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De Bruijn sequences and complexity of symmetric functions

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Abstract

A multivalued function is a function from a set \(E_{q}^{n}\) to a set E m , where E k is a set which contains k elements. These functions are used in cryptography: cipher design, hash function design and in theoretical computer science. In this paper, we study the representation of these functions with Multivalued Decision Diagrams (MDD). This representation can be used both to measure complexity and to implement efficiently the functions in hardware. We are especially interested in symmetric functions. We show that symmetric functions MDDs have much lower size than classical functions MDDs. One major result is to determine exactly their MDD’s maximum size. Notably, we highlight the links between De Bruijn sequences and the most complex symmetric functions and new functions are exhibited in the case q = 2 and any m. Enumeration of these functions are supplied, they are shown to be sufficiently numerous to allow many applications.

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Acknowledgements

We thank Boris Batteux for his computations on functions enumeration. We also thank the anonymous referees for excellent suggestions which greatly improved the clarity of this paper.

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Authors and Affiliations

  1. Cyber Security Customer Solutions Centre, CASSIDIAN, Elancourt, France

    Christelle Rovetta & Marc Mouffron

Authors
  1. Christelle Rovetta
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  2. Marc Mouffron
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Correspondence to Marc Mouffron.

Appendices

Appendix A: Algebraic degree distribution

These tables give the distribution of the algebraic degree of super hard symmetric functions and hard symmetric functions.

Table 1 Algebraic degree of super hard symmetric functions for n = 3, ..., 23
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Table 2 Algebraic degree of hard symmetric functions for n = 2,...,20
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Appendix B: Maximum complexities of symmetric functions from \(E_2^n\) to E 2

This table gives the cardinal of the sets HSM n (q, m), SHSM n (q, m) and HSM n (q, m) ∩ SHSM n (q, m) for any n up to 35. The special cases n = a + 2a − 2 are in bold.

Table 3 Complexity of symmetric functions from \(E_2^n\) to E 2
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Appendix C: Number of balanced symmetric Boolean functions hard and super hard

This table gives the number of balanced hard and super hard symmetric Boolean functions for any odd n up to 53. There is no balanced hard nor super hard symmetric Boolean functions when n is even except 2 for n = 2.

Table 4 Number of balanced symmetric Boolean functions of maximum complexity
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Appendix D: Boolean symmetric functions’ complexity c R (f) distribution n = 1 to 33

Table 5 Boolean symmetric functions’ complexity c R (f) n = 1–33
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Rovetta, C., Mouffron, M. De Bruijn sequences and complexity of symmetric functions. Cryptogr. Commun. 3, 207–225 (2011). https://doi.org/10.1007/s12095-011-0054-2

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  • DOI: https://doi.org/10.1007/s12095-011-0054-2

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