Paper 2013/131

Two is the fastest prime: lambda coordinates for binary elliptic curves

Thomaz Oliveira, Julio López, Diego F. Aranha, and Francisco Rodríguez-Henríquez


In this work, we present new arithmetic formulas for a projective version of the affine point representation $(x,x+y/x),$ for $x\ne 0,$ which leads to an efficient computation of the scalar multiplication operation over binary elliptic curves.A software implementation of our formulas applied to a binary Galbraith-Lin-Scott elliptic curve defined over the field $\mathbb{F}_{2^{254}}$ allows us to achieve speed records for protected/unprotected single/multi-core random-point elliptic curve scalar multiplication at the 127-bit security level. When executed on a Sandy Bridge 3.4GHz Intel Xeon processor, our software is able to compute a single/multi-core unprotected scalar multiplication in $69,500$ and $47,900$ clock cycles, respectively; and a protected single-core scalar multiplication in $114,800$ cycles. These numbers are improved by around 2\% and 46\% on the newer Ivy Bridge and Haswell platforms, respectively, achieving in the latter a protected random-point scalar multiplication in 60,000 clock cycles.

Note: Extended version of CHES 2013 to appear in JCEN.

Available format(s)
Publication info
Published elsewhere. Journal of Cryptographic Engineering
elliptic curve cryptographyGLS curvesscalar multiplication
Contact author(s)
francisco @ cs cinvestav mx
2014-01-31: last of 10 revisions
2013-03-07: received
See all versions
Short URL
Creative Commons Attribution


      author = {Thomaz Oliveira and Julio López and Diego F.  Aranha and Francisco Rodríguez-Henríquez},
      title = {Two is the fastest prime: lambda coordinates for binary elliptic curves},
      howpublished = {Cryptology ePrint Archive, Paper 2013/131},
      year = {2013},
      doi = {10.1007/s13389-013-0064-4},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.