# How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the order of $E(\mathbb F_q)$ is in the range $[q+1-2\sqrt{q}, q+1+2\sqrt{q}]$.

Is it possible to determine the order of $E(\mathbb F_q)$ given $a, b, q$ without enumerating the points?

• check 'algorithm of schoof' :) (sry not time for long answer) – Fleeep Sep 3 '15 at 15:37

There is a rather deep polynomial-time algorithm for counting the $$\mathbb F_q$$-rational points of an elliptic curve published by René Schoof in 1985 (with subsequent improvements by Noam Elkies and A. O. L. Atkin). It is based on two core ideas:

• The number of points is closely linked to a functional equation $$\varphi^2-t\varphi+q = 0 \qquad\in\operatorname{End}(E)$$ that the Frobenius endomorphism $$\varphi\colon\;E\to E,\;\begin{cases}\mathcal O&\mapsto \mathcal O\\(x,y)&\mapsto (x^q,y^q) \end{cases}$$ satisfies in the endomorphism ring of $$E$$. If $$t\in\mathbb Z$$ is chosen such that this equation holds, it is called the trace of Frobenius and one can show that $$\#E(\mathbb F_q)=q+1-t \text.$$ (The reason $$\varphi$$ has anything to do with point counting is that it leaves exactly the points with coordinates in $$\mathbb F_q$$ invariant.)
• For odd $$l$$, there exist division polynomials $$\psi_l\in\mathbb F_q[x]$$ which vanish precisely on the $$x$$-coordinates of the finite $$l$$-torsion points of $$E$$. Therefore, one can compute $$t\bmod l$$ by checking for which $$k\in\{0,\dots,l-1\}$$ the functional equation $$\varphi^2-t\varphi+q=0$$ holds modulo $$\psi_l$$. Computing modulo $$\psi_l$$ greatly improves the complexity (compared to expanding the terms without any reduction). Having computed $$t\bmod l$$ for a sufficiently large (according to Hasse's theorem, $$\prod_{l\in L}l>4\sqrt q$$ is enough) set $$L$$ of odd primes, the Chinese remainder theorem can be used to reconstruct the trace of Frobenius $$t$$, thereby yielding the number of $$\mathbb F_q$$-rational points $$\#E(\mathbb F_q)$$ on $$E$$.

Of course, there are implementations of (the improved variants of) this algorithm in most computer algebra systems. Using sage, the number of $$\mathbb F_{p^n}$$-rational points of an elliptic curve $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ defined over $$\mathbb F_p$$ can be computed as:

EllipticCurve(GF(p), [a1, a2, a3, a4, a6]).cardinality(extension_degree = n)


As yyyyyyy mentioned for counting number of points on elliptic curve over $\mathbb F_p$ we can use Elkies method. But for extension of fields use of this theorem make it so easy:

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 $n ≥ 2$.