Advertisement

Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 356–374 | Cite as

Recent Results on Constructing Boolean Functions with (Potentially) Optimal Algebraic Immunity Based on Decompositions of Finite Fields

  • Zhuojun Liu
  • Baofeng WuEmail author
Article

Abstract

Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. (2017), and Lou, et al. (2015). Motivated by their pioneering work, the authors and their collaborators have done a series of work, obtaining some more general constructions of OAI functions based on decompositions of finite fields. In this survey article, the authors review our work in this field in the past few years, illustrating the ideas for the step-by-step generalizations of previous constructions and recalling several new observations on a combinatorial conjecture on binary strings known as the Tu-Deng conjecture. In fact, the authors have obtained some variants or more general forms of Tu-Deng conjecture, and the optimal algebraic immunity of certain classes of functions we constructed is based on these conjectures.

Keywords

Additive decomposition algebraic immunity Boolean function multiplicative decomposition Tu-Deng conjecture 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The year of 2019 will be the centenary of Professor Wen-tsün Wu’s birth. He is one outstanding Chinese mathematician and passed away on May 7th, 2017. We are writing this paper to commemorate him. As some of Wu’s disciples, we learned a lot from him in the past days, especially got many benefits from the mathematics mechanization initiated by him. Into his 90’s, Prof. Wu showed great interests in cryptography and had worked very hard on such important cryptographic problems as factorization of big integers. Inspired by his spirits, we also carried out some researches in the field of cryptography in recent years, and a part of our results form this paper. We will forever be grateful for the knowledge and spiritual heritage that Prof. Wu had passed on to us.

References

  1. [1]
    Meier M, Pasalic E, and Carlet C, Algebraic attacks and decomposition of boolean functions, Proc. Adv. Cryptol. — EUROCRYPT’04 (Eds. by Cachin C and Camenisch J), Switzerland, 2004.Google Scholar
  2. [2]
    Courtois N and Meier M, Algebraic attack on stream ciphers with linear feedback, Proc. Adv. Cryptol. — EUROCRYPT’03 (Ed. by Biham E), Warsaw, 2003.Google Scholar
  3. [3]
    Carlet C, Dalai D, Gupta K, et al., Algebraic immunity for cryptographically significant boolean fucntions, analysis and construction, IEEE Trans. Inform. Theory, 2006, 52: 3105–3121.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Dalai D, Maitra S, and Sarkar S, Basic theory in construction of boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 2006, 40: 41–58.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Li N and Qi W, Construction and analysis of boolean functions of 2t + 1 variables with maximum algebraic immunity, Proc. Adv. Cryptol. — ASIACRYPT’06 (Eds. by Lai X and Chen K), Shanghai, 2006.Google Scholar
  6. [6]
    Li N, Qu L, Qi W, et al., On the construction of Boolean functions with optimal algebraic immunity, IEEE Trans. Inform. Theory, 2008, 54: 1330–1334.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Courtois N, Fast algebraic attacks on stream ciphers with linear feedback, Proc. Adv. Cryptol. — CRYPTO’03 (Ed. by Boneh D), California, 2003.Google Scholar
  8. [8]
    Carlet C and Feng K, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, Proc. Adv. Cryptol. — ASIACRYPT’08 (Ed. by Pieprzyk J), Melbourne, 2008.Google Scholar
  9. [9]
    Liu M, Zhang Y, and Lin D, Perfect algebraic immune functions, Proc. Adv. Cryptol. — ASIACRYPT’ 12 (Eds. Wang X and Sako K), Beijing, 2012.Google Scholar
  10. [10]
    Tu Z and Deng Y, A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity, Des. Codes Cryptogr., 2011, 60: 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Tang D, Carlet C and Tang X, Highly nonlinear Boolean functions with optimum algebraic immunity and good behavior against fast algebraic attacks, IEEE Trans. Inform. Theory, 2013, 59: 653–664.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Cohen G and Flori J P, On a generalized combinatorial conjecture involving addition mod 2k-1, Cryptology ePrint Archive, Report 2011/400, 2011, http://eprint.iacr.org/.Google Scholar
  13. [13]
    Han H and Tang C, New classes of even-variable Boolean functions with optimal algebraic immunity and very high nonlinearity, Int. J. Adv. Comput. Techn., 2013, 5(2): 419–428.MathSciNetGoogle Scholar
  14. [14]
    Lou Y, Han H, Tang C, et al., Constructing vectorial Boolean functions with high algebraic immunity based on group decomposition, Int. J. Comput. Math., 2015, 92(3): 451–462.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tu Z and Deng Y, Boolean functions optimizing most of the cryptographic criteria, Discrete Appl. Math., 2012, 160: 427–435.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Jin Q, Liu Z, Wu B, et al., A combinatorial condition and Boolean functions with optimal algebraic immunity, Journal of Systems Science & Complexity, 2015, 28(3): 725–742.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Wang T, Liu M, and Lin D, Construction of resilient and nonlinear boolean functions with almost perfect immunity to algebraic and fast algebraic attacks, Inscrypt 2012 (Eds. by Kutylowski M and Yung M), Beijing, 2012.Google Scholar
  18. [18]
    Zheng J, Wu B, Chen Y, et al., Constructing 2m-variable Boolean functions with optimal algebraic immunity based on polar decomposition of \(\mathbb{F}_{{2^{2m}}}^*\), Int. J. Found. Comput. Sci., 2014, 25(5): 537–551.CrossRefzbMATHGoogle Scholar
  19. [19]
    Khan M and Özbudak F, Hybrid classes of balanced Boolean functions with good cryptographic properties, Inform. Sci., 2014, 273: 319–328.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Wu B, Jin Q, and Liu Z, Constructing Boolean functions with potential optimal algebraic immunity based on additive decompositions of finite fields (extended abstract), Proceeding of 2014 IEEE International Symposium on Information Theory (Eds. by Høst-Madsen A, Kavcic A, and Veeravalli V), Honolulu, 2014.Google Scholar
  21. [21]
    Wang Q and Tan C, Properties of a Family of Cryptographic Boolean Functions, SETA 2014 (Eds. by Schmidt K U and Winterhof A), Melbourne, 2014.Google Scholar
  22. [22]
    Wu B, Zheng J, and Lin D, Constructing Boolean functions with (potentially) optimal alge braic immunity based on multiplicative decompositions of finite fields, Proceeding of 2015 IEEE International Symposium on Information Theory (Eds. by Tse D and Yeung R), Hong Kong, 2015.Google Scholar
  23. [23]
    Wang Z, Zhang X, Wang S, et al., Construction of Boolean functions with excellent cryptographic criteria using bivariate polynomial representation, International Journal of Computer Mathematics, 2016, 93(3): 425–444.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Liu M and Lin D, Results on highly nonlinear Boolean functions with provably good immunity to fast algebraic attacks, Inf. Sci., 2017, 421: 181–203.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Tang D, Carlet C, Tang X, et al., Construction of highly nonlinear 1-resilient Boolean functions with optimal algebraic immunity and provably high fast algebraic immunity, IEEE Trans. Inform. Theory, 2017, 63: 6113–6125.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Carlet C, Boolean functions for cryptography and error correcting codes, Monography Boolean Methods and Models (Eds. by Crama Y and Hammer P), Cambridge University Press, London, 2010.Google Scholar
  27. [27]
    Carlet C, On a weakness of the Tu-Deng function and its repair, Cryptology ePrint Archive, report 2009/606, 2009, http://eprint.iacr.org/.Google Scholar
  28. [28]
    Flori J P, Randriam H, Cohen G, et al., On a Conjecture about Binary Strings Distribution, Sequences and Their Applications — SETA 2010 (Eds. by Carlet C and Pott A), Paris, 2010.Google Scholar
  29. [29]
    Cusick T, Li Y, and Stănică P, On a combinatorial conjecture, Integers, 2011, 11(2): 185–203.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Cheng K, Hong S, and Zhong Y, A note on the Tu-Deng conjecture, Journal of Systems Science and Complexity, 2015, 28(3): 702–724.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Qarboua S, Schrek J, and Fontaine C, New results about Tu-Deng’s conjecture, 2016 IEEE International Symposium on Information Theory (ISIT) (Eds. by Fàbregas A, Martinez A, and Verdú S), Barcelona, 2016.Google Scholar
  32. [32]
    Spiegelhofer L and Wallner M, The Tu-Deng conjecture holds almost surely, arXiv: 1707.07945v2 [math.CO], 2017, https://arxiv.org/pdf/1707.07945.pdf.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina

Personalised recommendations