Journal of Cryptographic Engineering

, Volume 6, Issue 3, pp 187–199 | Cite as

Inversion-free arithmetic on elliptic curves through isomorphisms

  • Raveen R. Goundar
  • Marc Joye
Regular Paper


This paper presents inversion-free formulas for the efficient implementation of a scalar multiplication over elliptic curves. Specifically, it proposes to make use of curve isomorphisms as a way to avoid the computation of inverses in point addition formulas. Interestingly, the presented techniques are independent of the model used to represent the elliptic curve and of the coordinate system used to represent the points. In particular, they apply to affine representations. Further, whereas certain inversion-free techniques are mostly limited to specific scalar multiplication algorithms, the proposed techniques apply to all scalar multiplication algorithms. The so-obtained formulas are well suited to embedded systems and can easily be combined with existing countermeasures to provide secure implementations.


Elliptic curves Scalar multiplication Isomorphisms Affine coordinates 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.ColebeeAustralia
  2. 2.TechnicolorLos AltosUSA

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