# 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, (2006), 147. Google Scholar [2] C. Carlet, A method of construction of balanced functions with optimum algebraic immunity,, in, (2008). Google Scholar [3] C. Carlet, Boolean functions for cryptography and error correcting codes,, in, (2010), 257. 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, (2008), 425. Google Scholar [5] N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback,, in, (2003), 345. 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, (2005), 98. 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. 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. doi: 10.1007/s10623-008-9228-0. Google Scholar [9] R. G. Gallager, "Information Theory and Reliable Communication,'', John Wiley and Sons Inc., (1968). Google Scholar [10] M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions,, in, (2012), 172. Google Scholar [11] F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'', Amsterdam, (1977). Google Scholar [12] W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions,, in, (2004), 474. 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. 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, (2006), 147. Google Scholar [2] C. Carlet, A method of construction of balanced functions with optimum algebraic immunity,, in, (2008). Google Scholar [3] C. Carlet, Boolean functions for cryptography and error correcting codes,, in, (2010), 257. 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, (2008), 425. Google Scholar [5] N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback,, in, (2003), 345. 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, (2005), 98. 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. 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. doi: 10.1007/s10623-008-9228-0. Google Scholar [9] R. G. Gallager, "Information Theory and Reliable Communication,'', John Wiley and Sons Inc., (1968). Google Scholar [10] M. Liu, Y. Zhang and D. Lin, Perfect algebraic immune functions,, in, (2012), 172. Google Scholar [11] F. J. Macwilliams And N. J. Sloane, "The theory of Error-Correcting Codes,'', Amsterdam, (1977). Google Scholar [12] W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions,, in, (2004), 474. 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. doi: 10.1109/18.133259. Google Scholar
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