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English version: Journal of Applied and Industrial Mathematics, 2020, 14:1, 162–175 |
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Volume 27, No 1, 2020, P. 88-109 UDC 519.16
Keywords: Boolean polynomial, symmetric polynomial, annihilator, linear operator, cryptosystem. DOI: 10.33048/daio.2020.27.646 Vladimir K. Leontiev 1,2 Received January 24, 2019 References[1] I. A. Pankratova, Boolean Functions in Cryptography (Tomsk. Gos. Univ., Tomsk, 2014).[2] N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in Advances in Cryptology – EUROCRYPT 2003 (Proc. Int. Conf. Theory and Applications of Cryptography Techniques, Warsaw, Poland, May 4–8, 2003) (Springer, Heidelberg, 2003), pp. 345–359. [3] F. Didier, A new upper bound of the block error probability after decoding over the erasure channel, IEEE Trans. Inform. Theory. 52 (10), 4496–4503 (2006). [4] K. Feng, Q. Liao, and J. Yang, Maximal values of generalized algebraic immunity, Des. Codes Cryptogr. 50, 243–252 (2009). [5] C. Carlet and B. Merabet, Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions, Adv. Math. Commun. 7 (2), 197–217 (2013). [6] M. S. Lobanov, Exact ratios between nonlinearity and algebraic immunity, Diskretn. Anal. Issled. Oper. 15 (6), 34–47 (2008). [7] M. S. Lobanov, About a method for obtaining some lower estimates of nonlinearity of a Boolean function, Mat. Zametki 93 (5), 741–745 (2013). [8] M. S. Lobanov, An exact ratio between nonlinearity and algebraic immunity, Diskretn. Mat. 18 (3), 152–159 (2006). [9] V. K. Leont’ev, Boolean polynomials and linear transformations, Dokl. Ross. Akad. Nauk 425 (3), 320–322 (2009). [10] M. E. Tuzhilin, Algebraic immunity of Boolean functions, Prikl. Diskretn. Mat., No. 2, 18–22 (2008). [11] P. Rizomiliotis, Improving the high order nonlinearity of Boolean functions with prescribed algebraic immunity, Discrete Appl. Math. 158 (18), 2049–2055 (2010). [12] S. Mesnager, Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity, IEEE Trans. Inform. Theory 54 (8), 3656–3662 (2008). [13] S. Mesnager and G. Gohen, Fast algebraic immunity of Boolean functions, Adv. Math. Commun. 11 (2), 373–377 (2017). [14] Q. Wang and T. Johansson, On equivalence classes of Boolean functions, in Information Security and Cryptology (Rev. Sel. Pap. 13th Int. Conf., Seoul, Korea, Dec. 1–3, 2010) (Springer, Heidelberg, 2011), [15] J. Peng and H. Kan, Constructing rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E95-A (6), 1056–1064 (2012). [16] L. Sun and F.-W. Fu, Constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Theor. Comput. Sci. 738, 13–24 (2018). [17] L. Sun and F.-W. Fu, Constructions of even-variable RSBFs with Optimal algebraic immunity and high nonlinearity, J. Appl. Math. Comput. 56 (1–2), 593–610 (2018). [18] F. U. Shaojing, D. U. Jiao, Q. U. Longjiang, and L. I. Chao, Construction of odd-variable rotation symmetric Boolean functions with maximum algebraic immunity, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E99-A (4), 853–855 (2016). [19] Q. Wang, C. H. Tan, and P. Stanica, Concatenations of the hidden weighted bit function and their cryptographic properties, Adv. Math. Com?mun. 8 (2), 153–165 (2014). [20] V. K. Leont’ev, Symmetrical Boolean polynomials, Zh. Vychisl. Mat. Mat. Fiz. 50 (8), 1520–1531 (2010). [21] C. Carlet, G. Gao, and W. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Comb. Theory, Ser. A, 127, 161–175 (2014). [22] S. Su and X. Tang, Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Des. Codes Cryptogr. 71, 1567–1580 (2014). [23] V. K. Leont’ev, Combinatorics and Information, Vol. 1: Combinatorial Analysis (MFTI, Moscow, 2015). [24] V. K. Leont’ev and O. Moreno, About zeros of Boolean polynomials, Zh. Vychisl. Mat. Mat. Fiz. 38 (9), 1608–1615 (1998). [25] V. K. Leont’ev and E. N. Gordeev, On number of zeros of Boolean polynomials, Zh. Vychisl. Mat. Mat. Fiz. 68 (7), 1235–1245 (2018). [26] E. N. Gordeev, V. K. Leont’ev, and N. V. Medvedev, On properties of Boolean polynomials important for cryptosystems, Vopr. Kiberbezopasnosti, No. 3, 63–69 (2017). |
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