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Differential Addition in Generalized Edwards Coordinates

  • Benjamin Justus
  • Daniel Loebenberger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6434)

Abstract

We use two parametrizations of points on elliptic curves in generalized Edwards form x 2 + y 2 = c 2 (1 + d x 2 y 2) that omit the x-coordinate. The first parametrization leads to a differential addition formula that can be computed using 6M + 4S, a doubling formula using 1M + 4S and a tripling formula using 4M + 7S. The second one yields a differential addition formula that can be computed using 5M + 2S and a doubling formula using 5S. All formulas apply also for the case c ≠ 1 and arbitrary curve parameter d. This generalizes formulas from the literature for the special case c = 1 or d being a square in the ground field.

For both parametrizations the formula for recovering the missing X-coordinate is also provided.

Keywords

Elliptic curve Edwards form addition formula differential addition 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Benjamin Justus
    • 1
  • Daniel Loebenberger
    • 1
  1. 1.Bonn-Aachen International Center for Information TechnologyUniversität BonnBonnGermany

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