Introduction to Cryptography with Coding Theory, 3rd Edition

21.4 Elliptic Curves in Characteristic 2

Many applications use elliptic curves mod 2, or elliptic curves defined over the finite fields $GF(2^n)$ (these are described in Section 3.11). This is often because mod 2 adapts well to computers. In 1999, NIST recommended 15 elliptic curves for cryptographic uses (see [FIPS 186-2]). Of these, 10 are over finite fields $GF(2^n)\text{.}$

If we’re working mod 2, the equations for elliptic curves need to be modified slightly. There are many reasons for this. For example, the derivative of $y^2$ is $2y{y}^{\prime }=0\text{,}$ since $2$ is the same as 0. This means that the tangent lines we compute are vertical, so $2P=\text{∞}$ for all points $P\text{.}$ A more sophisticated explanation is that the curve $y^2\equiv x^3+bx+c(\mathrm{m}\mathrm{o}\mathrm{d}2)$ has singularities (points where the ...