Construction of Vectorial Boolean Function Based on T-D Conjecture
Abstract: An improvement has been made on the construction method of Boolean Functions and the relevant conclusions of combinatorial conjecture proposed by Ziran Tu. We generalized their results and extended to the vectorial case. A class of bent Boolean functions F with the maximum algebraic immunity is presented by a more general construction method. Then by modifying F, we get new vectorial balanced functions with optimum algebraic degree, good nonlinearity and good algebraic immunity even maximum algebraic immunity for some cases.
文章引用: 陈怡然 , 周 梦 (2014) T-D猜想上多输出布尔函数构造。 应用数学进展， 3， 62-69. doi: 10.12677/AAM.2014.32010
 Armknecht, F. (2004) Improving fast algebraic attacks: FSE 2004. Springer Verlag, 65-82.
 Batten, L.M. (2004) Algebraic attacks over GF(q): Cryptology-INDOCRYPT 2004. Springer Verlag, 84-91.
 Courtois, N. and Meier, W. (2003) Algebraic attacks on stream ciphers with linear feedback: Cryptology-EURO- CRYPT 2003. Springer Verlag, 345-359.
 Courtois, N. (2003) Fast algebraic attacks on stream ciphers with linear feedback: Advances in Cryptology-CRYPTO 2003. Springer Verlag, 176-194.
 Meier, W., Pasalic, E. and Carlet, C. (2004) Algebraic attacks and decomposition of Boolean functions: Cryptology- EUROCRYPT 2004. Springer Verlag, 474-491.
 Rothaus, O.S. (1976) On bent functions. Journal of Combinatorial Theory A, 20,300-305.
 Tu, Z. and Deng, Y. (2010) A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity. Designs, Codes and Cryptography, 1-14.
 Tang, D., Carlet, C. and Tang, X. Highly nonlinear Boolean functions with optimum algebraic immunity and good be- havior against fast algebraic attacks. Cryptology ePrint Archive. http://eprint.iacr.org/2011/366.pdf
 Cohen, G. and Flori, J.P. On a generalized combinatorial conjecture involving addition mod 2k-1. Cryptology ePrint Archive. http://eprint.iacr.org/2011/400.pdf
 Jin, Q., Liu, Z., Wu, B. and Zhang, X. A general conjecture similar to T-D conjecture and its applications in constructing Boolean functions with optimal algebraic immunity. Cryptology ePrint Archive. http://eprint.iacr.org/2011/515.pdf
 Feng, K., Liao, Q. and Yang, J. (2009) Maximal values of generalized algebraic immunity. Designs, Codes and Cryp- tography, 50, 243-252.
 MacWilliams, F.J. and Sloane, N.J.A. (1977) The Theory of Error-Correcting Codes. North-Holland, Amsterdam.
 Dillon, J.F. (1974) Elementary hadamard difference sets. Ph.D. Thesis, University of Maryland, College Park.
 Feng,K.,Yang,J.(2011)Vectorial Boolean Functions With Good Cryptographic Properties.International Journal of Foundations of Computer Science,Vol. 22, No. 6 , 1271–1282.