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International Symposium on Mathematical Foundations of Computer Science

MFCS 2006: Mathematical Foundations of Computer Science 2006 pp 179–189Cite as

Concrete Multiplicative Complexity of Symmetric Functions

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4162))

Abstract

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n − H ℕ(n), where H ℕ(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.

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Author information

Authors and Affiliations

  1. Dept. of Math. and Computer Science, University of Southern Denmark,  

    Joan Boyar

  2. Security Division, Information Technology Laboratory, NIST,  

    René Peralta

Authors
  1. Joan Boyar
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  2. René Peralta
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia

    Rastislav Královič

  2. Institute of Informatics, University of Warsaw, Poland

    Paweł Urzyczyn

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© 2006 Springer-Verlag Berlin Heidelberg

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Boyar, J., Peralta, R. (2006). Concrete Multiplicative Complexity of Symmetric Functions. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_16

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  • DOI: https://doi.org/10.1007/11821069_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37791-7

  • Online ISBN: 978-3-540-37793-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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