Skip to main content
SpringerLink
Log in

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field \(\mathbb{F}_q\) are permutations if and only if q ≡ 3 (mod 4).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beth T, Ding C. On almost perfect nonlinear permutations. In: Advances in Cryptology Eurocrypt’93. Lecture Notes on Computer Science, vol. 765. Berlin: Springer-Verlag, 1994, 65–76

    Google Scholar 

  2. Bierbrauer J. New semifields, PN and APN functions. Des Codes Cryptogr, 2010, 54: 189–200

    Article  MathSciNet  MATH  Google Scholar 

  3. Biham E, Shamir A. Differential cryptanalysis of DES-like cryptosystems. J Cryptol, 1991, 4: 3–72

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Browning K, Dillon J, Kibler R, et al. APN polynomials and related codes. J Combin Inform Sys Sci, 2009, 34: 135–159

    Google Scholar 

  6. Browning K, Dillon J, McQuistan M, et al. An APN permutation in dimension six. In: Proceedings of the 9th Conference on Finite Fields and Applications FQ9. Contemp Math, 2010, 518: 33–42

    Article  MathSciNet  Google Scholar 

  7. Budaghyan L, Helleseth T. New perfect nonlinear multinomials over \(\mathbb{F}_{p^{2k} }\) for any odd prime p. In: Proceedings of International Conference on Sequences and Their Applications SETA 2008. Lecture Notes on Computer Science, vol. 5203. Berlin: Springer-Verlag, 2008, 403–414

    Chapter  Google Scholar 

  8. Budaghyan L, Carlet C, Pott A. New classes of almost bent and almost perfect nonlinear functions. IEEE Trans Inform Theory, 2006, 52: 1141–1152

    Article  MathSciNet  MATH  Google Scholar 

  9. Budaghyan L, Helleseth T. New commutative semifields defined by new PN multinomials. Cryptogr Commun, 2011, 3: 1–16

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlet C. Vectorial Boolean functions for cryptography. In: Crama Y, Hammer P, eds. Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Cambridge: Cambridge University Press, 2010: 398–472

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Charpin P, Kyureghyan G. When does G(x) + γTr(H(x)) permute \(\mathbb{F}_{p^n }\)? Finite Fields Appl, 2009, 15: 615–632

    Article  MathSciNet  MATH  Google Scholar 

  13. Coulter R, Matthews R. Planar functions and planes of Lenz-Barlotti class II. Des Codes Cryptogr, 1997, 10: 167–184

    Article  MathSciNet  MATH  Google Scholar 

  14. Daemen J, Rijmen V. The design of rijndael. In: AES-The Advanced Encryption Standard. Berlin: Springer-Verlag, 2002, 221–227

  15. Dembowshi P, Ostrom T. Planes of order n with collineation groups of order n2. Math Z, 1968, 103: 239–258

    Article  MathSciNet  Google Scholar 

  16. Ding C, Yuan J. A new family of skew Paley-Hadamard difference sets. J Combin Theory Ser A, 2006, 113: 1526–1535

    Article  MathSciNet  MATH  Google Scholar 

  17. Felke P. Computing the uniformity of power mappings: a systematic approach with the multi-variate method over finite fields of odd characteristic. Ph.D. Dissertation. University of Bochum, Bochum, Germany, 2005

    Google Scholar 

  18. Helleseth T, Kholosha A. Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans Inform Theory, 2006, 52: 2018–2032

    Article  MathSciNet  MATH  Google Scholar 

  19. Helleseth T, Rong C, Sandberg D. New families of almost perfect nonlinear power mappings. IEEE Trans Inform Theory, 1999, 45: 475–485

    Article  MathSciNet  MATH  Google Scholar 

  20. Helleseth T, Sandberg D. Some power mappings with low differential uniformity. Appl Algebra Engrg Comm Comput, 1997, 8: 363–370

    Article  MathSciNet  MATH  Google Scholar 

  21. Hollmann H, Xiang Q. A class of permutation polynomials of \(\mathbb{F}_{2^m }\) related to Dickson polynomials. Finite Fields Appl, 2005, 11: 111–122

    Article  MathSciNet  MATH  Google Scholar 

  22. Hou X D. Two classes of permutation polynomials over finite fields. J Combin Theory Ser A, 2011, 118: 448–454

    Article  MathSciNet  MATH  Google Scholar 

  23. Kyureghyan G, Pott A. Some theorems on planar mappings. In: Proceedings of WAIFI 2008. Lecture Notes on Computer Science, vol. 5130. Berlin: Springer-Verlag, 2008, 115–122

    Google Scholar 

  24. Nyberg K. Differentially uniform mappings for cryptography. In: Advances in Cryptology Eurocrypt’93. Lecture Notes on Computer Science, vol. 765. Berlin: Springer-Verlag, 1994, 55–64

  25. Weng G B, Dong C P. A note on permutation polynomials over ℤn. IEEE Trans Inform Theory, 2008, 54: 4388–4390

    Article  MathSciNet  Google Scholar 

  26. Weng G B, Zeng X Y. Further results on planar DO functions and commutative semifields. Des Codes Cryptogr, 2012, 63: 413–423

    Article  MathSciNet  MATH  Google Scholar 

  27. Yuan J, Ding C. Four classes of permutation polynomials of \(\mathbb{F}_{2^m }\). Finite Fields Appl, 2007, 13: 869–876

    Article  MathSciNet  MATH  Google Scholar 

  28. Yuan J, Ding C, Wang H, et al. Permutation polynomials of the form (xp − x + δ)s + L(x). Finite Fields Appl, 2008, 14: 482–493

    Article  MathSciNet  MATH  Google Scholar 

  29. Zha Z B, Hu L. Two classes of permutation polynomials over finite fields. Finite Fields Appl, 2012, 18: 781–790

    Article  MathSciNet  MATH  Google Scholar 

  30. Zha Z B, Kyureghyan G, Wang X. Perfect nonlinear binomials and their semifields. Finite Fields Appl, 2009, 15: 125–133

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Hubei University, Wuhan, 430062, China

    WenJie Jia & XiangYong Zeng

  2. Department of Informatics, University of Bergen, Bergen, N-5020, Norway

    ChunLei Li & Tor Helleseth

  3. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    XiangYong Zeng & Lei Hu

Authors
  1. WenJie Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XiangYong Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. ChunLei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tor Helleseth
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lei Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XiangYong Zeng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jia, W., Zeng, X., Li, C. et al. Permutation polynomials with low differential uniformity over finite fields of odd characteristic. Sci. China Math. 56, 1429–1440 (2013). https://doi.org/10.1007/s11425-013-4599-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4599-8

Keywords

MSC(2010)

Search

Navigation