# Four-Dimensional Gallant–Lambert–Vanstone Scalar Multiplication

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## Abstract

The GLV method of Gallant, Lambert, and Vanstone (CRYPTO 2001) computes any multiple for some explicit constant for some explicit

*kP*of a point*P*of prime order*n*lying on an elliptic curve with a low-degree endomorphism*Φ*(called GLV curve) over \(\mathbb{F}_{p}\) as$$kP = k_1P + k_2\varPhi(P) \quad\text{with } \max \bigl\{ |k_1|,|k_2| \bigr\} \leq C_1\sqrt{n} $$

*C*_{1}>0. Recently, Galbraith, Lin, and Scott (EUROCRYPT 2009) extended this method to all curves over \(\mathbb{F}_{p^{2}}\) which are twists of curves defined over \(\mathbb{F}_{p}\). We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over \(\mathbb{F}_{p^{2}}\), a four-dimensional decomposition together with fast endomorphisms*Φ*,*Ψ*over \(\mathbb{F}_{p^{2}}\) acting on the group generated by a point*P*of prime order*n*, resulting in a proven decomposition for any scalar*k*∈[1,*n*] given by$$kP=k_1P+ k_2\varPhi(P)+ k_3\varPsi(P) + k_4\varPsi\varPhi(P) \quad \text{with } \max_i \bigl(|k_i| \bigr)< C_2\, n^{1/4} $$

*C*_{2}>0. Remarkably, taking the best*C*_{1},*C*_{2}, we obtain*C*_{2}/*C*_{1}<412, independently of the curve, ensuring in theory an almost constant relative speedup. In practice, our experiments reveal that the use of the merged GLV–GLS approach supports a scalar multiplication that runs up to 1.5 times faster than the original GLV method. We then improve this performance even further by exploiting the Twisted Edwards model and show that curves originally slower may become extremely efficient on this model. In addition, we analyze the performance of the method on a multicore setting and describe how to efficiently protect GLV-based scalar multiplication against several side-channel attacks. Our implementations improve the state-of-the-art performance of scalar multiplication on elliptic curves over large prime characteristic fields for a variety of scenarios including side-channel protected and unprotected cases with sequential and multicore execution.## Key words

Elliptic curves GLV–GLS method Scalar multiplication Twisted Edwards curve Side-channel protection Multicore computation## Notes

### Acknowledgements

We thank the reviewers, Mike Scott and Dan Bernstein for their helpful comments. Also, we would like to thank Diego Aranha for his advice on multicore programming, Joppe Bos for his help on looking for efficient chains for implementing modular inversion, and Craig Costello and Kristin Lauter for helping us to detect a typo on a curve parameter in a previous paper version.

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© International Association for Cryptologic Research 2013