Abstract
Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to 6 variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to 20 variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.
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References
Bertoni, G., Daemen, J., Peeters, M., Assche, G.V.: The Keccak reference, January 2011. http://keccak.noekeon.org/
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press (2010)
Formenti, E., Imai, K., Martin, B., Yunès, J.-B.: Advances on random sequence generation by uniform cellular automata. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Computing with New Resources. LNCS, vol. 8808, pp. 56–70. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13350-8_5
Ghoshal, A., Sadhukhan, R., Patranabis, S., Datta, N., Picek, S., Mukhopadhyay, D.: Lightweight and side-channel secure 4 \(\times \) 4 s-boxes from cellular automata rules. IACR Trans. Symmetric Cryptol. 2018(3), 311–334 (2018)
Leporati, A., Mariot, L.: Cryptographic properties of bipermutive cellular automata rules. J. Cell. Autom. 9(5–6), 437–475 (2014)
Manzoni, L., Mariot, L.: Cellular automata pseudo-random number generators and their resistance to asynchrony. In: Mauri, G., El Yacoubi, S., Dennunzio, A., Nishinari, K., Manzoni, L. (eds.) ACRI 2018. LNCS, vol. 11115, pp. 428–437. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99813-8_39
Mariot, L., Gadouleau, M., Formenti, E., Leporati, A.: Mutually orthogonal Latin squares based on cellular automata. Des. Codes Cryptogr. 88(2), 391–411 (2019). https://doi.org/10.1007/s10623-019-00689-8
Mariot, L., Leporati, A.: Sharing secrets by computing preimages of bipermutive cellular automata. In: Wąs, J., Sirakoulis, G.C., Bandini, S. (eds.) ACRI 2014. LNCS, vol. 8751, pp. 417–426. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11520-7_43
Mariot, L., Leporati, A.: Inversion of mutually orthogonal cellular automata. In: Mauri, G., El Yacoubi, S., Dennunzio, A., Nishinari, K., Manzoni, L. (eds.) ACRI 2018. LNCS, vol. 11115, pp. 364–376. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99813-8_33
Mariot, L., Leporati, A., Dennunzio, A., Formenti, E.: Computing the periods of preimages in surjective cellular automata. Nat. Comput. 16(3), 367–381 (2016). https://doi.org/10.1007/s11047-016-9586-x
Mariot, L., Picek, S., Leporati, A., Jakobovic, D.: Cellular automata based s-boxes. Cryptogr. Commun. 11(1), 41–62 (2019)
del Rey, Á.M., Mateus, J.P., Sánchez, G.R.: A secret sharing scheme based on cellular automata. Appl. Math. Comput. 170(2), 1356–1364 (2005)
Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20(3), 300–305 (1976)
Seredynski, F., Bouvry, P., Zomaya, A.Y.: Cellular automata computations and secret key cryptography. Parallel Comput. 30(5–6), 753–766 (2004)
Szaban, M., Seredynski, F.: Cryptographically strong s-boxes based on cellular automata. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) ACRI 2008. LNCS, vol. 5191, pp. 478–485. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-79992-4_62
Wolfram, S.: Cryptography with cellular automata. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 429–432. Springer, Heidelberg (1986). https://doi.org/10.1007/3-540-39799-X_32
Acknowledgements
The authors wish to thank Claude Carlet and Stjepan Picek for useful comments on a preliminary version of this work. This research was partially supported by FRA 2020 - UNITS.
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Appendix: Source Code and Experimental Data
Appendix: Source Code and Experimental Data
The source code of the search algorithm and the experimental data are available at https://github.com/rymoah/ca-boolfun-construction.
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Mariot, L., Saletta, M., Leporati, A., Manzoni, L. (2021). Exploring Semi-bent Boolean Functions Arising from Cellular Automata. In: Gwizdałła, T.M., Manzoni, L., Sirakoulis, G.C., Bandini, S., Podlaski, K. (eds) Cellular Automata. ACRI 2020. Lecture Notes in Computer Science(), vol 12599. Springer, Cham. https://doi.org/10.1007/978-3-030-69480-7_7
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