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Computación y Sistemas

Print version ISSN 1405-5546

Comp. y Sist. vol.12 n.3 México Jan./Mar. 2009

 

Artículos

 

Algebraic Immunity of Boolean Functions – Analysis and Construction*

 

Inmunidad Algebraica de Funciones Booleanas –Análisis y Construcción

 

Deepak Kumar Dalai1 and Subhamoy Maitra2

 

1 Department of Mathematics National Institute of Science Education and Research, Sachivalay Marg, Bhubaneswar 751005 INDIA. E–mail: deepak@iopb.res.in

2 Applied Statistics Unit, Indian Statistical Institute 203 B T Road, Calcutta 700108, INDIA. E–mail: subho@isical.ac.in

 

Article received on March 1, 2008
Accepted on June 14, 2008

 

Abstract

In this paper, we first analyse the method of finding algebraic immunity of a Boolean function. Given a Boolean function f on n–variables, we identify a reduced set of homogeneous linear equations by solving which one can decide whether there exist annihilators of f at a specific degree. Moreover, we analyse how an affine transformation on the input variables of f can be exploited to achieve further reduction in the set of homogeneous linear equations. Next, from the design point of view, we construct balanced Boolean functions with maximum possible AI with an additional property which is necessary to resist the fast algebraic attack.

Keywords: Algebraic Attacks, Algebraic Normal Form, Annihilators, Boolean Functions, Fast Algebraic Attacks, Homogeneous Linear Equations.

 

Resumen

En este artículo, analizamos primero el método que permite encontrar la inmunidad algebraica de una función Booleana. Dada una función Booleana f de n variables, identificamos un conjunto reducido de ecuaciones lineales homogéneas resolviendo cuál de ellas puede ser usada para determinar si existen nulificadores de f de un grado específico. Además analizamos cómo una transformación afín de las variables de entrada de f puede ser aplicada para alcanzar una mayor reducción en el conjunto de ecuaciones lineales homogéneas. En seguida, y analizando desde el punto de vista de diseño, construimos funciones Booleanas balanceadas con inmunidad algebraica máxima y una propiedad adicional necesaria para resistir versiones rápidas de ataques algebraicos.

Palabras Claves: Ataques algebraicos, froma normal algebraica, nulificadores, funciones Booleanas, ataques algebrados rápidos, ecuaciones lineales homogéneas.

 

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References

1. Armknecht, F. (2004). Improving fast algebraic attacks. In Fast Software Encryptions 2004, Proceedings, Volume 3017 of Lecture Notes in Computer Science, pp. 65–82. Springer.        [ Links ]

2. Armknecht, F., C. Carlet, P. Gaborit, S. Kuenzli, W. Meier, and O. Ruatta (2006). Efficient computation of algebraic immunity for algebraic and fast algebraic attacks. In Advances in Cryptology – EUROCRYPT 2006, Proceedings, Volume 4004 of Lecture Notes in Computer Science, pp. 147–164. Springer.        [ Links ]

3. Armknecht, F. and M. Krause (2003). Algebraic attacks on combiners with memory. In Advances in Cryptology – CRYPTO 2003, Proceedings, Volume 2729 of Lecture Notes in Computer Science, pp. 162–175. Springer.        [ Links ]

4. Armknecht, F. and M. Krause (2006). Constructing single– and multi–output boolean functions with maximal immunity. In 33rd International Colloquium on Automata, Languages and Programming(ICALP) 2006, Proceedings, Volume 4052 of Lecture Notes in Computer Science, pp. 180–191. Springer.        [ Links ]

5. Ars, G. and J. Faugére (2005). Algebraic immunities of functions over finite fields. INRIA Techno report.        [ Links ]

6. Batten, L. M. (2004). Algebraic attacks over gf(g). In Progress in Cryptology – INDOCRYPT 2004, Proceedings, Volume 3348 of Lecture Notes in Computer Science, pp. 84–91. Springer.        [ Links ]

7. Braeken, A., J. Lano, N. Mentens, B. Praneel, and I. Verbauwhede (2005). Sfinks: A synchronous stream cipher for restricted hardware environments. In SKEW – Symmetric Key Encryption Workshop, 2005, Proceedings.        [ Links ]

8. Braeken, A., J. Lano, and B. Praneel (2005). Evaluating the resistance of filters and combiners against fast algebraic attacks. Eprint on ECRYPT http://eprint.iacr.org/.        [ Links ]

9. Braeken, A., J. Lano, and B. Praneel (2006). Evaluating the resistance of stream ciphers with linear feedback against fast algebraic attacks. In 11th Australasian Conference on Information Security and Privacy(ACISP) 2006, Proceedings, Volume 4058 of Lecture Notes in Computer Science, pp. 40–51. Springer.        [ Links ]

10. Braeken, A. and B. Praneel (2005). On the algebraic immunity of symmetric boolean functions. In Progress in Cryptology – INDOCRYPT 2005, Proceedings, Volume 3797 of Lecture Notes in Computer Science, pp. 35–48. Springer. Also available at Cryptology ePrint Archive, http://eprint.iacr.org/, No. 2005/245,26 July, 2005.        [ Links ]

11. Canteaut, A. (2005). Open problems related to algebraic attacks on stream ciphers. In International Workshop on Coding and Cryptography (WCC) 2005. Proceedings, pp. 1–10.        [ Links ]

12. Carlet, C, D. K. Dalai, K. C. Gupta, and S. Maitra (2006). Algebraic immunity for cryptographically significant boolean functions: Analysis and construction. IEEE Transactions on Information Theory 52(7), 3105–3121.        [ Links ]

13. Cheon, J. H. and D. H. Lee (2004). Resistance of s–boxes against algebraic attacks. In Fast Software Encryptions 2004, Proceedings, Volume 3017 of Lecture Notes in Computer Science, pp. 83–94. Springer.        [ Links ]

14. Cho, J. Y. and J. Pieprizyk (2004). Algebraic attacks on sober–t32 and sober–128. In Fast Software Encryptions 2004, Proceedings, Volume 3017 of Lecture Notes in Computer Science, pp. 49–64. Springer.        [ Links ]

15. Comtet, L. (1974). Advanced combinatorics. Reidel Publication.        [ Links ]

16. Coppersmith, D. and S. Winograd (1990). Matrix multiplication via arithmatic programming. Journal of Symbolic Computation 9(3), 251–280.        [ Links ]

17. Courtois, N. (2003). Fast algebraic attacks on stream ciphers with linear feedback. In Advances in Cryptology –CRYPTO 2003, Proceedings, Volume 2729 of Lecture Notes in Computer Science, pp. 176–194. Springer.        [ Links ]

18. Courtois, N. (2005). Cryptanalysis of sfinks. In 8th International Conference on Information Security and Cryptology (ICISC) 2005, Proceedings, Volume 3935 of Lecture Notes in Computer Science, pp. 261–269. Springer.Also available at Cryptology ePrint Archive, http://eprint.iacr.org/, Report 2005/243,2005.        [ Links ]

19. Courtois, N., B. Debraize, and E. Garrido (2005). On exact algebraic (non–)immunity of s–boxes based on power functions. In 11th Australasian Conference on Information Security and Privacy (ACISP) 2006, Proceedings, Volume 4058 of Lecture Notes in Computer Science, pp. 76–86. Springer.        [ Links ]

20. Courtois, N. and W. Meier (2003). Algebraic attacks on stream ciphers with linear feedback. In Advances in Cryptology – EUROCRYPT 2003, Proceedings, Volume 2656 of Lecture Notes in Computer Science, pp. 345–359. Springer.        [ Links ]

21. Courtois, N. and J. Pieprzyk (2002). Cryptanalysis of block ciphers with overdefined systems of equations. In Advances in Cryptology – ASIACRYPT 2002, Proceedings, Volume 2501 of Lecture Notes in Computer Science, pp. 267–287. Springer.        [ Links ]

22. Dalai, D. K., K. C. Gupta, and S. Maitra (2004). Results on algebraic immunity for cryptographically significant boolean functions. In Progress in Cryptology – INDOCRYPT 2004, Proceedings, Volume 3348 of Lecture Notes in Computer Science, pp. 92–106. Springer.        [ Links ]

23. Dalai, D. K., K. C. Gupta, and S. Maitra (2005). Cryptographically significant boolean functions: Construction and analysis in terms of algebraic immunity. In Fast Software Encryptions 2005, Proceedings, Volume 3557 of Lecture Notes in Computer Science, pp. 98–111. Springer.        [ Links ]

24. Dalai, D. K., K. C. Gupta, and S. Maitra (2006). Notion of algebraic immunity and its evaluation related to fast algebraic attacks. In Second Workshop on Boolean Functions: Cryptography and Applications (BFCA 2006). Proceedings. .Also available at Cryptology ePrint Archive, http://eprint.iacr.org/, No. 2006/018.        [ Links ]

25. Dalai, D. K., S. Maitra, and S. Sarkar (2006). Basic theory in construction of boolean functions with maximum possible annihilator immunity. Design, Codes and Cryptography 40(1), 41–58.        [ Links ]

26. Didier, F. (2006). Using wiedemann's algorithm to compute the immunity against algebraic and fast algebraic attacks. In Progress in Cryptology – INDOCRYPT 2006, Proceedings, Volume 4329 of Lecture Notes in Computer Science, pp. 236–250. Springer.        [ Links ]

27. Didier, F. and J. Tillich (2006). Computing the algebraic immunity efficiently. In Fast Software Encryptions 2006, Proceedings, Volume 4047 of Lecture Notes in Computer Science, pp. 359–374. Springer.        [ Links ]

28. Lee, D. H., J. Kim, J. Hong, J. W. Han, and D. Moon (2004). Algebraic attacks on summation generators. In Fast Software Encryptions 2004, Proceedings, Volume 3017 of Lecture Notes in Computer Science, pp. 34–48. Springer.        [ Links ]

29. Lobanov, M. (2005). Tight bound between nonlinearity and algebraic immunity. Cryptology ePrint Archive, Report 2005/441. http://eprint.iacr.org/.        [ Links ]

30. Meier, W., E. Pasalic, and C. Carlet (2004). Algebraic attacks and decomposition of boolean functions. In Advances in Cryptology – EUROCRYPT 2004, Proceedings, Volume 3027 of Lecture Notes in Computer Science, pp. 474–491. Springer.        [ Links ]

31. Nawaz, Y., G. Gong, and K. C. Gupta (2006). Upper bounds on algebraic immunity of power functions. In Fast Software Encryptions 2006, Proceedings, Volume 4047 of Lecture Notes in Computer Science, pp. 375–389. Springer.        [ Links ]

32. Strassen, V. (1969). Guassian elimination is not optimal. Numerische Mathematik 13, 354–356.        [ Links ]

 

Note

* This is a substantially revised and merged version of two conference papers, (i) "Reducing the Number of Homogeneous Linear Equations in Finding Annihilators", in Sequences and Their Applications, SETA '06, pages 376–390, volume 4086, Lecture Notes in Computer Science, Springer Verlag, 2006. Section 3.1 and Appendix A are added over the conference version, (ii) "Balanced Boolean Functions with (more than) Maximum Algebraic Immunity", in International Workshop on Coding and Cryptography, WCC '07, pages 99–108, INRIA, Rocquencourt, France in April 16–20, 2007. The proceedings of WCC '07 is only a workshop record and it is not printed by any publisher.

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