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Constructing Elliptic Curves for the GLV Method with Low-cost Decomposition

Wroński M., Dryło R., Kijko T., Bora P.

Fundamenta Informaticae

2017

Vol. 153, nr 4

399--413

The GLV method allows to improve scalar multiplication on an elliptic curve E/Fqwith an efficiently computable endomorphism Φ : E → E over Fq. For points in a subgroup of large prime order r this requires decomposition of scalar k = k0 + k1λ mod r, where Φ acts on the subgroup of order r as multiplication by λ ∊ Fr and k0, k1 are integers O(√r) . In this note we consider the case when λ is of the form λ = 2s + a, where a is a small integer and λ=O(√r), which allows very easy and fast decomposition of k especially in hardware implementations. We give a method to construct such elliptic curves based on the complex multiplication method, and give examples of elliptic curves for λ ∊ {2s, 2s - 1} and various security levels.