Vectorial resilient PC(l) of order k Boolean functions from AG-codes

Abstract

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 1, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2 ^m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.

Keywords

Cryptography / Boolean function / Algebraic-geometric code

References

[1] Carlet C.. On the propagation criterion of degree l and order k, Advances in Cryptology, 1998, Berlin, Heidelberg, New York: Springer-Verlag 462-474

[2] Carlet C.. Crama Y., Hammer P.. Boolean Functions for Cryptography and Error Correcting Codes, Boolean Methods and Models, 2010, Cambridge: Cambridge University Press

[3] Carlet, C., Vectorial Boolean functions for cryptography, Boolean Methods and Models, Y. Crama and P. Hammer (eds.), Cambridge University Press, Cambridge, to appear.

[4] Cheon J. H.. Nonlinear vector Boolean functions, Advances in Cryptology, 2001, Berlin, Heidelberg, New York: Springer-Verlag 458-469

[5] Garcia A., Stichtenoth H.. On the asymptotic behaviour of some towers of function fields over finite fields. J. Number Theory, 1996, 61: 248-273

[6] Grassl M., Geiselmann W., Beth T.. Fossoreier M., Imai H., Lin S., Poli A.. Quantum Reed-Solomon codes, Proc. AAECC 13, 1996, New York: Springer-Verlag 231-244

[7] Johansson T., Pasalic E.. A construction of resilient functions with high nonlinearity. IEEE Trans. Inf. Theory, 2002, 49(2): 494-501

[8] Kurosawa K., Satoh T.. Design of SAC/PC(l) of order k Boolean functions and three other cryptographic criteria, Advances in Cryptology, 1997, Berlin, Heidelberg, New York: Springer-Verlag 434-449

[9] Matsumoto, R., Kurosawa, K., Itoh, T., et al, Primal-dual distance bounds of linear codes with applications to cryptography, IEEE Trans. Inf. Theory, to appear. Cryptology e-print 194/2005

[10] Meier W., Staffelbach O.. Nonlinearity criteria for cryptographic functions, Advances in Cryptology, 1989, Berlin, Heidelberg, New York: Springer-Verlag 549-562

[11] Preneel B., Govaerts R., Vandevalle J.. Boolean functions satisfying high order propagation criteria, Advances in Cryptology, 1990, Berlin, Heidelberg, New York: Springer-Verlag 161-173

[12] Stichtenoth H.. Algebraic Function Fields and Codes, 1993, Berlin: Springer-Verlag

[13] Tsfasman M. A., Vladut S. G.. Algebraic-Geometric Codes, 1991, Dordrecht: Kluwer

[14] van der Geer, G. and van der Vludgt, M., Tables of curves with many points. http://www.science.uva.nl/geer/

[15] van Lint J. H.. Introduction to coding theory, 1999 3rd ed. Berlin: Springer-Verlag

[16] Webster A., Tavares S.. On the design of S-boxes, Advances in Cryptology, 1985, Berlin, Heidelberg, New York: Springer-Verlag 523-534

[17] Zhang X. M., Zheng Y.. Cryptographically resilient functions. IEEE Transactions on Information Theory, 1997, 43(5): 1740-1747