A characterization of balanced Boolean functions with optimal algebraic immunity

Abstract In this paper, we characterize balanced Boolean functions with optimal algebraic immunity by Schur functions. By applying this characterization, three classes of balanced Boolean functions with optimal algebraic immunity are constructed. Some examples of the constructed functions with other desired cryptographic properties are also presented.

[1]  Chik How Tan,et al.  Properties of a Family of Cryptographic Boolean Functions , 2014, SETA.

[2]  Claude Carlet.  A method of construction of balanced functions with optimum algebraic immunity , 2006, IACR Cryptol. ePrint Arch..

[3]  Jean-Pierre Tillich,et al.  Computing the Algebraic Immunity Efficiently , 2006, FSE.

[4]  Haibin Kan,et al.  Constructions of Cryptographically Significant Boolean Functions Using Primitive Polynomials , 2010, IEEE Transactions on Information Theory.

[5]  Bin Zhang,et al.  Revised Algorithms for Computing Algebraic Immunity against Algebraic and Fast Algebraic Attacks , 2014, ISC.

[6]  Chik How Tan,et al.  A Family of Cryptographically Significant Boolean Functions Based on the Hidden Weighted Bit Function , 2013, ICISC.

[7]  Subhamoy Maitra,et al.  Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity , 2006, Des. Codes Cryptogr..

[8]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[9]  Chao Li,et al.  Generalized Construction of Boolean Function with Maximum Algebraic Immunity Using Univariate Polynomial Representation , 2013, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[10]  Bart Preneel,et al.  On the Algebraic Immunity of Symmetric Boolean Functions , 2005, INDOCRYPT.

[11]  Lei Hu,et al.  Further properties of several classes of Boolean functions with optimum algebraic immunity , 2009, Des. Codes Cryptogr..

[12]  Jing Yang,et al.  Maximal values of generalized algebraic immunity , 2009, Des. Codes Cryptogr..

[13]  Yongzhuang Wei,et al.  Efficient probabilistic algorithm for estimating the algebraic properties of Boolean functions for large n , 2016, Inf. Sci..

[14]  Claude Carlet,et al.  Algebraic Attacks and Decomposition of Boolean Functions , 2004, EUROCRYPT.

[15]  Lei Hu,et al.  Two constructions of balanced Boolean functions with optimal algebraic immunity, high nonlinearity and good behavior against fast algebraic attacks , 2015, Des. Codes Cryptogr..

[16]  Subhamoy Maitra,et al.  Cryptographically Significant Boolean Functions: Construction and Analysis in Terms of Algebraic Immunity , 2005, FSE.

[17]  Xiaohu Tang,et al.  Highly Nonlinear Boolean Functions With Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks , 2013, IEEE Transactions on Information Theory.

[18]  Yingpu Deng,et al.  A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity , 2011, Des. Codes Cryptogr..

[19]  Nicolas Courtois.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

[20]  Yuan Li,et al.  A Note on "On the Construction of Boolean Functions with Optimal Algebraic Immunity" , 2008, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[21]  Dongdai Lin,et al.  Perfect Algebraic Immune Functions , 2012, ASIACRYPT.

[22]  Wen-Feng Qi,et al.  Construction and Analysis of Boolean Functions of 2t+1 Variables with Maximum Algebraic Immunity , 2006, ASIACRYPT.

[23]  Claude Carlet,et al.  An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity , 2008, ASIACRYPT.

[24]  Panagiotis Rizomiliotis,et al.  On the Resistance of Boolean Functions Against Algebraic Attacks Using Univariate Polynomial Representation , 2010, IEEE Transactions on Information Theory.

[25]  Claude Carlet,et al.  Algebraic immunity for cryptographically significant Boolean functions: analysis and construction , 2006, IEEE Transactions on Information Theory.

[26]  Lei Hu,et al.  More Balanced Boolean Functions With Optimal Algebraic Immunity and Good Nonlinearity and Resistance to Fast Algebraic Attacks , 2011, IEEE Transactions on Information Theory.

[27]  Anne Canteaut,et al.  Open Problems Related to Algebraic Attacks on Stream Ciphers , 2005, WCC.