Constructions of almost optimal resilient Boolean functions on large even number of variables

IEEE Transactions on Information Theory

Volumn 55, Issue 12, 2009, Pages 5822-5831

  Zhang, Wei Guo ^a   Xiao, Guo Zhen ^a  

^a XIDIAN UNIVERSITY   (China)

Author keywords

Algebraic degree; Boolean function; Disjoint spectra functions; Nonlinearity; Resiliency; Stream cipher

Indexed keywords

ALGEBRAIC DEGREES; NON-LINEARITY; RESILIENCY; SPECTRA FUNCTIONS; STREAM CIPHERS;

ALGEBRA; CRYPTOGRAPHY; HYDRAULICS; NUMBER THEORY; OPTIMIZATION;

BOOLEAN FUNCTIONS;

EID: 77955618991     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2009.2032736     Document Type: Article

References (32)

1. P. Camion, C. Carlet, P. Charpin, and N. Sendrier, "On correlation-immune functions," in Advances in Cryptology-CRYPTO'91, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1992, vol.547, pp. 86-100.
2. C. Carlet, "A larger class of cryptographic Boolean functions via a study of the Maiorana-Mcfarland constructions," in Advances in Cryptology- CRYPTO 2002, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2002, vol.2442, pp. 549-564.
3. S. Chee, S. Lee, D. Lee, and S. H. Sung, "On the correlation immune functions and their nonlinearity," in Advances in Cryptology-Asiacrypt' 96, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1997, vol.1163, pp. 232-243.
4. J. Clark, J. Jacob, S. Stepney, S. Maitra, and W. Millan, "Evolving Boolean functions satisfying multiple criteria," in Progress in INDOCRYPT 2002, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2002, vol.2551, pp. 246-259.
5. J. F. Dillon, "Elementary Hadamard Difference Set," Ph.D. dissertation, University of Maryland, College Park, 1974.
6. H. Dobbertin, "Construction of bent functions and balanced Boolean functions with high nonlinearity," in Proc. Workshop on Fast Software Encryption (FES 1994), Berlin, Germany, 1995, vol.1008, pp. 61-74.
7. M. Fedorova and Y. V. Tarannikov, "On the constructing of highly nonlinear resilient Boolean functions by means of special matrices," in Progress in Cryptology-INDOCRYPT 2001, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2001, vol.2247, pp. 254-266.
8. S. Kavut, S. Maitra, and M. D. Yücel, "Search for Boolean functions with excellent profiles in the rotation symmetric class," IEEE Trans. Inf. Theory, vol.53, no.5, pp. 1743-1751, 2007.
9. K. Khoo, G. Gong, and H.-K. Lee, "The rainbow attack on stream ciphers based on Maiorana-McFarland functions," in Appiled Cryptography and Network Security-ACNS 2006, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2006, vol.3989, pp. 194-209.
10. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
11. n/2-2" Discr. Appl. Math., vol.154, pp. 357-369, 2006.
12. S. Maitra and E. Pasalic, "Further constructions of resilient Boolean functions with very high nonlinearity," IEEE Trans. Inf. Theory, vol.52, no.5, pp. 2269-2270, 2006.
13. W. Meier and O. Staffelbach, "Nonlinearity criteria for cryptographic functions," in Advances in Cryptology-EUROCRYPT'89, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1990, vol.434, pp. 549-562.
14. W. Millan, A. Clark, and E. Dawson, "An effective genetic algorithm for finding highly nonlinear Boolean functions," in Proc. First Int. Conf. Inf. Commun. Security, Berlin, Germany, 1997, vol.1334, pp. 149-158.
15. W. Millan, A. Clark, and E. Dawson, "Boolean function design using hill climbing methods," in Proc. 4th Australasian Conf. Inf. Security Privacy, Berlin, Germany, 1999, vol.1587, pp. 1-11.
16. W. Millan, A. Clark, and E. Dawson, "Heuristic design of cryptographically strong balanced Boolean functions," in Advances in Cryptology-EUROCRYPT'98, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1998, vol.1403, pp. 489-499.
17. E. Pasalic, "Maiorana-McFarland class: Degree optimization and algebraic properties," IEEE Trans. Inf. Theory, vol.52, no.10, pp. 4581-4594, 2006.
18. E. Pasalic, S. Maitra, T. Johansson, and P. Sarkar, "New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity," in Workshop on Coding and Cryptography (WCC 2001), Paris, France, Jan. 2001.
19. Published in Electronic Notes in Discrete Mathematics. Amsterdam, The Netherlands: Elsevier, 2001, vol. 6, pp. 158-167.
20. O. S. Rothaus, "On 'bent' functions," J. Combin. Theory, ser. A, vol.20, pp. 300-305, 1976.
21. Z. Saber, M. F. Uddin, and A. Youssef, "On the existence of (9, 3, 5, 240) resilient functions," IEEE Trans. Inf. Theory, vol.48, no.7, pp. 1825-1834, 2002.
22. P. Sarkar and S. Maitra, "Construction of nonlinear Boolean functions with important cryptographic properties," in Advances in Cryptology-EUROCRYPT 2000, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 2000, vol.1807, pp. 485-506.
23. P. Sarkar and S. Maitra, "Efficient implementation of cryptographically useful large Boolean functions," IEEE Trans. Comput., vol.52, no.4, pp. 410-417, 2003.
24. P. Sarkar and S. Maitra, "Nonlinearity bounds and constructions of resilient Boolean functions," in Advances in Cryptology-CRYPTO2000, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer- Verlag, 2000, vol.1880, pp. 515-532.
25. P. Sarkar and S. Maitra, "Construction of nonlinear resilient Boolean functions using small affine functions," IEEE Trans. Inf. Theory, vol.50, no.9, pp. 2185-2193, 2004.
26. J. Seberry, X.-M. Zhang, and Y. Zheng, "Nonlinearity and propagation characteristics of balanced Boolean functions," Inf. Computat., vol.119, pp. 1-13, 1995.
27. J. Seberry, X.-M. Zhang, and Y. Zheng, "Nonlinearly balanced Boolean functions and their propagation characteristics," in Advances in Cryptology-CRYPTO'93, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1994, vol.773, pp. 49-60.
28. J. Seberry, X.-M. Zhang, and Y. Zheng, "On constructions and nonlinearity of correlation immune Boolean functions," in Advances in Cryptography- EUROCRYPT'93 (Lecture Notes in Computer Science). Berlin, Germany: Springer-Verlag, 1984, vol.765, pp. 181-199.
29. T. Siegenthaler, "Correlation-immunity of nonlinear combining functions for cryptographic applications," IEEE Trans. Inf. Theory, vol.30, no.5, pp. 776-780, 1984.
30. Y. V. Tarannikov, "On resilient Boolean functions with maximum possible nonlinearity," in Progress in Cryptology-INDOCRYPT 2000, ser. Lecture Notes in Computer Science. Berlin, Germany: Springer Verlag, 2000, vol.1977, pp. 19-30.
31. Y. V. Tarannikov, "New constructions of resilient Boolean functions with maximal nonlinearity," in Proc. Workshop on Fast Software Encryption (FSE 2001), Berlin, Germany, 2001, vol.2355, pp. 66-77.
32. G. Z. Xiao and J. L. Massey, "A spectral characterization of correlation- immune combining functions," IEEE Trans. Inf. Theory, vol.34, no.3, pp. 569-571, 1988.