Welcome to the resource topic for
**2007/455**

**Title:**

Analysis and optimization of elliptic-curve single-scalar multiplication

**Authors:**
Daniel J. Bernstein, Tanja Lange

**Abstract:**

Let P be a point on an elliptic curve over a finite field of large characteristic. Exactly how many points 2P,3P,5P,7P,9P,\ldots,mP should be precomputed in a sliding-window computation of nP? Should some or all of the points be converted to affine form, and at which moments during the precomputation should these conversions take place? Exactly how many field multiplications are required for the resulting computation of nP? The answers depend on the size of n, the \inversions/\mults ratio, the choice of curve shape, the choice of coordinate system, and the choice of addition formulas. This paper presents answers that, compared to previous analyses, are more carefully optimized and cover a much wider range of situations.

**ePrint:**
https://eprint.iacr.org/2007/455

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