Бент-функции: результаты и приложения. Обзор работ | Прикладная дискретная математика. 2009. № 1(3).

Yu N. Y., Gong G. Constructions of quadratic bent functions in polynomial forms // IEEE Trans. Inform. Theory 2006. V. 52. No. 7. P. 3291-3299.

available at <http://eprint.iacr.org/>.

Youssef A., Gong G.Hyper-bent functions// 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).

Yang M., Meng Q., Zhang H. Evolutionary design of trace form bent functions // Cryptology ePrint Archive, Report 2005/322,

Wang L., Zhang J. A best possible computable upper bound on bent functions // J. West of China. 2004. V. 33. No. 2. P. 113-115.

Schmidt K-U. Quaternary Constant-Amplitude Codes for Multicode CDMA // Available at <http://arxiv.org/abs/cs.IT/0611162>.

Rothaus O. On bent functions // J. Combin. Theory. Ser. A. 1976. V. 20. No. 3. P. 300-305.

Rothaus O. On bent functions // IDA CRD W.P. No. 169. 1966.

Rodier F. Private Communication. 2008.

Preprint is available at <http://iml.univ-mrs.fr/editions/preprint2003/files/RodierFoncBool.pdf>

Rodier F. Asymptotic nonlinearity of Boolean functions// Designs, Codes and Cryptography. 2006. V. 40. No. 1. P. 59-70.

Riera G., Parker M. G. Generalised Bent Criteria for Boolean Functions (I) // IEEE Trans. Inform. Theory 2006. V. 52. No. 9. P. 4142-4159.

Qu C., Seberry J., Pieprzyk J. Homogeneous bent functions // Discrete Appl. Math. 2000. V. 102. No. 1-2. P. 133-139.

Preneel В. Analysis and design of cryptographic hash functions //Ph. D. thesis, Katholieke Universiteit Leuven, 3001 Leuven, Belgium. 1993.

Preneel В., Van Leekwijck W., Van Linden L., Govaerts R., Vandevalle J. Propogation characteristics of Boolean functions // Advances in cryptology - EUROCRYPT'1990. Int. conference on the theory and application of cryptographic techniques (Aarhus, Denmark, May 21-24, 1990). Proc. Berlin: Springer, 1991. P. 161-173 (Lecture Notes in Comput. Sci. V. 473).

Paters on K. G. Sequences For OFDM and Multi-code CDMA: two problems in algebraic Coding Theory // Sequences and their applications. - Seta 2001. Second Int. Conference (Bergen, Norway, May 13-17, 2001). Proc. Berlin: Springer, 2002. P. 46-71.

Parker M. G., Pott A. On Boolean functions which are bent and negabent // Sequences, Subsequences, and Consequences - SSC 2007 - International Workshop. (Los Angeles, СА, USA, May 31 - June 2, 2007). Proc. Berlin: Springer, 2007. P. 9-23 (Lecture Notes in Comput. Sci. V. 4893).

Parker M. G. The constabent properties of Golay-Davis-Jedwab sequences // IEEE International Symposium on Information Theory - ISIT'2000. (Sorrento, Italy, June 25-30, 2000). Proc. 2000. P. 302.

Olsen J.D., Scholtz R. A., Welch L. R. Bent-function sequences // IEEE Trans. Inform. Theory. 1982. V. 28. No. 6. P. 858-864.

Olejdr D., Stanek M. On cryptographic properties of random Boolean functions // J. Universal Computer Science. 1998. V. 4. No. 8. P. 705-717.

Nyberg K. Differentially uniform mappings for cryptography // Advances in cryptology - EUROCRYPT'1993. Int. conference on the theory and application of cryptographic techniques (Lofthus, Norway. May 23-27, 1993). Proc. Berlin: Springer, 1994. P. 55-64 (Lecture Notes in Comput. Sci. V. 765).

Nyberg K. Perfect nonlinear S-boxes // 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).

Millan W., Clark A., Dawson E. Smart hill climbing finds better Boolean functions // Workshop on Selected Areas in Cryptology. 1997. Workshop record. P. 50-63.

Meng Q., Zhang H., Wang Z. Designing bent functions using evolving computing // Acta Electronica Sinica. 2004. No. 11. P. 1901-1903.

Millan W., Clark A., Dawson E. An effective genetic algorithm for finding highly nonlinear Boolean functions // First Int. conference on Information and Communications Security -ICICS'97. (Beijing, China, November 11-14, 1997). Proc. Springer Verlag, 1997. P. 149-158 (Lecture Notes in Comput. Sci. V. 1334).

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., Yang M. G., Zhang H., Cui J.-S. A novel algorithm enumerating bent functions // Cryptology ePrint Archive, Report 2004/274, available at <http://eprint.iacr.org/>.

Matsui M. Linear cryptanalysis method for DES cipher // Advances in Cryptology -EUROCRYPT'93. Workshop on the theory and application of cryptographic techniques (Lofthus, Norway, May 23-27, 1993). Proc. Berlin: Springer, 1994. P. 386-397 (Lecture Notes in Comput. Sci. V. 765).

Maitra S., Sarkar P. Maximum Nonlinearity of Symmetric Boolean Functions on Odd Number of Variables // IEEE Trans. Inform. Theory. 2002. V. 48. No. 9. P. 2626-2630.

Leander N. G. Monomial bent functions // IEEE Trans. Inform. Theory. 2006. V. 52. No. 2. P. 738-743.

Leander N. G., Langevin P. On exponents with highly divisible Fourier coefficients and conjectures of Niho and Dobbertin // Algebraic Geometry and its applications (France, May 7-11, 2007) Proc. 2008. P. 410-418.

Langevin P., Rabizzoni P., Veron P., Zanotti J.-P. On the number of bent functions with 8 variables // Second International Conference BFCA - Boolean Functions: Cryptography and Applications (Rouen, France, March 13-15, 2006). Proc. 2006. P. 125-135.

Langevin P., Leander G. Monomial bent functions and Stickelberger's theorem // Finite Fields and Applications. 2008. V. 14. P. 727-742.

<http://langevin.univ-tln.fr/project/quartics/>Classification of Boolean Quartics Forms in eight Variables (Langevin P.). 2008.

Krotov D. S., Avgustinovich S. V. On the Number of 1-Perfect Binary Codes: A Lower Bound // IEEE Trans. Inform. Theory. 2008. V. 54. No. 4. P. 1760-1765.

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.

Khoo K., Gong G., Stinson D. R. A new characterization of semi-bent and bent functions on finite fields // Designs, Codes and Cryptography. 2006. V. 38. No. 2. P. 279-295.

Kerdock A. M. A class of low-rate non-linear binary codes // Inform. Control. 1972. V. 20. No. 2. P. 182-187.

Khoo K., Gong G., Stinson D. R. A new family of Gold-like sequences // ISIT - IEEE Int. Symposium on Information Theory (Lausanne, Switzerland, June 30-July 5, 2002). Proc. 2002. P. 181.

Kavut S., Maitra S., Yucel M. D. Search for Boolean functions with excellent profiles in the rotation symmetric class // IEEE Trans. Inform. Theory. 2007. V. 53. No. 5. P. 1743-1751.

Available at http://darkwing.uoregon.edu/~kantor/.

Kantor W. M. Codes, Quadratic Forms and Finite Geometries // Proceedings of Symposia in Applied Math. 1995. V. 50. P. 153-177.

Hu H., Feng D. On quadratic bent functions in polynomial forms // IEEE Trans. Inform. Theory 2007. V. 53. No. 7. P. 2610-2615.

Hou X.-D., Langevin P. Results on bent functions // J. Comb. Theory, Series A. 1997. V. 80. P. 232-246.

Hou X.-D. Cubic bent functions // Discrete Mathematics. 1998. V. 189. P. 149-161.

Available at <http://www.itl.waw.pl/czasopisma/JTIT/2003/4/19.pdf>.

Grocholewska-Czuryio A. A study of differences between bent functions constructed using Rothaus method and randomly generated bent functions // J. Telecommunications and Information Technology. 2003. No. 4. P. 19-24.

Available at <http://www.math.tugraz>.at/~cecc08/abstracts/cecc08_abstract_25.pdf.

Gangopadhyay S., Sharma D., Sarkar S., Maitra S. On Affine (Non) Equivalence of Bent Functions // CECC'08 - Central European Conference on Cryptography (Graz, Austria, July 2-4, 2008). Proc. 2008.

Fuller J. E., Dawson E., Millan W. Evolutionary generation of bent functions for cryptography // The 2003 Congress on Evolutionary Computation. 2003. CEC apos;03. V. 3. P. 1655-1661.

Available at <http://eprints.qut.edu.au/15828/>.

Available at <http://www-rocq.inria.fr/secret/Anne.Canteaut/Publications/index-pub.html>.

Fuller J. E. Analysis of affine equivalent Boolean functions for cryptography // Ph. D. thesis, Queensland University of Technology. Brisbane, Australia. 2003.

available at <http://eprint.iacr.org/>.

Dobbertin H., Leander G., Canteaut A., et al. Construction of Bent Functions via Niho Power Functions // J. Combin. Theory. Ser. A. 2006. V. 113. No. 5. P. 779-798.

Dobbertin H., Leander G. Cryptographer's Toolkit for Construction of 8-Bit Bent Functions // Cryptology ePrint Archive, Report 2005/089,

Dobbertin H., Leander G. A survey of some recent results on bent functions // 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 Н. Almost perfect nonlinear power functions over GF(2n): a new case for n divisible by 5 // Finite Fields and Applications FQ5 (Augsburg, Germany, August 2-6, 2000). Proc. Springer / Eds. D. Jungnickel, H. Niederreiter. 2000. P. 113-121.

Dobbertin H. Almost perfect nonlinear power functions over GF(2n): the Niho case // Inform. and Comput. 1999. V. 151. No. 1-2. P. 57-72.

Dobbertin H. Construction of bent functions and balanced Boolean functions with high nonlinearity // Fast Software Encryption, Second International Workshop - FSE'95. (Leuven, Belgium, December 14-16, 1994) Proc. Berlin: Springer, 1995. P. 61-74 (Lecture Notes in Comput. Sci. V. 1008).

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 // Ph. D. Thesis. Univ. of Maryland, 1974.

<http://www.mathematik.uni-kl.de/~dempw/>-Homepage of U. Dempwolff. See the section «Bent Functions in Dimensions 8,10,12». 2009.

Delsarte P. An algebraic approach to the association schemes of coding theory // Ph. D. Thesis, Univ. Catholique de Louvain, 1973.

Dempwolff U. Automorphisms and equivalence of bent functions and of difference sets in elementary Abelian 2-groups // Communications in Algebra. 2006. V. 34. No. 3. P. 1077-1131.

Van Dam E. R., Fon-Der-Flaass D. G. Codes, graphs, and schemes from nonlinear functions // European J. Combinatorics, 2003. V. 24. No. 1. P. 85-98.

Van Dam E. R., Fon-Der-Flaass D. G. Uniformly Packed Codes and More Distance Regular Graphs from Crooked Functions // J. Algebraic Combinatorics. 2000. V. 12. No. 2. P. 115-121.

Climent J.-J., Garcia F. J., Requena V. On the construction of bent functions of n + 2 variables from bent functions of n variables. // Advances in Math. of Communications. 2008. V. 2. No. 4. P. 421-431.

Clark J. A., Jacob J. L., Maitra S., Stanica P. Almost Boolean Functions: the Design of Boolean Functions by Spectral Inversion. // Computational Intelligence. Special Issue on Evolutionary Computing in Cryptography and Security. 2004. V. 20. No. 3. P. 450-462.

Clark J. A., Jacob J.L. Two-stage optimisation in the design of Boolean functions // 5th Australian Conference on Information Security and Privacy. (Brisbane, Australia, July 10-12, 2000). Proc. Springer-Verlag, 2000. P. 242-254 (Lecture Notes in Comput. Sci. V. 1841).

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).

Chase P. J., Dillon J. F., Lerche K. D. Bent functions and difference sets // NSA R41 Technical Paper. September, 1970.

Charpin P., Pasalic E., Tavernier С. On bent and semi-bent quadratic Boolean functions // IEEE Trans. Inform. Theory. 2005. V. 51. No. 12. P. 4286-4298.

<http://www.faqs.org/rfcs/rfc2144.html-CAST-128>. Rfc 2144 - the cast-128 encryption algorithm-1997.

Chabaud F., Vaudenay S. Links between Differential and Linear Cryptanalysis // Advances in Cryptology - EUROCRYPT '94, International Conference on the Theory and Application of Cryptographic Techniques. (Perugia, Italy. May 9-12, 1994) Proc. Springer, 1995. P. 356-365 (Lecture Notes in Comput. Sci. V. 950).

The full version will appear in Lecture Notes dedicated to Philippe Delsarte. Available at http://www.cs <http://www.es>.engr.uky.edu/~klapper/ps/bent.ps.

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).Ргос. 2002. Р. 307-314.

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., Ding C., Niederreiter H. Authentication schemes from highly nonlinear functions // Designs, Codes and Cryptography. 2006. V. 40. No. 1. P. 71-79.

Carlet C., Guillot P. A characterization of binary bent functions // J. Combin. Theory. Ser. A. 1996. V. 76. No. 2. P. 328-335.

Carlet C., Ding С. Highly nonlinear mappings // J. Complexity. 2004. V. 20. No. 2-3. P. 205-244.

Carlet С., Danielsen L.-E., Parker M. G., Sole P. Self Dual Bent Functions // Fourth International Conference BFCA - Boolean Functions: Cryptography and Applications (Copenhagen, Denmark, May 19-21, 2008). Proc. to appear. P. 39-52.

Carlet C., Charpin P., Zinoviev V. Codes, bent functions and permutations suitable for DES-like cryptosystems // Designs, Codes and Cryptography. 1998. V. 15. No. 2. P. 125-156.

Carlet С. Vectorial Boolean Functions for Cryptography //Chapter of the monograph «Boolean Methods and Models», CambridgeUniv. Press (P. Hammer, Y. Crama eds.), to appear. Prelim, version isavailable at www-rocq.<inria.fr/secret/Claude.Carlet/chap-vectorial-fcts.pdf>.

Carlet С. Boolean Functions for Cryptography and Error Correcting Codes // Chapter of the monograph «Boolean Methods and Models», Cambridge Univ. Press (P. Hammer, Y. Crama eds.), to appear. Prelim, version is available at www-rocq.<inria.fr/secret/Claude.Carlet/chap-fcts-Bool>.pdf.

Carlet С. On the confusion and diffusion properties of Maiorana-McFarland's and extended Maiorana-McFarland's functions // Special Issue «Complexity Issues in Coding Theory and Cryptography» dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday, J. Complexity. 2004. V. 20. P. 182-204.

Carlet С. On cryptographic complexity of Boolean functions // Proc. of the Sixth Conference on Finite Fields with Applications to Coding Theory, Cryptography and Related Areas. Springer, G. L. Mullen, H. Stichtenoth and H. Tapia-Recillas Eds. 2002. P. 53-69.

Carlet С. A construction of bent functions // Finite Fields and Applications, London mathematical society. 1996. Lecture series 233. P. 47-58.

Canteaut A., Charpin P., Kuyreghyan G. A new class of monomial bent functions // Finite Fields and Applications. 2008. V. 14. No. 1. P. 221-241.

Carlet С. Generalized Partial Spreads // IEEE Trans. Inform. Theory. 1995. V. 41. No. 5. P. 1482-1487.

Byrne E., McGuire G. On the non-existence of crooked functions on finite fields // WCC - International Workshop on Coding and Cryptography (Bergen, Norway, March 14-18, 2005). Proc. 2005. P. 316-324.

Budaghyan L. Private Communication. 2008.

Budaghyan L., Pott A. On differential uniformity and nonlinearity of functions // Discrete Mathematics. 2009. V. 309. No. 1. P. 371-384.

Available at <http://www.сosic.esat.kuleuven.be/publications/thesis-129.pdf>

Budaghyan L., Carlet С., Leander G. On inequivalence between known power APN functions // Fourth International Conference BFCA - Boolean Functions: Cryptography and Applications (Copenhagen, Denmark, May 19-21, 2008). Proc. to appear. P. 3-15.

Braeken A. Cryptographic properties of Boolean functions and S-boxes // Ph. D. Thesis, Katholieke Univ. Leuven, 2006.

Bracken C., Leander G. New families of functions with differential uniformity of 4 //Fourth International Conference BFCA - Boolean Functions: Cryptography and Applications (Copenhagen, Denmark, May 19-21, 2008). Proc. to appear. P. 190-194.

Bernasconi A., Codenotti В., 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.

Biham E., Shamir A.Differential cryptanalysis of DES-like cryptosystems // J. Cryptology. 1991. V. 4. No. 1. P. 3-72.

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.

Available at <http://arxiv.org/abs/0804.0209>.

Bending T. D., Fon-Der-Flaass D. G. Crooked Functions, Bent Functions and Distance Regular Graphs // Electronic J. Combinatorics. 1998. No. 5 (R34).

Available at <http://arxiv.org/abs/math/0502087>.

Agievich S. V. On the affine classification of cubic bent functions // Cryptology ePrint Archive, Report 2005/044,

available at <http://eprint.iacr.org/>.

Agievich S. V. Bent rectangles // NATO Advanced Study Institute on Boolean Functions in Cryptology and Information Security (Zvenigorod, Russia. September 8-18, 2007). Proc: Netherlands, IOS Press, 2008. P. 3-22.

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.

Adams С. On immunity against Biham and Shamir's «differential cryptanalysis» // Information Processing Letters. 1992. V. 41. P. 77-80.

Ященко В. В. О двух характеристиках нелинейности булевых отображений // Дискрет. анализ и исслед. операций. Сер. 1. 1998. Т. 5. № 2. С. 90-96.

Ященко В. В. О критерии распространения для булевых функций и о бент-функциях // Пробл. передачи информации. 1997. Т. 33. Вып. 1. С. 75-86.

Черемушкин А. В. Методы аффинной и линейной классификации булевых функций // Труды по дискретной математике. М.: Физматлит, 2001. Т. 4. С. 273-314.

www.math.nsc.ru/~tokareva.

Холл М. Комбинаторика. М.: Мир, 1970. 424 с.

Токарева Н. Н. Обобщения бент-функций. Обзор // Дискрет. анализ и исслед. операций. 2009. Т. 16

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.

Солодовников В. И. Бент-функции из конечной абелевой группы в конечную абелеву группу // Дискретная математика. 2002. Т. 14. № 1. С. 99-113.

Сиделъников В. М. Об экстремальных многочленах, используемых при оценках мощности кода // Проблемы передачи информации. 1980. Т. 14. Вып. 3. С. 17-30.

Сиделъников В. М. О взаимной корреляции последовательностей // Проблемы кибернетики. 1971. Т. 24. С. 15-42.

Нигматуллин Р. Г. Сложность булевых функций. М.: Наука, 1991. 240 с.

Мак-Вилъямс Ф. Дж., Слоэн Н.Дж.А. Теория кодов, исправляющих ошибки. М.: Связь, 1979. 745 с.

Молдовян А. А., Молдовян Н. А., Еремеев М. А. Криптография: от примитивов к синтезу алгоритмов. СПб.: БХВ-Петербург, 2004.

Логачев О. А., Сальников А. А., Ященко В. В. Булевы функции в теории кодирования и криптологии. М.: Московский центр непрерывного математического образования, 2004. 470 с.

Логачев О. А., Сальников А. А., Ященко В. В. Некоторые характеристики «нелинейности» групповых отображений // Дискрет. анализ и исслед. операций. Сер. 1. 2001. Т. 8. № 1. С. 40-54.

Логачев О. А., Сальников А. А., Ященко В. В. Бент-функции на конечной абелевой группе // Дискретная математика. 1997. Т. 9. № 4. С. 3-20.

Кузьмин А. С., Марков В. Т., Нечаев А. А. и др. Бент-функции и гипербент-функции над полем из 2l элементов // Проблемы передачи информации. 2008. Т. 44. Вып. 1. С. 15-37.

Кузнецов Ю. В., Шкарин С. А. Коды Рида-Маллера (обзор публикаций) // Математические вопросы кибернетики. 1996. Вып. 6. С. 5-50.

Амбросимов А. С. Свойства бент-функций q-значной логики над конечными полями // Дискретная математика. 1994. Т. 6. № 3. С. 50-60.

Васильев Ю. Л. О негрупповых плотно упакованных кодах // Проблемы кибернетики. 1962. Вып. 8. С. 337-339.

Августинович С. В. Об одном свойстве совершенных двоичных кодов // Дискрет. анализ и исслед. операций. 1995. Т. 2. № 1. С. 4-6.