# American Institute of Mathematical Sciences

May  2013, 7(2): 197-217. doi: 10.3934/amc.2013.7.197

## Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions

 1 LAGA, Universities of Paris 8 and Paris 13, CNRS, France 2 Department of Algebraic and number theory, University of Sciences and Technology, Houari Boumedienne, Algiers, Algeria, and University of Kasdi Merbah, Ouargla, Algeria

Received  January 2013 Published  May 2013

This paper extends the work of F. Didier (IEEE Transactions on Information Theory, Vol. 52(10): 4496-4503, October 2006) on the algebraic immunity of random balanced Boolean functions, into an asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions.
Citation: Claude Carlet, Brahim Merabet. Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 197-217. doi: 10.3934/amc.2013.7.197
##### References:
 [1] F. Armknecht, C. Carlet, P. Gaborit, S. Kunzli, W. Meier and O. Ruatta, Efficient computation of algebraic immunity for algebraic and fast algebraic attacks, in "Advances in Cryptology-EUROCRYPT 2006,'' Springer-Verlag, (2006), 147-164. [2] C. Carlet, A method of construction of balanced functions with optimum algebraic immunity, in "Proceedings of the International Workshop on Coding and Cryptography,'' World Scientific Publishing Co., 2008. [3] C. Carlet, Boolean functions for cryptography and error correcting codes, in "Boolean Models and Methods in Mathematics, Computer Science, and Engineering'' (eds. Y. Crama and P. Hammer), Cambridge Univ. Press, (2010), 257-397. [4] C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in "Advances in Cryptology-ASIACRYPT 2008,'' Springer-Verlag, (2008), 425-440. [5] N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in "Proceedings of Eurocrypt 2003,'' Springer, (2003), 345-359; An extended version is available at http://www.cryptosystem.net/stream/ [6] D. K. Dalai, K. C. Gupta and S. Maitra, Cryptographically significant Boolean functions: Construction and analysis in terms of algebraic immunity, in "Fast Software Encryption 2005,'' Springer, (2005), 98-111. doi: 10.1007/11502760_7. [7] F. Didier, A new bound on the block error probability after decoding over the erasure channel, IEEE Trans. Inform. Theory, 52 (2006), 4496-4503. doi: 10.1109/TIT.2006.881719. [8] K. Feng, Q. Liao and J. Yang, Maximal values of generalized algebraic immunity, Des. Codes Crypt., 50 (2009), 243-252. doi: 10.1007/s10623-008-9228-0. [9] R. G. Gallager, "Information Theory and Reliable Communication,'' John Wiley and Sons Inc., New York, 1968. [10] M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions, in "Advances in Cryptology,'' Springer, (2012), 172-189. [11] F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'' Amsterdam, North Holland, 1977. [12] W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in "Advances in Cryptology EUROCRYPT 2004,'' Springer-Verlag, (2004), 474-491. doi: 10.1007/978-3-540-24676-3_28. [13] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.

show all references

##### References:
 [1] F. Armknecht, C. Carlet, P. Gaborit, S. Kunzli, W. Meier and O. Ruatta, Efficient computation of algebraic immunity for algebraic and fast algebraic attacks, in "Advances in Cryptology-EUROCRYPT 2006,'' Springer-Verlag, (2006), 147-164. [2] C. Carlet, A method of construction of balanced functions with optimum algebraic immunity, in "Proceedings of the International Workshop on Coding and Cryptography,'' World Scientific Publishing Co., 2008. [3] C. Carlet, Boolean functions for cryptography and error correcting codes, in "Boolean Models and Methods in Mathematics, Computer Science, and Engineering'' (eds. Y. Crama and P. Hammer), Cambridge Univ. Press, (2010), 257-397. [4] C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in "Advances in Cryptology-ASIACRYPT 2008,'' Springer-Verlag, (2008), 425-440. [5] N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in "Proceedings of Eurocrypt 2003,'' Springer, (2003), 345-359; An extended version is available at http://www.cryptosystem.net/stream/ [6] D. K. Dalai, K. C. Gupta and S. Maitra, Cryptographically significant Boolean functions: Construction and analysis in terms of algebraic immunity, in "Fast Software Encryption 2005,'' Springer, (2005), 98-111. doi: 10.1007/11502760_7. [7] F. Didier, A new bound on the block error probability after decoding over the erasure channel, IEEE Trans. Inform. Theory, 52 (2006), 4496-4503. doi: 10.1109/TIT.2006.881719. [8] K. Feng, Q. Liao and J. Yang, Maximal values of generalized algebraic immunity, Des. Codes Crypt., 50 (2009), 243-252. doi: 10.1007/s10623-008-9228-0. [9] R. G. Gallager, "Information Theory and Reliable Communication,'' John Wiley and Sons Inc., New York, 1968. [10] M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions, in "Advances in Cryptology,'' Springer, (2012), 172-189. [11] F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'' Amsterdam, North Holland, 1977. [12] W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in "Advances in Cryptology EUROCRYPT 2004,'' Springer-Verlag, (2004), 474-491. doi: 10.1007/978-3-540-24676-3_28. [13] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.
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