Lower Bounds on the Algebraic Immunity of Boolean Functions

From the motivation of algebraic attacks to stream and block ciphers([1,2,7,13,14,15]), the concept of {\em algebraic immunity} (AI) was introduced in [21] and studied in [3,5,10,11,17,18,19,20,21]. High algebraic immunity is a necessary condition for resisting algebraic attacks. In this paper, we give some lower bounds on algebraic immunity of Boolean functions. The results are applied to give lower bounds on AI of symmetric Boolean functions and rotation symmetric Boolean functions. Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value $\lceil \frac{n}{2}\rceil$ are constructed.

[1]  Thomas W. CusickPantelimon Stùanicùa.  Fast Evaluation, Weights and Nonlinearity of Rotation-Symmetric Functions , 2000 .

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

[3]  Frederik Armknecht,et al.  Improving Fast Algebraic Attacks , 2004, FSE.

[4]  Frederik Armknecht,et al.  Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks , 2006, EUROCRYPT.

[5]  Josef Pieprzyk,et al.  Fast Hashing and Rotation-Symmetric Functions , 1999 .

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

[7]  Josef Pieprzyk,et al.  Rotation-Symmetric Functions and Fast Hashing , 1998, J. Univers. Comput. Sci..

[8]  Bart Preneel,et al.  Evaluating the Resistance of Stream Ciphers with Linear Feedback Against Fast Algebraic Attacks , 2006, ACISP.

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

[10]  Claude Carlet,et al.  On the Higher Order Nonlinearities of Algebraic Immune Functions , 2006, CRYPTO.

[11]  Pantelimon Stanica,et al.  Rotation symmetric Boolean functions - Count and cryptographic properties , 2003, Discret. Appl. Math..

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

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

[14]  Subhamoy Maitra,et al.  Results on Algebraic Immunity for Cryptographically Significant Boolean Functions , 2004, INDOCRYPT.

[15]  Guang Gong.  On Existence and Invariant of Algebraic Attacks , 2004 .

[16]  Pantelimon Stanica,et al.  A constructive count of rotation symmetric functions , 2003, Inf. Process. Lett..

[17]  Josef Pieprzyk,et al.  Cryptanalysis of Block Ciphers with Overdefined Systems of Equations , 2002, ASIACRYPT.

[18]  Frederik Armknecht.  On the Existence of low-degree Equations for Algebraic Attacks , 2004, IACR Cryptol. ePrint Arch..

[19]  Jovan Dj. Golic.  Vectorial Boolean functions and induced algebraic equations , 2006, IEEE Transactions on Information Theory.

[20]  Anne Canteaut,et al.  Symmetric Boolean functions , 2005, IEEE Transactions on Information Theory.

[21]  Willi Meier,et al.  Fast Algebraic Attacks on Stream Ciphers with Linear Feedback , 2003, CRYPTO.

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