*4.5. The Elliptic Curve Discrete Logarithm Problem (ECDLP)
Unlike the finite field DLP, there are no general-purpose subexponential algorithms to solve the ECDLP. Though good algorithms are known for certain specific types of elliptic curves, all known algorithms that apply to general curves take fully exponential time. The square root methods of Section 4.4 are the fastest known methods for solving the ECDLP over an arbitrary curve. As a result, elliptic curves are gaining popularity for building cryptosystems. The absence of subexponential algorithms implies that smaller fields can be chosen compared to those needed for cryptosystems based on the (finite field) DLP. This, in particular, results in smaller sizes of keys.
We start with Menezes, ...
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