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Parallelized Software Implementation of Elliptic Curve Scalar Multiplication

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Information Security and Cryptology (Inscrypt 2014)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8957))

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Abstract

Recent developments of multicore architectures over various platforms (desktop computers and servers as well as embedded systems) challenge the classical approaches of sequential computation algorithms, in particular elliptic curve cryptography protocols. In this work, we deploy different parallel software implementations of elliptic curve scalar multiplication of point, in order to improve the performances in comparison with the sequential counter parts, taking into account the multi-threading synchronization, scalar recoding and memory management issues. Two thread and four thread algorithms are tested on various curves over prime and binary fields, they provide improvement ratio of around 15 % in comparison with their sequential counterparts.

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Acknowledgement

We would like to thank Christophe Nègre for his valuable and helpful comments.

This work has been suported by a PHD grant from PAVOIS project (ANR 12 BS02 002 01).

Author information

Authors and Affiliations

  1. Team DALI, Université de Perpignan, Perpignan, France

    Jean-Marc Robert

  2. LIRMM, UMR 5506, Université Montpellier 2 and CNRS, Montpellier, France

    Jean-Marc Robert

Authors
  1. Jean-Marc Robert
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Corresponding author

Correspondence to Jean-Marc Robert .

Editor information

Editors and Affiliations

  1. SKLOIS, Institute of Engineering, Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. Computer Science Department, Columbia University, New York, New York, USA

    Moti Yung

  3. Infocomm Security Department, Institute for Infocomm Research, Singapore, Singapore

    Jianying Zhou

A Appendix: Curve Parameters

A Appendix: Curve Parameters

1.1 A.1 Elliptic Curves over Binary Field

The curve equation is:

$$y^2+xy=x^3+x^2+b \text { where } b \in \mathbb {F}_{2^m}. $$

The parameters are for B233:

$$ \begin{array}{rcl} a &{}=&{} 1, \\ h &{}=&{} 2,\\ f(x) &{}=&{} x^{233} + x^{74} + 1, \\ b &{}=&{} \mathtt{0x00000066 ~\mathtt 647ede6c ~\mathtt 332c7f8c ~\mathtt 0923bb58 ~\mathtt 213b333b ~ \mathtt 20e9ce42 ~\mathtt 81fe115f ~\mathtt 7d8f90ad },\\ r &{}=&{} \mathtt{0x00000100 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 0013e974 ~\mathtt e72f8a69 ~\mathtt 22031d26 ~\mathtt 03cfe0d7 }.\\ \end{array} $$

where the order of the curve is \(n \times h\). For B409 we have:

$$ \begin{array}{rcl} a &{}=&{} 1, \\ h &{}=&{} 2, \\ f(x) &{}=&{} x^{409} + x^{87} + 1,\\ b &{}=&{} \mathtt{0x0021a5c2 ~\mathtt c8ee9feb ~\mathtt 5c4b9a75 ~\mathtt 3b7b476b ~\mathtt 7fd6422e ~\mathtt f1f3dd67 ~\mathtt 4761fa99 ~\mathtt d6ac27c8 }\\ &{} &{}\mathtt{a9a197b2 ~\mathtt 72822f6c ~\mathtt d57a55aa ~\mathtt 4f50ae31 ~\mathtt 7b13545f, }\\ r &{}=&{} \mathtt{0x01000000 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 00000000 ~\mathtt 000001e2 ~\mathtt aad6a612 }\\ &{} &{}\mathtt{f33307be ~\mathtt 5fa47c3c ~\mathtt 9e052f83 ~\mathtt 8164cd37 ~\mathtt d9a21173. }\\ \end{array} $$

1.2 A.2 Weierstrass Curve over Prime Field

The curve equation is:

$$y^2=x^3-3x+b \text { where } b \in \mathbb {F}_p. $$

The parameters are:

$$\begin{array}{lcl} p &{}=&{} 2^{255}-19\\ b &{}=&{} \mathtt{0x1d09bac9ffe9e7f8284aed0442552779bcdef2e62b9cb1d568513fa798b94003 }\\ r &{}=&{} \mathtt{0x800000000000000000000000000000012c18945a05ad7f2edf026258ea5288ef } \end{array}$$

\(r\) is the prime order of \(P\).

1.3 A.3 Jacobi Quartic Curve over Prime Field

The curve equation is:

$$y^2=x^4-\frac{3}{2}\theta x^2+1, \theta \in \mathbb {F}_p. $$

The parameters are:

$$\begin{array}{ll} \theta &{}= \mathtt{0x1731beeea2156180446f9e5ab64af78d4ed3e0eb68d5070c10ef2468b910d5f7 }\\ {\text {number}}&{} {\text {of points:}}\\ h\times r &{}= \mathtt{0x800000000000000000000000000000002672bdbb41f31390c5527cab6e282744 }\\ &{}=4\cdot \mathtt{0x20000000000000000000000000000000099caf6ed07cc4e431549f2adb8a09d1 } \end{array}$$

The Jacobi Quartic curve is isomorphic to the following Weierstrass elliptic curve:

$$y^2=x^3+ax+b$$

where: \(a=(-16-3\theta ^2)/4\) and \(b=-\theta ^3-a\theta \). Hence, in our case:

$$\begin{array}{lcl} a &{}=&{} \mathtt{0xc500be2450246d16c114830a5d1aef9c2b80c567b4fd87562c69db659713ad2, }\\ b &{}=&{} \mathtt{0xa38f53e5d27462dcdada9a78b9eac482ef06e855af92ca704060c551a9a5854. } \end{array}$$

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Robert, JM. (2015). Parallelized Software Implementation of Elliptic Curve Scalar Multiplication. In: Lin, D., Yung, M., Zhou, J. (eds) Information Security and Cryptology. Inscrypt 2014. Lecture Notes in Computer Science(), vol 8957. Springer, Cham. https://doi.org/10.1007/978-3-319-16745-9_24

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  • DOI: https://doi.org/10.1007/978-3-319-16745-9_24

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-16745-9

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