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Partially APN functions with APN-like polynomial representations

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Abstract

In this paper we investigate several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field \(\mathbb {F}_2\). We also investigate the differential uniformity of some binomial partial APN functions. Furthermore, the partial APN-ness for some classes of multinomial functions is investigated. We show also that the size of the pAPN spectrum is preserved under CCZ-equivalence.

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References

  1. Berlekamp E.R., Rumsey H., Solomon G.: On the solutions of algebraic equations over finite fields. Inf. Control 10, 553–564 (1967).

    Article  MathSciNet  Google Scholar 

  2. Bracken C., Leander G.: A highly nonlinear differentially \(4\) uniform power mapping that permutes fields of even degree. Finite Fields Appl. 16(4), 231–242 (2010).

    Article  MathSciNet  Google Scholar 

  3. Budaghyan L.: Construction and Analysis of Cryptographic Functions. Springer, New York (2014).

    Book  Google Scholar 

  4. Budaghyan L., Carlet C.: Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Trans. Inform. Theory 54(5), 2354–2357 (2008).

    Article  MathSciNet  Google Scholar 

  5. Budaghyan L., Carlet C., Helleseth T., Li N., Sun B.: On upper bounds for algebraic degrees of APN functions. IEEE Trans. Inform. Theory 64(6), 4399–4411 (2018).

    Article  MathSciNet  Google Scholar 

  6. Budaghyan, L., Kaleyski, N., Kwon, S., Riera, C., Stănică, P.: Partially APN Boolean functions and classes of functions that are not APN infinitely often. In: Proceedings of the Cryptography and Communication, 2019; preliminary version as Partially APN Boolean functions, Sequences and Their Applications, SETA, Hong Kong (2018)

  7. Carlet C.: Boolean functions for cryptography and error correcting codes. In: Crama Y., Hammer P. (eds.) Boolean Methods and Models, pp. 257–397. Cambridge University Press, Cambridge (2010).

    MATH  Google Scholar 

  8. Carlet C.: Vectorial Boolean functions for cryptography. In: Crama Y., Hammer P. (eds.) Boolean Methods and Models, pp. 398–472. Cambridge University Press, Cambridge (2010).

    MATH  Google Scholar 

  9. Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable For DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998).

    Article  MathSciNet  Google Scholar 

  10. Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis, advances in cryptology-EUROCRYPT’94. LNCS 950, 356–365 (1995).

    MATH  Google Scholar 

  11. Cusick T.W., Stănică P.: Cryptographic Boolean Functions and Applications. Academic Press, San Diego (2017).

    MATH  Google Scholar 

  12. Dillon J.F.: APN Polynomials and Related Codes. Polynomials Over Finite Fields and Applications. Banff International Research Station, Banff (2006).

    Google Scholar 

  13. Hou X., Mullen G.L., Sellers J.A., Yucas J.: Reversed Dickson polynomials over finite fields. Finite Fields Appl. 15, 748–773 (2009).

    Article  MathSciNet  Google Scholar 

  14. Hughes D.R.: Collineation groups and non-Desarguesian planes. Am. J. Math 81, 921–938 (1959).

    Article  Google Scholar 

  15. Hughes D.R.: Collineation groups and non-Desarguesian planes. Am. J. Math 82, 113–119 (1959).

    Article  Google Scholar 

  16. Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (1997).

    Google Scholar 

  17. Rodier, F.: Borne sur le degré des polynômes presque parfaitement non-linéaires, Arithmetic, Geometry, Cryptography and Coding Theory, G. Lachaud, C. Ritzenthaler and M.Tsfasman eds., Contemporary Math. no 487, AMS, Providence (RI), USA, pp. 169–181 (2009)

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Acknowledgements

The authors express their deep appreciation to the editor, as well as to the three anonymous referees, whose thorough reading and constructive comments have greatly improved the paper. The paper was started while the fourth named author visited Selmer center at University of Bergen and Western Norway University of Applied Sciences in the Spring of 2019. This author thanks these institutions for the excellent working conditions. The research of the first two named authors was supported by Trond Mohn foundation.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Lilya Budaghyan & Nikolay Kaleyski

  2. Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, 5020, Bergen, Norway

    Constanza Riera

  3. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943-5212, USA

    Pantelimon Stănică

Authors
  1. Lilya Budaghyan
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  2. Nikolay Kaleyski
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  3. Constanza Riera
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  4. Pantelimon Stănică
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Corresponding author

Correspondence to Pantelimon Stănică.

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Communicated by C. Carlet.

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Budaghyan, L., Kaleyski, N., Riera, C. et al. Partially APN functions with APN-like polynomial representations. Des. Codes Cryptogr. 88, 1159–1177 (2020). https://doi.org/10.1007/s10623-020-00739-6

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  • DOI: https://doi.org/10.1007/s10623-020-00739-6

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