Advertisement

Natural Computing

, Volume 17, Issue 3, pp 487–498 | Cite as

A cryptographic and coding-theoretic perspective on the global rules of cellular automata

  • Luca Mariot
  • Alberto Leporati
Article

Abstract

Cellular Automata (CA) have widely been studied to design cryptographic primitives such as stream ciphers and pseudorandom number generators, focusing in particular on the properties of the underlying local rules. On the other hand, there have been comparatively fewer works concerning the applications of CA to the design of S-boxes and block ciphers, a task that calls for a study of CA global rules in terms of vectorial boolean functions. The aim of this paper is to analyze some of the most basic cryptographic criteria of the global rules of CA. We start by observing that the algebraic degree of a CA global rule equals the degree of its local rule. Then, we characterize the Walsh spectrum of CA induced by permutive local rules, from which we derive a formula for the nonlinearity of such CA. Additionally, we prove that the 1-resiliency property of bipermutive local rules transfers to the corresponding global rules. This result leads us to consider CA global rules from a coding-theoretic point of view: in particular, we show that linear CA are equivalent to linear cyclic codes, observing that the syndrome computation process corresponds to the application of the CA global rule, while the error-correction capability of the code is related to the resiliency order of the global rule.

Keywords

Cellular automata Boolean functions S-boxes Nonlinearity Resiliency Cyclic codes 

Mathematics Subject Classification

37B15 68Q80 94B15 94A55 

References

  1. Bertoni G, Daemen J, Peeters M, Assche GV (2013) Keccak. In: Proceedings of advances in cryptology—EUROCRYPT 2013, 32nd annual international conference on the theory and applications of cryptographic techniques, Athens, Greece, May 26–30, 2013, pp 313–314Google Scholar
  2. Carlet C (2010a) Boolean functions for cryptography and error correcting codes. Boolean Models Methods Math Comput Sci Eng 2:257–397CrossRefzbMATHGoogle Scholar
  3. Carlet C (2010b) Vectorial boolean functions for cryptography. Boolean Models Methods Math Comput Sci Eng 134:398–469CrossRefzbMATHGoogle Scholar
  4. Daemen J, Rijmen V (2002) The design of Rijndael. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  5. Daemen J, Govaerts R, Vandewalle J (1994) An efficient nonlinear shift-invariant transformation. In: Macq B (ed) Proceedings of the 15th symposium on information theory in the Benelux, Werkgemeenschap voor Informatie-en Communicatietheorie, Citeseer, pp 108–115Google Scholar
  6. Formenti E, Imai K, Martin B, Yunès J (2014) Advances on random sequence generation by uniform cellular automata. In: Computing with new resources—essays dedicated to Jozef Gruska on the occasion of his 80th birthday, pp 56–70Google Scholar
  7. Kari J (2012) Basic concepts of cellular automata. In: Handbook of natural computing, pp 3–24Google Scholar
  8. Koc CK, Apohan A (1997) Inversion of cellular automata iterations. IEE Proc Comput Digital Tech 144(5):279–284CrossRefGoogle Scholar
  9. Leporati A, Mariot L (2014) Cryptographic properties of bipermutive cellular automata rules. J Cell Autom 9(5–6):437–475MathSciNetzbMATHGoogle Scholar
  10. Mariot L, Leporati A (2015) On the periods of spatially periodic preimages in linear bipermutive cellular automata. In: Proceedings of cellular automata and discrete complex systems—21st IFIP WG 1.5 international workshop, AUTOMATA 2015, Turku, Finland, June 8–10, 2015, pp 181–195Google Scholar
  11. Mariot L, Leporati A (2016) Resilient vectorial functions and cyclic codes arising from cellular automata. In: Proceedings of cellular automata—12th international conference on cellular automata for research and industry, ACRI 2016, Fez, Morocco, September 5–8, 2016, pp 34–44Google Scholar
  12. Martin B (2008) A walsh exploration of elementary CA rules. J Cell Autom 3(2):145–156MathSciNetzbMATHGoogle Scholar
  13. McEliece R (2002) The theory of information and coding. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  14. Meier W, Staffelbach O (1991) Analysis of pseudo random sequence generated by cellular automata. In: Proceedings of advances in cryptology—EUROCRYPT ’91, workshop on the theory and application of of cryptographic techniques, Brighton, UK, April 8–11, 1991, pp 186–199Google Scholar
  15. Nyberg K (1994) S-boxes and round functions with controllable linearity and differential uniformity. In: Proceedings of fast software encryption: second international workshop. Leuven, Belgium, 14–16 December 1994, pp 111–130Google Scholar
  16. Rijmen V, Barreto PSLM, Filho DLG (2008) Rotation symmetry in algebraically generated cryptographic substitution tables. Inf Process Lett 106(6):246–250MathSciNetCrossRefzbMATHGoogle Scholar
  17. Shannon CE (1949) Communication theory of secrecy systems. Bell Labs Tech J 28(4):656–715MathSciNetCrossRefzbMATHGoogle Scholar
  18. Siegenthaler T (1985) Decrypting a class of stream ciphers using ciphertext only. IEEE Trans Comput 34(1):81–85CrossRefGoogle Scholar
  19. Stinson DR (1995) Cryptography—theory and practice. Discrete mathematics and its applications series. CRC Press, Boca RatonGoogle Scholar
  20. Stinson DR (2004) Combinatorial designs—constructions and analysis. Springer, BerlinzbMATHGoogle Scholar
  21. Ulam S (1952) Random processes and transformations. Proc Int Congr Math 2:264–275MathSciNetzbMATHGoogle Scholar
  22. Von Neumann J (1966) Theory of self-reproducing automata. Edited by Burks, Arthur W. University of Illinois Press, ChampaignGoogle Scholar
  23. Wolfram S (1983) Statistical mechanics of cellular automata. Rev Mod Phys 55(3):601MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wolfram S (1985) Cryptography with cellular automata. In: Proceedings of advances in cryptology—CRYPTO ’85, Santa Barbara, California, USA, August 18–22, 1985, pp 429–432Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi Milano-BicoccaMilanoItaly

Personalised recommendations