Counting points on elliptic curve over binary field How to count number of rational points on elliptic curve over binary field?
Counting number of points on elliptic curve over $\mathbb F_2$ is very easy.For extension of fields we can use of this theorem:

Theorem : Let $E$ be an elliptic curve defined over $F_q$, and let $\#E(F_q ) = q +1−t$.
Then $\#E(F_{q^n} ) = q^n + 1 − V_n$ for all $n ≥ 2$, where $\{V_n\}$ is the sequence defined
recursively by $V_0 = 2, V_1 = t$, and $V_n = V_1V_{n−1}−qV_{n−2}$ for $n ≥ 2$.