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International Workshop on Lightweight Cryptography for Security and Privacy

LightSec 2014: Lightweight Cryptography for Security and Privacy pp 21–33Cite as

The Multiplicative Complexity of Boolean Functions on Four and Five Variables

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8898))

Abstract

A generic way to design lightweight cryptographic primitives is to construct simple rounds using small nonlinear components such as 4\(\,\times \,\)4 S-boxes and use these iteratively (e.g., PRESENT [1] and SPONGENT [2]). In order to efficiently implement the primitive, efficient implementations of its internal components are needed. Multiplicative complexity of a function is the minimum number of AND gates required to implement it by a circuit over the basis (AND, XOR, NOT). It is known that multiplicative complexity is exponential in the number of input bits \(n\). Thus it came as a surprise that circuits for all \(65 536\) functions on four bits were found which used at most three AND gates [3]. In this paper, we verify this result and extend it to five-variable Boolean functions. We show that the multiplicative complexity of a Boolean function with five variables is at most four.

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Notes

  1. 1.

    Lest the reader think this easy, he/she may attempt to compute the function \(f(x_1,x_2,x_3,x_4,x_5) = x_1 x_2 x_3 x_4 x_5+x_1 x_2 x_3+x_1 x_2 x_4+x_2 x_3 x_4+x_1 x_2+ x_1 x_3+x_1 x_4+x_2 x_4+x_3 x_4\) using only four AND gates.

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Acknowledgments

We thank Çağdaş Çalık, Joan Boyar, and Magnus Find for helpful discussions and suggestions. We also thank our colleagues Yi-Kai Liu, Ray Perlner, Lily Chen, and the anonymous reviewers for their useful comments.

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Authors and Affiliations

  1. National Institute of Standards and Technology, Gaithersburg, MD, USA

    Meltem Turan Sönmez & René Peralta

  2. Dakota Consulting Inc., Silver Spring, MD, USA

    Meltem Turan Sönmez

Authors
  1. Meltem Turan Sönmez
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  2. René Peralta
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Corresponding author

Correspondence to Meltem Turan Sönmez .

Editor information

Editors and Affiliations

  1. Worcester Polytechnic Institute, Worcester, Massachusetts, USA

    Thomas Eisenbarth

  2. Istanbul Commerce University, Istanbul, Turkey

    Erdinç Öztürk

Appendix

Appendix

Table 5. Circuits for \(n=5\).
Full size table

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Turan Sönmez, M., Peralta, R. (2015). The Multiplicative Complexity of Boolean Functions on Four and Five Variables. In: Eisenbarth, T., Öztürk, E. (eds) Lightweight Cryptography for Security and Privacy. LightSec 2014. Lecture Notes in Computer Science(), vol 8898. Springer, Cham. https://doi.org/10.1007/978-3-319-16363-5_2

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  • DOI: https://doi.org/10.1007/978-3-319-16363-5_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16362-8

  • Online ISBN: 978-3-319-16363-5

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