nach oben

Designs, Codes and Cryptography

Erschienen in:

01.11.2014

A matrix approach for constructing quadratic APN functions

verfasst von: Yuyin Yu, Mingsheng Wang, Yongqiang Li

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2/2014

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A one to one correspondence is given between quadratic homogeneous APN functions and a special kind of matrices which we call as QAM’s. By modifying the elements of a known QAM, new quadratic APN functions can be constructed. Based on the nice mathematical structures of the QAM’s, an efficient algorithm for constructing quadratic APN functions is proposed. On \(\mathbb {F}_{2^7}\), we have found 471 new CCZ-inequivalent quadratic APN functions, which is 20 times more than the number of the previously known ones. Before this paper, It is only found 23 classes of CCZ-inequivalent APN functions on \(\mathbb {F}_{2^8}\). With the method of this paper, we have found 2,252 new CCZ-inequivalent quadratic APN functions, and this number is still increasing.

Vorheriger Artikel On transform-domain error and erasure correction by Gabidulin codes

Nächster Artikel A new large class of functions not APN infinitely often

Beth T., Ding C.: On almost perfect nonlinear permutations. In: Advances in Cryptology—EUROCRYPT’93. LNCS, vol. 765, pp. 65–76. Springer, New York (1994).

Bracken C., Byrne E., Markin N., McGuire G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields Appl. 14(3), 703–714 (2008).

Bracken C., Byrne E., Markin N., McGuire G.: A few more quadratic APN functions. Cryptogr. Commun. 3(3), 43–53 (2011).

Browning K., Dillon J.F., McQuistan M.: APN polynomials and related codes. J. Comb. Inf. Syst. Sci., 34(1–4), 135–159, (2009) (Special volume honoring the 75-th birthday of Prof. D.K.Ray-Chaudhuri).

Browning K., Dillon J.F., McQuistan M., Wolfe A.J.: An APN permutation in dimension six. Contemaray Math. 58, 33–42 (2010).

Budaghyan L., Carlet C., Pott A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006).

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

Budaghyan L, Carlet C., Leander G.: Constructing new APN functions from known ones. Finite Fields Appl. 15(2), 150–159 (2009).

Budaghyan L., Carlet C., Leander G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inf. Theory 54(9), 4218–4229 (2008).

10.

Carlet C.: Vectorial Boolean functions for cryptography, In: Crama Y., Hammer P. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398–469. Cambridge University Press, Cambridge. http://www.math.univ-paris13.fr/~carlet/pubs.html (2014). Accessed 25 Aug 2013.

11.

Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998).

12.

Daemen J., Rijmen V.: The Design of Rijndael. Springer (2002).

13.

Dillon J.F.: APN polynomials: an update, Fq9, In: The 9th International Conference on Finite Fields and Their Applications, Dublin (2009).

14.

Edel Y.: Geoemetric and combinatorial aspects of APN functions. In: Contact Forum: Coding Theory and Cryptography III, Brussels. http://cage.ugent.be/~ls/website2009/abstracts/slidesyvesedel.pdf (2009). Accessed 20 Aug 2013.

15.

Edel Y.: Quadratic APN functions as subspaces of alternating bilinear forms. In: Proceedings of the Contact Forum Coding Theory and Cryptography III, Belgium 2009, pp. 11–24 (2011).

16.

Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009).

17.

Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inf. Theory 52(2), 744–747 (2006).

18.

Dempwolff U., Edel Y.: Dimensional dual hyperovals and APN functions with translation group. J. Algebr. Comb. 39, 457–496. http://link.springer.com/article/10.1007 (2014).

19.

Edel Y.: On quadratic APN functions and dimensional dual hyperovals. Des. Codes Cryptogr. 57(1), 35–44 (2010).

20.

Gold R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968).

21.

Lidl R., Niederreiter H.: Finite Fields, pp. 58. Cambridge University Press, Cambridge (1983).

22.

Ling S., Qu L.J.: A note on linearized polynomials and the dimension of their kernels. Finite Fields Appl. 18(1), 56–62 (2012).

23.

Nyberg K., Knudsen L.R.: Provable security against differential cryptanalysis. In: CRYPTO 92. LCNS, vol. 740, pp. 566–574. Springer, New York (1993).

24.

Yoshiara S.: Equivalences of quadratic APN functions. J. Algebr. Comb. 35(3), 461–475 (2012).

25.

Yu Y., Wang M., Li Y.: A matrix approach for constructing quadratic APN functions. Cryptology ePrint Archive. Report (2013/2007).

Titel: A matrix approach for constructing quadratic APN functions
verfasst von: Yuyin Yu
Mingsheng Wang
Yongqiang Li
Publikationsdatum: 01.11.2014
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2/2014
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9955-3

Springer Professional

A matrix approach for constructing quadratic APN functions

Abstract

Premium Partner

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 2/2014

Generation of full cycles by a composition of NLFSRs

Distinguisher-based attacks on public-key cryptosystems using Reed–Solomon codes

Editorial: special issue on coding and cryptography

On transform-domain error and erasure correction by Gabidulin codes

Multi-trial Guruswami–Sudan decoding for generalised Reed–Solomon codes

Wiretap lattice codes from number fields with no small norm elements

Premium Partner