary constructions for all \(n\) and better weightwise nonlinearities. In: Isobe, T., Sarkar, S. (eds.) Progress in Cryptology (INDOCRYPT 2022). LNCS, vol. 13774, pp. 492–514. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22912-1_22

  11. Guo, X., Sihong, S.: Construction of weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables. Discret. Appl. Math. 307, 102–114 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, J., Mesnager, S.: Weightwise perfectly balanced functions with high weightwise nonlinearity profile. Des. Codes Cryptogr. 87(8), 1797–1813 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, J., Sihong, S.: Construction of weightwise perfectly balanced Boolean functions with high weightwise nonlinearity. Discret. Appl. Math. 279, 218–227 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Millan, W., Clark, A., Dawson, E.: An effective genetic algorithm for finding highly nonlinear Boolean functions. In: Han, Y., Okamoto, T., Qing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 149–158. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0028471

    Chapter  Google Scholar 

  15. Méaux, P.: On the fast algebraic immunity of threshold functions. Cryptogr. Commun. 13(5), 741–762 (2021). https://doi.org/10.1007/s12095-021-00505-y

    Article  MathSciNet  MATH  Google Scholar 

  16. Mesnager, S.: Bent Functions. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32595-8

    Book  MATH  Google Scholar 

  17. Méaux, P., Journault, A., Standaert, F.-X., Carlet, C.: Towards stream ciphers for efficient FHE with low-noise ciphertexts. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 311–343. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_13

    Chapter  Google Scholar 

  18. Mandujano, S., Ku Cauich, J.C., Lara, A.: Studying special operators for the application of evolutionary algorithms in the seek of optimal Boolean functions for cryptography. In: Pichardo Lagunas, O., Martínez-Miranda, J., Martínez Seis, B. (eds.) Advances in Computational Intelligence (MICAI 2022). LNCS, vol. 13612, pp. 383–396. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-19493-1_30

  19. Maitra, S., Mandal, B., Martinsen, T., Roy, D., Stănică, P.: Tools in analyzing linear approximation for Boolean functions related to FLIP. In: Chakraborty, D., Iwata, T. (eds.) INDOCRYPT 2018. LNCS, vol. 11356, pp. 282–303. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-05378-9_16

    Chapter  MATH  Google Scholar 

  20. Maitra, S., Mandal, B., Roy, M.: Modifying Bent functions to obtain the balanced ones with high nonlinearity. In: Isobe, T., Sarkar, S. (eds.) Progress in Cryptology (INDOCRYPT 2022). LNCS, vol. 13774, pp. 449–470. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22912-1_20

  21. Mariot, L., Picek, S., Jakobovic, D., Djurasevic, M., Leporati, A.: Evolutionary construction of perfectly balanced Boolean functions. In: 2022 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE (2022)

    Google Scholar 

  22. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, 2nd edn. North-Holland Publishing Company (1978)

    Google Scholar 

  23. Mesnager, S., Su, S.: On constructions of weightwise perfectly balanced Boolean functions. Cryptogr. Commun. 13(6), 951–979 (2021). https://doi.org/10.1007/s12095-021-00481-3

    Article  MathSciNet  MATH  Google Scholar 

  24. Mesnager, S., Su, S., Li, J.: On concrete constructions of weightwise perfectly balanced functions with optimal algebraic immunity and high weightwise nonlinearity. Boolean Funct. Appl. (2021)

    Google Scholar 

  25. Mesnager, S., Sihong, S., Li, J., Zhu, L.: Concrete constructions of weightwise perfectly balanced (2-rotation symmetric) functions with optimal algebraic immunity and high weightwise nonlinearity. Cryptogr. Commun. 14(6), 1371–1389 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  26. Picek, S., Carlet, C., Guilley, S., Miller, J.F., Jakobovic, D.: Evolutionary algorithms for Boolean functions in diverse domains of cryptography. Evol. Comput. 24(4), 667–694 (2016)

    Article  Google Scholar 

  27. Qu, L., Feng, K., Liu, F., Wang, L.: Constructing symmetric Boolean functions with maximum algebraic immunity. IEEE Trans. Inf. Theory 55(5), 2406–2412 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rothaus, O.S.: On "bent" functions. J. Comb. Theory Ser. A. 20(3), 300–305 (1976)

    Google Scholar 

  29. Sarkar, P., Maitra, S.: Balancedness and correlation immunity of symmetric Boolean functions. Discret. Math. 307(19–20), 2351–2358 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Seberry, J., Zhang, X.-M., Zheng, Y.: Nonlinearly balanced Boolean functions and their propagation characteristics. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 49–60. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48329-2_5

    Chapter  Google Scholar 

  31. The Sage Developers: SageMath, the Sage mathematics software system (Version 8.1) (2017). https://www.sagemath.org/

  32. Tang, D., Liu, J.: A family of weightwise (almost) perfectly balanced Boolean functions with optimal algebraic immunity. Cryptogr. Commun. 11(6), 1185–1197 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tokareva, N.: Bent Functions: Results and Applications to Cryptography. Academic Press, Cambridge (2015)

    Book  MATH  Google Scholar 

  34. Varrette, S., Bouvry, P., Cartiaux, H., Georgatos, F.: Management of an academic HPC cluster: the UL experience. In: 2014 International Conference on High Performance Computing and Simulation (HPCS), pp. 959–967. IEEE (2014)

    Google Scholar 

  35. Zhang, R., Su, S.: A new construction of weightwise perfectly balanced Boolean functions. Adv. Math. Commun. (2021)

    Google Scholar 

  36. Zhu, L., Sihong, S.: A systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables. Discret. Appl. Math. 314, 181–190 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The two authors were supported by the ERC Advanced Grant no. 787390.

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Authors and Affiliations

  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Agnese Gini & Pierrick Méaux

Authors
  1. Agnese Gini
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  2. Pierrick Méaux
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Corresponding author

Correspondence to Agnese Gini .

Editor information

Editors and Affiliations

  1. Mathematics Department, Faculty of Sciences, Mohammed V University, Rabat, Morocco

    Said El Hajji

  2. University of Paris VIII, Paris, France

    Sihem Mesnager

  3. Computer Science Department, Faculty of Sciences, Mohammed V University, Rabat, Morocco

    El Mamoun Souidi

A Approximation of \(\mathfrak {N}_{16}\)

A Approximation of \(\mathfrak {N}_{16}\)

Table 8. Approximation of \(\mathfrak {N}_{16}\) via Algorithm 2 with \(s=18110464>2^{24}\) (Part 1). See Fig. 3.
Full size table
Table 9. Approximation of \(\mathfrak {N}_{16}\) via Algorithm 2 with \(s=18110464>2^{24}\) (Part 2). See Fig. 3.
Full size table

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Gini, A., Méaux, P. (2023). Weightwise Perfectly Balanced Functions and Nonlinearity. In: El Hajji, S., Mesnager, S., Souidi, E.M. (eds) Codes, Cryptology and Information Security. C2SI 2023. Lecture Notes in Computer Science, vol 13874. Springer, Cham. https://doi.org/10.1007/978-3-031-33017-9_21

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  • DOI: https://doi.org/10.1007/978-3-031-33017-9_21

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  • Online ISBN: 978-3-031-33017-9

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