Elliptic curve cryptographic schemes require calculations of the type
where k is a large integer and the addition is over the elliptic curve (see elliptic curves). The operation is known as scalar or point multiplication, and dominates the execution time of signature and encryption schemes based on elliptic curves. Double-and-add variations of familiar square-and-multiply methods (see binaryexponentiation) for modular exponentiation are commonly used to find kP. Windowing methods can significantly reduce the number of point additions required, but the number of point doubles remains essentially unchanged.
Among techniques to reduce the cost of the point doubles in point multiplication, perhaps the best known is illustrated in the case of Koblitz curves (elliptic curves over the field \(\Bbb{F}_{2^m}\) with coefficients in \(\Bbb{F}_2\); see [8]), where point doubling is replaced by inexpensive field squarings. Knudsen [4] and Schroeppel [6, 7]...
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Hankerson, D., Menezes, A. (2005). Elliptic Curve Point Multiplication Using Halving. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_136
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