Abstract
In recent years, several classes of Boolean functions with good cryptographic properties have been constructed by using univariate (or bivariate) polynomial representation of Boolean functions over finite fields. The estimation of an incomplete additive character sum plays an important role in analyzing the nonlinearity of these functions. In this paper, we consider replacing this character sum with another incomplete additive character sum, whose estimation was firstly given by A.Winterhof in 1999. Based on Winterhof’s estimation, we try to modify two of these functions and obtain better nonlinearity bound of them.
This work is supported by Funds of Key Lab of Fujian Province University Network Security and Cryptology (2011008) and National Natural Science Foundations of China (Grant No. 61070168, Grant No. 10971246).
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Carlet, C., Feng, K.: An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)
Wang, Q., Peng, J., Kan, H., Xue, X.: Constructions of cryptographically significant Boolean functions using primitive polynomials. IEEE Trans. Inform. Theory 56(6), 3048–3053 (2010)
Tu, Z., Deng, Y.: A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity. Designs, Codes and Cryptography 60(1), 1–14 (2011)
Tu, Z., Deng, Y.: A Conjecture on Binary String and Its Applications on Constructing Boolean Functions of Optimal Algebraic Immunity. Cryptology ePrint Archive, http://eprint.iacr.org/2009/272.pdf
Tu, Z., Deng, Y.: Boolean functions with all main cryptographic properties. Cryptology ePrint Archive, http://eprint.iacr.org/2010/518.pdf
Tang, X., Tang, D., Zeng, X., Hu, L.: Balanced Boolean functions with (almost) optimal algebraic immunity and very high nonlinearity. Cryptology ePrint Archive, http://eprint.iacr.org/2010/443
Rizomiliotis, P.: On the Resistance of Boolean Functions Against Algebraic Attacks Using Univariate Polynomial Representation. IEEE Trans. Inform. Theory 56(8), 4014–4024 (2010)
Zeng, X., Carlet, C., Shan, J., Hu, L.: Balanced Boolean Functions with Optimum Algebraic Immunity and High Nonlinearity. Cryptology ePrint Archive, http://eprint.iacr.org/2010/606
Dillon, J.F.: Elementary Hadamard Difference Sets. PhD thesis, University of Maryland (1974)
Winterhof, A.: Incomplete Additive Character Sums and Applications. In: Jungnickel, D., Niederreiter, H. (eds.) The Fifth International Conference on Finite Fields and Applications Fq5 1999, pp. 462–474. Springer, Berlin (2001)
Courtois, N., Meier, W.: Algebraic Attacks on Stream Ciphers with Linear Feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)
Meier, W., Pasalic, E., Carlet, C.: Algebraic Attacks and Decomposition of Boolean Functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)
Dobbertin, H.: Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 61–74. Springer, Heidelberg (1995)
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Du, Y., Zhang, F. (2011). Two Applications of an Incomplete Additive Character Sum to Estimating Nonlinearity of Boolean Functions. In: Qing, S., Susilo, W., Wang, G., Liu, D. (eds) Information and Communications Security. ICICS 2011. Lecture Notes in Computer Science, vol 7043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25243-3_16
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DOI: https://doi.org/10.1007/978-3-642-25243-3_16
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