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On Algebraic Immunity and Annihilators

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Information Security and Cryptology – ICISC 2006 (ICISC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 4296))

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Abstract

Algebraic immunity AI(f) defined for a boolean function f measures the resistance of the function against algebraic attacks. Currently known algorithms for computing the optimal annihilator of f and AI(f) are inefficient. This work consists of two parts. In the first part, we extend the concept of algebraic immunity. In particular, we argue that a function f may be replaced by another boolean function f c called the algebraic complement of f. This motivates us to examine AI(f c). We define the extended algebraic immunity of f as AI *(f)= min {AI(f), AI(f c)}. We prove that 0≤AI(f)–AI *(f)≤1. Since AI(f)–AI *(f)= 1 holds for a large number of cases, the difference between AI(f) and AI *(f) cannot be ignored in algebraic attacks. In the second part, we link boolean functions to hypergraphs so that we can apply known results in hypergraph theory to boolean functions. This not only allows us to find annihilators in a fast and simple way but also provides a good estimation of the upper bound on AI *(f).

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Author information

Authors and Affiliations

  1. Centre for Advanced Computing – Algorithms and Cryptography, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

    Xian-Mo Zhang & Josef Pieprzyk

  2. Department of Software & Information Systems, The University of North Carolina at Charlotte, 9201 University City Blvd, Charlotte, NC, 28223-0001, USA

    Yuliang Zheng

Authors
  1. Xian-Mo Zhang
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  2. Josef Pieprzyk
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  3. Yuliang Zheng
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Editor information

Editors and Affiliations

  1. Dankook University, San 29, Anseo-dong, Cheonan-shi, 330-714, Chungnam, Korea

    Min Surp Rhee

  2. Department of Information Security, Joongbu University, 101 Daehak-Ro, Chubu-Myeon, Geumsan-Gun,, 312-702, Chungnam, Korea

    Byoungcheon Lee

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© 2006 Springer-Verlag Berlin Heidelberg

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Zhang, XM., Pieprzyk, J., Zheng, Y. (2006). On Algebraic Immunity and Annihilators. In: Rhee, M.S., Lee, B. (eds) Information Security and Cryptology – ICISC 2006. ICISC 2006. Lecture Notes in Computer Science, vol 4296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11927587_8

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  • DOI: https://doi.org/10.1007/11927587_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49112-5

  • Online ISBN: 978-3-540-49114-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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