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On the Algebraic Normal Form and Walsh Spectrum of Symmetric Functions over Finite Rings

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Arithmetic of Finite Fields (WAIFI 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7369))

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Abstract

A function over finite rings is a function from a ring \(E_{q}^{n}\) to a ring E r , where E k is ℤ /k ℤ. These functions are well used in cryptography: cipher design, hash function design and in theoretical computer science. In this paper, we are especially interested in symmetric functions. We give practical ways of computing their ANF and their Walsh Spectrum in \(\mathcal{O}\left({ n+q-1 \choose q-1 }^2\right)\) using linear algebra. Thus, we achieve a better complexity both in time and memory than the fast Fourier transform which is in \(\mathcal{O}\left( q^nn\log(q) \right)\).

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Authors and Affiliations

  1. CASSIDIAN, Cyber Security, France

    Boris Batteux

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  1. Boris Batteux
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Editor information

Editors and Affiliations

  1. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

  2. Computer Science Section, CINVESTAV, Mexico DF, Mexico

    Francisco Rodríguez-Henríquez

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Batteux, B. (2012). On the Algebraic Normal Form and Walsh Spectrum of Symmetric Functions over Finite Rings. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_7

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  • DOI: https://doi.org/10.1007/978-3-642-31662-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31661-6

  • Online ISBN: 978-3-642-31662-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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