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Separation results for boolean function classes

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Abstract

We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In particular, we show (almost) separation of several classes of Boolean functions which have been studied in coding theory and cryptography from classes which have been studied in combinatorics and complexity theory.

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References

  1. Boppana, R. B.: The average sensitivity of bounded-depth circuits. Inf. Process. Lett. 63(5), 257–261 (1997). https://doi.org/10.1016/S0020-0190(97)00131-2

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruck, J.: Harmonic analysis of polynomial threshold functions. SIAM J. Discret. Math. 3(2), 168–177 (1990). https://doi.org/10.1137/0403015

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlet, C.: On the nonlinearity of monotone Boolean functions. Cryptogr. Commun. 10(6), 1051–1061 (2018). https://doi.org/10.1007/s12095-017-0262-5

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlet, C., Joyner, D., Stănică, P., Tang, D.: Cryptographic properties of monotone Boolean functions. J. Math. Cryptol. 10 (1), 1–14 (2016). http://www.degruyter.com/view/j/jmc.2016.10.issue-1/jmc-2014-0030/jmc-2014-0030.xmlhttp://www.degruyter.com/view/j/jmc.2016.10.issue-1/jmc-2014-0030/jmc-2014-0030.xml

    Article  MathSciNet  Google Scholar 

  5. Celerier, C., Joyner, D., Melles, C., Phillips, D.: On the Walsh-Hadamard transform of monotone Boolean functions. Tbilisi Math. J. 5(2), 19–35 (2012). https://projecteuclid.org/euclid.tbilisi/1528768900

    Article  MathSciNet  Google Scholar 

  6. Chakraborty, S., Karmalkar, S., Kundu, S., Lokam, S.V., Saurabh, N.: Fourier entropy-influence conjecture for random linear threshold functions. In: Latin American Symposium on Theoretical Informatics, vol. 10807, pp. 275–289. Springer. https://doi.org/10.1007/978-3-319-77404-6_21 (2018)

  7. Chow, C.K.: On the characterization of threshold functions. In: 2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961). pp. 34–38. IEEE. https://ieeexplore.ieee.org/document/5397305 (1961)

  8. Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Academic Press, Cambridge (2017). https://www.elsevier.com/books/cryptographic-boolean-functions-and-applications/cusick/978-0-12-811129-1

    MATH  Google Scholar 

  9. Diakonikolas, I., Raghavendra, P., Servedio, R.A., Tan, L.: Average sensitivity and noise sensitivity of polynomial threshold functions. SIAM J. Comput. 43(1), 231–253 (2014). https://doi.org/10.1137/110855223

    Article  MathSciNet  Google Scholar 

  10. Gangopadhyay, S., Stănică, P.: Fourier entropy-influence conjecture for cryptographic Boolean functions. Trans. Adv. Res. (Special issue on Advances in Cryptology and Information Security) 12(2), 8–14 (2016). http://tar.ipsitransactions.org/

    Google Scholar 

  11. Gotsman, C., Linial, N.: Spectral properties of threshold functions. Combinatorica 14(1), 35–50 (1994). https://doi.org/10.1007/BF01305949

    Article  MathSciNet  Google Scholar 

  12. Håstad, J.T.: Computational limitations for small-depth circuits. MIT press, Cambridge (1987). https://mitpress.mit.edu/books/computational-limitations-small-depth-circuits

    Google Scholar 

  13. Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions (extended abstract). In: [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pp. 68–80. IEEE Computer Society. https://doi.org/10.1109/SFCS.1988.21923 (1988)

  14. Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform, and learnability. J. ACM (JACM) 40(3), 607–620 (1993). https://doi.org/10.1145/174130.174138

    Article  MathSciNet  Google Scholar 

  15. Preneel, B., Leekwijck, W.V., Linden, L.V., Govaerts, R., Vandewalle, J.: Propagation characteristics of Boolean functions. In: Advances in Cryptology - EUROCRYPT ’90, Workshop on the Theory and Application of of Cryptographic Techniques, vol. 473, pp. 161–173. Springer. https://doi.org/10.1007/3-540-46877-3_14 (1990)

  16. Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A. 20 (3), 300–305 (1976). https://doi.org/10.1016/0097-3165(76)90024-8

    Article  Google Scholar 

  17. Shalev, G.: On the Fourier entropy influence conjecture for extremal classes. arXiv:1806.03646 (2018)

  18. Webster, A.F., Tavares, S.E.: On the design of S-boxes. In: Advances in Cryptology - CRYPTO ’85, Conference on the theory and application of cryptographic techniques, vol. 218, pp. 523–534. Springer. https://doi.org/10.1007/3-540-39799-X_41(1985)

  19. Zhang, S.: Note on the average sensitivity of monotone Boolean functions. http://www.cse.cuhk.edu.hk/syzhang/papers/Sensitivity.pdf (2004)

  20. Zheng, Y., Zhang, X.M.: Relationships between bent functions and complementary plateaued functions. In: International Conference on Information Security and Cryptology, pp. 60–75. Springer. https://link.springer.com/chapter/10.1007/10719994_6 (1999)

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Acknowledgements

We thank the reviewers for their kind comments.

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  1. Indian Statistical Institute 203, B.T.Road, Kolkata, 700108, India

    Aniruddha Biswas & Palash Sarkar

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  1. Aniruddha Biswas
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Appendix A: Auto-correlation, influence and average sensitivity

Appendix A: Auto-correlation, influence and average sensitivity

Given an n-variable Boolean function f, its auto-correlation function \(C_{f}: \{-1, 1\}^{n} \rightarrow [-2^{n}, 2^{n}]\) is defined to be

$$ C_{f}(\mathbf{u}) = \underset{\mathbf{x} \in \{-1,1\}^{n}}{\sum}f(\mathbf{x})f(\mathbf{x}\odot \mathbf{u}) $$

where x ⊙u represents the pointwise multiplication of vectors x and u. The auto-correlation function is related to the influence in the following manner. For \(S\subseteq [n]\),

$$ \textsf{Inf}_{S}(f) = \frac{1}{2} - \frac{1}{2^{n+1}}C_{f}(\mathbf{w}_{S}). $$

where \(\mathbf {w}_{S}=(w_{1},\ldots ,w_{n}) \in \{-1,1\}^{n}\) is such that wi = − 1 if i ∈ S and wi = 1 otherwise.

The sensitivity s(f,x) of a Boolean function f on input \(\mathbf {x} = (x_{1},x_{2},{\ldots } x_{n}) \in \{-1, 1\}^{n}\) is defined in the following manner.

$$ s(f, \mathbf{x}) = \#\{i \in [n] : f(\mathbf{x}) \neq f(\mathbf{x^{\oplus i}})\}. $$

The average sensitivity s(f) of f is given by

$$ s(f) =\frac{1}{2^{n}} {\sum}_{\mathbf{x} \in \{-1,1\}^{n}} s(f,\mathbf{x}). $$

It is easy to see that I(f) = s(f) [14].

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Biswas, A., Sarkar, P. Separation results for boolean function classes. Cryptogr. Commun. 13, 451–458 (2021). https://doi.org/10.1007/s12095-021-00488-w

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