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

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

**Abstract: **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.

**Category / Keywords: **elliptic curve cryptography, GLS curves, scalar multiplication

**Original Publication**** (in the same form): **Journal of Cryptographic Engineering
**DOI: **10.1007/s13389-013-0064-4

**Date: **received 5 Mar 2013, last revised 30 Jan 2014

**Contact author: **francisco at cs cinvestav mx

**Available format(s): **PDF | BibTeX Citation

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

**Version: **20140131:014434 (All versions of this report)

**Short URL: **ia.cr/2013/131

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