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The GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) computes anymultiple $kP$ of a point $P$ of prime order $n$ lying on an elliptic curve witha low-degree endomorphism $\Phi$ (called GLV curve) over $\mathbb{F}_p$ as [kP= k_1P + k_2\Phi(P), \quad\text{with} \max{|k_1|,|k_2|}\leq C_1\sqrt n] forsome explicit constant $C_1>0$. Recently, Galbraith, Lin and Scott (EUROCRYPT2009) extended this method to all curves over $\mathbb{F}_{p^2}$ which aretwists of curves defined over $\mathbb{F}_p$. We show in this work how to mergethe two approaches in order to get, for twists of any GLV curve over$\mathbb{F}_{p^2}$, a four-dimensional decomposition together with fastendomorphisms $\Phi, \Psi$ over $\mathbb{F}_{p^2}$ acting on the groupgenerated by a point $P$ of prime order $n$, resulting in a proveddecomposition for any scalar $k\in[1,n]$ $$ kP=k_1P+ k_2\Phi(P)+ k_3\Psi(P) +k_4\Psi\Phi(P)\quad \text{with} \max_i (|k_i|)< C_2\, n^{1/4} $$ for someexplicit $C_2>0$. Furthermore, taking the best $C_1, C_2$, we get$C_2/C_1<408$, independently of the curve, ensuring a constant relativespeedup. We also derive new families of GLV curves, corresponding to thosecurves with degree 3 endomorphisms.

Four-Dimensional Gallant-Lambert-Vanstone Scalar Multiplication