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

Question

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?

Answer 1

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)

Answer 2

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 $n ≥ 2$.