On Efficient Computations of $y^2=x^3+b/\mathbb{F}_p$ for Primes $p\equiv 1 \mod 3$

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Since its introduction, Solinas' window $\tau$-NAF algorithm has been a landmark method for accelerating scalar multiplication on Koblitz curves over binary fields. A long-standing open problem has been to identify a suitable family of elliptic curves over prime fields for which the window $\tau$-NAF approach can be effectively extended. In this paper, we settle this problem by establishing such an extension for the family $E_b: y^2=x^3+b$ over $\mathbb{F}_p$ with prime $p\equiv1\pmod 3$. This f