Bent functions and their generalizations
Thelecture is about bent functions - Boolean functions on the maximal possible distancesfrom the set of all affine functions. In the compact form we present main properties,constructions, generalizations of bent functions and discuss main open problems in thisarea.
Download file
Counter downloads: 233
Keywords
бент-функция, обобщения бент-функцийAuthors
| Name | Organization | |
| Tokareva N. N. | Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk | tokareva@math.nsc.ru |
References
Амбросимов А. С. Свойства бент-функций q-значной логики над конечными полями / / Дискретная математика. 1994. Т 6. №3. С. 50-60.
Логачев О. А., Сальников А. А., Ященко В. В. Бент-функции на конечной абелевой группе / / Дискретная математика. 1997. Т. 9. №4. С. 3-20.
Солодовников В. И. Бент-функции из конечной абелевой группы в конечную абелеву группу / / Дискретная математика. 2002. Т. 14. №1. С. 99-113.
Токарева Н. Н. Бент-функции с более сильными свойствами нелинейности: k-бент- функции / / Дискрет. анализ и исслед. операций. Сер. 1. 2007. Т. 14. №4. С. 76-102.
Токарева Н. Н. Бент-функции: результаты и приложения. Обзор работ / / Прикладная дискретная математика. 2009. Т. 2. №1. С. 15-37. Доступен на http://mi.mathnet.ru/pdm50.
Токарева Н. Н. Обобщения бент-функций. Обзор работ / / Дискрет. анализ и исслед. операций. 2009. Т. 16.
Ященко В. В. О критерии распространения для булевых функций и о бент-функциях / / Пробл. передачи информации. 1997. Т. 33. Вып. 1. С. 75-86.
http://www.math.cornell.edu/News/AnnRep/AR2002-2003.pdf - Cornell University. Department of Mathematics. Annual Report 2002-2003.
Agievich S. V. On the representation of bent functions by bent rectangles / / Fifth Int. Petrozavodsk conf. on probabilistic methods in discrete mathematics (Petrozavodsk, Russia, June 1-6, 2000). Proc. Boston: VSP, 2000. P. 121-135. Available at http://arxiv.org/abs/math/0502087.
Bernasconi A., Codenotti B. Spectral analysis of Boolean functions as a graph eigenvalue problem / / IEEE Trans. Computers. 1999. V. 48. No. 3. P. 345-351.
Bernasconi A., Codenotti B., VanderKam J. M. A characterization of bent functions in terms of strongly regular graphs / / IEEE Trans. Computers. 2001. V. 50. No. 9. P. 984-985.
Carlet C. Partially-bent functions / / Designs, Codes and Cryptography. 1993. V. 3. No. 2. P. 135-145.
Carlet C. On the higher order nonlinearities of Boolean functions and S-boxes, and their generalizations / / The Fifth Int. Conf. on Sequences and Their Applications - SETA'2008 Proc. (Lexington, Kentucky, USA. September 14-18, 2008). Berlin: Springer, 2008. P. 345-367 (Lecture Notes in Comput. Sci. V. 5203).
Carlet C., Ding C. Highly nonlinear mappings / / J. Complexity. 2004. V. 20. No. 2-3. P. 205-244.
Carlet C., Guillot P. An alternate characterization of the bentness of binary functions, with uniqueness / / Designs, Codes and Cryptography. 1998. V. 14. P. 133-140.
Carlet C., Klapper A. Upper bounds on the numbers of resilient functions and of bent functions / / 23rd Symposium on Information Theory (Benelux, Belgium. May, 2002). Proc. 2002. P. 307-314. The full version will appear in Lecture Notes dedicated to Philippe Delsarte. Available at http://www.cs.engr.uky.edu/~klapper/ps/bent.ps.
Chee S., Lee S., Kim K. Semi-bent Functions / / Advances in Cryptology - ASIACRYPT '94 - 4th International Conference on the Theory and Applications of Cryptology. (Wollongong, Australia. November 28 - December 1, 1994). Proc. Berlin: Springer, 1995. P. 107-118 (Lecture Notes in Comput. Sci. V. 917).
Detombe J., Tavares S. Constructing large cryptographically strong S-boxes / / Advances in Cryptology - AUSCRYPT'92. (Gold Coast, Queensland, Australia. December 13-16, 1992) Proc. Berlin: Springer, 1993. P. 165-181 (Lecture Notes in Comput. Sci. V. 718).
Dillon J. F. A survey of bent functions / / The NSA Technical J. 1972. Special Issue. P. 191-215.
Dillon J. F. Elementary Hadamard Difference sets j j Ph. D. Thesis. Univ. of Maryland, 1974.
Dobbertin H., Leander G. A survey of some recent results on bent functions j j Sequences and their applications. - SETA 2004. Third Int. conference (Seoul, Korea, October 24-28, 2004). Revised selected papers. Berlin: Springer, 2005. P. 1-29 (Lecture Notes in Comput. Sci. V. 3486).
Dobbertin H., Leander G. Cryptographer's Toolkit for Construction of 8-Bit Bent Functions j j Cryptology ePrint Archive, Report 2005/089, available at h t tp ://e p r in t .ia c r .o r g /.
Dobbertin H., Leander G., Canteaut A., et al. Construction of Bent Functions via Niho Power Functions j j J. Combin. Theory. Ser. A. 2006. V. 113. No. 5. P. 779-798. Available at http://www-rocq.inria.fr/secret/Anne.Canteaut/Publications/index-pub.html.
Gong G., Golomb S. W. Transform Domain Analysis of DES j j IEEE Trans. Inform. Theory. 1999. V. 45. No. 6. P. 2065-2073.
Kumar P. V., Scholtz R. A., Welch L. R. Generalized bent functions and their properties / / J. Combin. Theory. Ser. A. 1985. V. 40. No. 1. P. 90-107.
h ttp ://la n g e v in .u n iv -tln .fr /p r o je c t/q u a r tic s / - Classification of Boolean Quartics Forms in eight Variables (Langevin P.). 2008.
Langevin P., Leander G. Counting all bent functions in dimension 8 / / Workshop on Coding and Cryptograpy. 2009. to appear.
Leveiller S., Zemor G., Guillot P., and Boutros J. A new cryptanalytic attack for PNgenerators filtered by a Boolean function j j Selected Areas of Cryptography - SAC 2002. Proc. P. 232-249 (Lecture Notes in Comput. Sci. V. 2595).
McFarland R. L. A family of difference sets in non-cyclic groups / / J. Combin. Theory. Ser. A. 1973. V. 15. No. 1. P. 1-10.
Meng Q., Zhang H., Yang M. C., Cui J. On the degree of homogeneous bent functions j j Available at h t tp ://e p r in t .ia c r .o r g , 2004j284.
Nyberg K. Perfect nonlinear S-boxes j j Advances in cryptology - EUROCRYPT'1991. Int. conference on the theory and application of cryptographic techniques (Brighton, UK, April 8-11, 1991). Proc. Berlin: Springer, 1991. P. 378-386 (Lecture Notes in Comput. Sci. V. 547).
Olejar D., Stanek M. On cryptographic properties of random Boolean functions j j J. Universal Computer Science. 1998. V. 4. No. 8. P. 705-717.
Poinsot L. Multidimensional bent functions GESTS International Transactions on Computer Science and Engeneering. 2005. V. 18. No. 1 P. 185-195.
Poinsot L., Harari S. Generalized Boolean bent functions j j Progress in Cryptology - Indocrypt 2004 (Chennai (Madras), India. December 20 - 22, 2004). Proc. Springer. P. 107-119 (Lecture Notes in Comput. Sci. V. 3348).
Qu C., Seberry J., Pieprzyk J. Homogeneous bent functions j j Discrete Appl. Math. 2000. V. 102. No. 1-2. P. 133-139.
Rothaus O. On bent functions j j IDA CRD W. P. 1966. No. 169.
Rothaus O. On bent functions j j J. Combin. Theory. Ser. A. 1976. V. 20. No.3. P. 300-305.
Schmidt K-U. Quaternary Constant-Amplitude Codes for Multicode CDMA j j IEEE International Symposium on Information Theory - ISIT'2007. (Nice, France. June 24-29, 2007). Proc. 2007. P. 2781-2785. Available at http://a rxiv.org/abs/cs.IT/0 6111 62.
Yang M., Meng Q., Zhang H. Evolutionary design of trace form bent functions j j Cryptology ePrint Archive, Report 2005j322, available at h t tp ://e p r in t .ia c r .o r g /.
Youssef A., Gong G. Hyper-bent functions j j Advances in cryptology - EUROCRYPT'2001. Int. conference on the theory and application of cryptographic techniques (Innsbruk, Austria, May 6-10, 2001). Proc. Berlin: Springer, 2001. P. 406-419 (Lecture Notes in Comput. Sci. V. 2045).
