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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

Claude Carlet 1, and  Brahim Merabet 2,

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.
Keywords: Reed-Muller codes, algebraic immunity., Boolean functions, erasure channel, vectorial functions, generalized Hamming distances.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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/  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. G. Gallager, "Information Theory and Reliable Communication,'' John Wiley and Sons Inc., New York, 1968. Google Scholar

[10]

M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions, in "Advances in Cryptology,'' Springer, (2012), 172-189. Google Scholar

[11]

F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'' Amsterdam, North Holland, 1977. Google Scholar

[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.  Google Scholar

[13]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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/  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. G. Gallager, "Information Theory and Reliable Communication,'' John Wiley and Sons Inc., New York, 1968. Google Scholar

[10]

M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions, in "Advances in Cryptology,'' Springer, (2012), 172-189. Google Scholar

[11]

F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'' Amsterdam, North Holland, 1977. Google Scholar

[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.  Google Scholar

[13]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.  Google Scholar

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