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Generating Complete Edwards Curves

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Information Security and Privacy (ACISP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10343))

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Abstract

Twisted Edwards curves are elliptic curves of the form \(ax^2 + y^2 = 1 + dx^2y^2\) for some constants a and d. The curves are called complete Edwards curves for the special case when \(a=1\) and d is not a square. Using complete Edwards curves for elliptic curve cryptography has many advantages as they have very efficient, complete, and unified point addition formula. In order to use complete Edwards curves for elliptic curve cryptography, we need to specify the curve as well as a point on the curve (typically of prime order). In this paper, we introduce some algorithms for generating complete Edwards curves over \(\mathbb {F}_p\) with \(4p_0\) number of points, where \(p_0\) is a prime and p is a prime of user-specified bit length. These algorithms are able to generate a complete Edwards curve over \(\mathbb {F}_p\) and a point of prime order on the curve in less than 3 (resp. 15, 35) minutes when p is a 256 (resp. 384, 512)-bit prime. These are much faster than the running time of the twisted Edwards curves generation algorithm proposed by Costello et al. in [4].

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Notes

  1. 1.

    Pseudo-Mersenne primes are primes of the form \(2^{\alpha } - \gamma \), while Montgomery-friendly primes are primes of the form \(2^{\alpha }(2^{\beta } - \gamma ) -1\), where \(\alpha , \beta , \gamma \) are integers. Primes of these forms are usually preferred for efficiency consideration [1].

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Author information

Authors and Affiliations

  1. Temasek Laboratories, National University of Singapore, 5A Engineering Drive 1, #09-02, Singapore, 117411, Singapore

    Theo Fanuela Prabowo & Chik How Tan

Authors
  1. Theo Fanuela Prabowo
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  2. Chik How Tan
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Corresponding author

Correspondence to Theo Fanuela Prabowo .

Editor information

Editors and Affiliations

  1. Queensland University of Technology, Brisbane, Queensland, Australia

    Josef Pieprzyk

  2. Queensland University of Technology, Brisbane, Queensland, Australia

    Suriadi Suriadi

Appendices

Appendix

A List of Discriminants and Their Hilbert Class Polynomials

figure f

B Sample Output of Algorithms  4 and 5

The sample output of Algorithms 4 and 5 are given in Tables 3, 4 and 5 below.

Table 3. Input (bit-length of p) = 256
Full size table
Table 4. Input (bit-length of p) = 384
Full size table
Table 5. Input (bit-length of p) = 512
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Prabowo, T.F., Tan, C.H. (2017). Generating Complete Edwards Curves. In: Pieprzyk, J., Suriadi, S. (eds) Information Security and Privacy. ACISP 2017. Lecture Notes in Computer Science(), vol 10343. Springer, Cham. https://doi.org/10.1007/978-3-319-59870-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-59870-3_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59869-7

  • Online ISBN: 978-3-319-59870-3

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