Paper 2012/670

Fast Cryptography in Genus 2

Joppe W. Bos, Craig Costello, Huseyin Hisil, and Kristin Lauter


In this paper we highlight the benefits of using genus 2 curves in public-key cryptography. Compared to the standardized genus 1 curves, or elliptic curves, arithmetic on genus 2 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus 2 based cryptography, which includes fast formulas on the Kummer surface and efficient 4-dimensional GLV decompositions. By studying different modular arithmetic approaches on these curves, we present a range of genus 2 implementations. On a single core of an Intel Core i7-3520M (Ivy Bridge), our implementation on the Kummer surface breaks the 125 thousand cycle barrier which sets a new software speed record at the 128-bit security level for constant-time scalar multiplications compared to all previous genus 1 and genus 2 implementations.

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Publication info
Published elsewhere. MAJOR revision.EUROCRYPT 2013
Genus 2JacobianKummer surfaceGLVDiffie-Hellmanspeed record
Contact author(s)
craigco @ microsoft com
2014-03-13: last of 5 revisions
2012-11-28: received
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      author = {Joppe W.  Bos and Craig Costello and Huseyin Hisil and Kristin Lauter},
      title = {Fast Cryptography in Genus 2},
      howpublished = {Cryptology ePrint Archive, Paper 2012/670},
      year = {2012},
      doi = {10.1007/978-3-642-38348-9_12},
      note = {\url{}},
      url = {}
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