L-notation

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The L-notation is widely used to express the computational complexity of an algorithm. By definition

[L_n[alpha, c] = O(e^)})],

where c is a positive constant, and [\alpha] is a constant 0 [< \alpha < 1].

When

[\alpha]

is 0, then

[L_n[0, c]]

is a polynomial function of [\ln n]. When [\alpha] is 1 then

[L_n[1, c]]

is a fully exponential function of [\ln n].

Example

For the elliptic curve discrete log problem, the fastest general purpose algorithm is the baby step giant step algorithm, which has a running time on the order of the square-root of the group order n. In L-notation this would be

[L_n[1, frac] = O(n^})].

Reference

Menezes, Alfred J., van Oorschot, Paul C., Vanstone, Scott A., Handbook of Applied Cryptography, CRC Press, Boca Raton, New York, London, Tokyo, 1996. ISBN 0-8493-8523-7.

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