Associativity of the group law on an elliptic curve via the Cayley-Bacharach theorem

We recount here an elementary proof of associativity for the group law on a non-singular elliptic curve. The principal ingredient is the Cayley-Bacharach theorem, which has a neat combinatorial proof using only a corollary of Bézout’s theorem (see “further reading” below).

Theorem (Cayley-Bacharach): Let $D, D’$ be two cubic curves intersecting in nine distinct points. If $D^″$ is a cubic curve through eight of the nine points, then it has the form $D^″ = aD + a’D’$ for some $(a:a’) \in \mathbb{P}_1(k)$ and in particular goes through the ninth point.

Note that “curve” in this context just means zero set in the projective plane of some homogeneous (not necessarily irreducible) polynomial. In particular, the union of three distinct lines in the projective plane is a cubic curve, by this definition, being the zero set of a product of three linear polynomials.

Theorem: Let $E$ be a non-singular elliptic curve with base point (=identity) $O \in E$ and addition defined (as usual) via $P + Q := O \circ (P \circ Q)$ where $P \circ Q$ denotes the third point of intersection of $E$ with the line through $P$ and $Q$, for all $P, Q \in E$. Then for all $P, Q, R \in E$ distinct we have $$ P + (Q + R) = (P + Q) + R.$$

Proof: First notice that since $$ P + (Q + R) = O \circ (P \circ (Q + R)) $$ and $$ (P + Q) + R = O \circ ((P + Q) \circ R), $$ the identity $$ O \circ (O \circ X) = X \quad \forall X \in E$$
implies that it is sufficient to show that $$ P \circ (Q + R) = (P + Q) \circ R.$$

Consider the following diagram. A red line and a blue line intersect the curve $E$ inside the dotted circle. Our goal is to show that these two intersections occur at the same point of $E$ (as indeed they appear to, in the diagram).

Let $l, l’, l^″$ be an enumeration of the red lines, $m, m’, m^″$ an enumeration of the blue lines as follows:

Define cubics $L = l l’ l^″$, $M = m m’ m^″$ (these are the red and blue triangles in the first diagram). Then $E$ and $L$ meet at exactly nine points.

We assume that the eight points $O, P, Q, R, P \circ Q, Q \circ R, P + Q, Q + R$ (in black on both diagrams) are distinct from one another and also from $P \circ (Q + R)$ and $(P + Q) \circ R$ (i.e. the points we are trying to show are equal).

Then $E$ and $L$ have nine points in common (the black points and $(P + Q) \circ R$), and moreover cannot share more than these, since otherwise Bézout’s Theorem (applied to $E$ and each component $l, l’, l^″$ of $L$) would imply that one of these components $l, l’, l^″$ belonged to $E$, thereby contradicting the assumed non-singularity of $E$ (since this implies irreducibility, see note below).

Similarly, $E$ and $M$ share precisely nine points, viz. the black points and $P \circ (Q + R)$. Now $M$ contains eight points (the black points) that are shared by $E$ and $L$, and hence contains also the ninth, by the Cayley-Bacharach Theorem, i.e. $(P + Q) \circ R \in M$.

So $E$ and $M$ share precisely nine points. On the other hand, we’ve shown that they share ten points: the eight black points, $P \circ (Q + R)$, and $(P + Q) \circ R$. Hence, in view of the point distinctness assumptions, the last two points must be equal.


  • Non-singular curves are irreducible since reducible curves are necessarily singular (since components must intersect by Bézout’s Theorem, and these intersection points are necessarily singularities).


  • The distinctness assumptions are sufficient in view of Zariski closure?

Further reading

  • Husemöller’s book “Elliptic Curves” (page 51, 2nd edition) proves this, as well as the Cayley-Bacharach theorem itself, along with the corollary of Bézout’s theorem needed for it.
  • Terence Tao’s blog covers the same material as above (and does a much better job of it).
  • Timothy Murphy’s 2016 lecture notes are great.

Pairings on elliptic curves: a toy example

Here is a nice simple example of bilinear pairing on an elliptic curve over a finite field. Different sorts of pairings exist – here we’ll construct a pairing that coincides with the Weil pairing. For simplicity, our construction will avoid talking about divisors on algebraic curves.

Consider the elliptic curve over $\mathbb{F}_7$ defined by the equation $$ y^2 = x^3 + 2.$$ There are nine solutions: $$ \{ \mathcal{O},\ (0, \pm 3),\ (3, \pm 1),\ (5, \pm 1),\ (6, \pm 1)\ \}. $$ The identity element $\mathcal O$ is a point at infinity corresponding to the vertical axis (it is a point of the projective plane). All the other points are elements of the affine plane $\mathbb{F}_7 \times \mathbb{F}_7$. These nine points form a group which we’ll denote by $\mathbb{G}$.

We can check that for any $P \in \mathbb{G}$, we have $3 P = \mathcal O$. The most elementary way to check this is by direct calculation: we need to check that $P + P = -P$ for each point $P$. Recalling the diagrammatic rule for computing $P+P$, we need to draw the tangent to the curve at $P$, and then find the other point of intersection with the curve (it turns out this geometric interpretation of point doubling works over prime fields as well). So let’s calculate the slope $s$ of the tangent at each point $P=(x,y)$:
$$ s = \frac{\frac{\partial}{\partial x} RHS}{\frac{\partial}{\partial y} LHS} = \frac{\frac{\partial}{\partial x} (x^3 + 2)}{\frac{\partial}{\partial y} y^2} = \frac{3x^2}{2y}.$$
This gives us the following diagram:

The dashed blue lines indicate the tangents to the curve at each point. Check for yourself that if you follow any of these tangents (remembering to wrap around to the opposite side when you get to an edge!) then you don’t hit any other points, i.e. the point of tangency $P$ is the only point of the elliptic curve on the tangent line, meaning that the “other” point of intersection must be $P$ itself (this means it’s an intersection of “higher order”). So $P + P$ is equal to the reflection of $P$ through the line $y=0$, i.e. $-P$, and so $3 P = \mathcal O$, as claimed.

In fact, it turns out that $\mathbb{G} \cong \mathbb{Z}_3 \times \mathbb{Z}_3$ as groups; you can construct such an isomorphism yourself by setting e.g. $g_1 = (0,3)$ and $g_2 = (3, 1)$ and taking these as the generators of the two copies of $\mathbb{Z}_3$ in the direct product. Note that this choice of isomorphism is not canonical (i.e. there is nothing to justify this particular choice) but that’s fine for our purposes. All we really need is that any $g \in \mathbb{G}$ has a unique expression $g = a_1 g_1 + a_2 g_2$ as a $\mathbb{Z}_3$-linear combination of our chosen generators $g_1, g_2$.

We are finally in a position to define a bilinear pairing. There are various types of pairings. Here, we’ll construct the Weil pairing. Let $\mu$ be an abelian group, written multiplicatively. We want a map $ e: \mathbb{G} \times \mathbb{G} \to \mu $ that is bilinear, i.e. $$ e(g + g’, h) = e(g, h) \cdot e(g’, h), \quad e(g, h + h’) = e(g, h) \cdot e(g, h’), $$ for all $g, g’, h, h’ \in \mathbb{G}$, and alternating, i.e. $$ e(g, h) = e(h, g)^{-1} \quad \forall g, h \in \mathbb{G}.$$ Some further properties follow immediately. For example, using that $\mathcal{O} + \mathcal{O} = \mathcal{O}$, it follows from bilinearity that $$ e(g, \mathcal{O}) = e(\mathcal{O}, h) = 1 \in \mu,$$ and from this last property and the fact that $3 P = \mathcal{O}$ for all $P \in \mathbb{G}$, it follows that $$ e(g, h)^3 = 1 \quad \forall g, h \in \mathbb{G},$$ and so we might as well take $\mu = \mu_3$ to be the group of third roots of unity over $\mathbb{F}_7$. Note $\mathbb{F}_7$ in fact contains all its third roots of unity, since $2^3 \equiv 1$ modulo $7$.

So how many alternating, bilinear forms can be there be on $\mathbb{G}$? Up to a choice of third root of unity, just one! To see this, just use the bilinearity and alternation properties to check that
$$ e(a g_1 + b g_2, c g_1 + d g_2) = e(g_1, g_2)^{ad – bc}, \quad \forall a,b,c,d \in \mathbb{Z}_3,$$
i.e. all such forms are given by some third root of unity $e(g_1, g_2)$ raised to the power of the determinant of the matrix of coefficients. And on the other hand, since we know that the determinant is an alternating multilinear function on matrix rows, we see that this definition really does give us an alternating bilinear form on $\mathbb{G}$. From uniqueness, we know that this pairing must in fact coincide (up to choice of root of unity) with the Weil pairing of our elliptic curve, since it too is alternating and bilinear.

What about the general case? The Weil pairing is defined on the “$r$-torsion” $E[r]$ of an elliptic curve, i.e. the subgroup of points $P$ such that $r P = \mathcal{O}$. In our example, $r=3$ and the $3$-torsion subgroup $E[3]$ coincides with the entire group $\mathbb{G}$ (this is rarely true). On the other hand, the group isomorphism $E[r] \cong \mathbb{Z}_r \times \mathbb{Z}_r$ holds in general, with the caveat that you’ll likely need to take an extension field (not a prime field) as your base field.

Is this construction useful? While the above construction might be instructive, it isn’t useful. In order to prove useful things about the pairing, the divisor-theoretic construction is much more useful, since it allows us to call on the theory of algebraic curves (and in fact this is needed to even prove that $\mathbb{G}$ is a group!). Furthermore, in order to actually calculate a Weil pairing using the determinant construction, it is necessary to express an arbitrary $g \in E[r]$ as a linear combination of the chosen generators (and this problem would be hard even if there was just one generator: that’s the discrete logarithm problem!). So there are good reasons for the definition of the Weil pairing in terms of divisors, after all.

It should also be noted that the (divisor-theoretic) construction of the Weil pairing is canonical, not requiring a choice of basis for $E[r]$ nor a choice of primitive root of unity.

Addendum #1: as an alternative to the slope/tangent method above for visually checking that each point has order $3$, you can check instead that the Hessian determinant $X (Y^2 – Z^2)$ of the homogenisation of the equation defining the elliptic cruve vanishes at each point, i.e. show that each point is a point of inflection. See Silverman’s The Arithmetic of Elliptic Curves, exercise III.3.9.

Addendum #2: The example above has been chosen so that everything can be calculated by hand. To work with other examples, or to check your work, Magma is a great help. Here is some example code for the above calculations, which you can plug into the online calculator:

K := GF(7);  // the finite field with 7 elements

// The EC with equation y^2 = x^3 + 0*x + 2 over K
E := EllipticCurve([K | 0, 2]);
print E, "\n";

print "Number of points on the curve", #E, "\n";

// Creation of a point with given projective coordinates
P := E ! [6, -1, 1];

print "The order of each element:";
for P in Points(E) do
    print "Element", P, "has order", Order(P);
end for;
print "\n";

print "Point doubling:";
for P in Points(E) do
    print "2", P, "=", P+P;
end for;
print "\n";

print "Find a decomposition of E as a product of finite cyclic groups:";
print "\n";

print "Calculate the Weil pairing:";
for P in Points(E) do
    for Q in Points(E) do
        P, "paired with", Q, "=", WeilPairing(P, Q, 3);
    end for;
end for;

What does an elliptic curve look like near the point at infinity (the identity)?

The appearance of an elliptic curve from the point of view of the affine $(x,y)$ plane is familiar to us, but leaves us wondering what the curve might look like near the point at infinity (i.e. the identity element $\mathcal{O}$). This is not merely of visual interest, as it allows one to see directly that e.g. a line that is vertical in the $(x,y)$ plane has a pole of order 2 at the identity (even though it also intersects the curve at that point). The divisors of such linear functions play a crucial role in e.g. the computation of the Weil pairing.

A toy elliptic curve over a finite field

Here is a first example of an elliptic curve over a finite field where you can work everything out by hand.

Consider the elliptic curve defined by the equation
$$ y^2 = (x-1)(x-2)(x-3) $$
over the field $\mathbb{F}_5$. Multiplying out the right hand side, we see that (over $\mathbb{F}_5$), the right hand side (“RHS”) is $x^3 – x^2 + x – 1$. So our equation is not in Weierstrass form, but that’s fine. It’s still defines an elliptic curve. Note that we know immediately that it is non-singular, since the roots of the RHS are distinct.

It’s easy to find all the solutions $(x,y)$ to our equation by hand – just consider each possible value for $x \in \mathbb{F}_5$, calculate the RHS, and set $y = \pm \sqrt{RHS}$, if such $y$ exist. So we first need to know which values in $\mathbb{F}_5$ are squares:

z & 0 & 1 & 2 & 3 & 4 \\
z^2 & 0 & 1 & 4 & 4 & 1

Thus the points on our elliptic curve are
$$ (0,2), (0,3), (1,0), (2,0), (3,0), (4,1), (4,4), \mathcal{O}.$$
The point $\mathcal{O}$ is the solution at infinity: this is the one extra solution $(0:0:1)$ that arises when the equation defining the elliptic curve is homogenized, and solutions in the projective plane are permitted. The other solutions live in the “affine” $(x,y)$ plane.

The nice thing about working over an unextended finite field like $\mathbb{F}_5$ is that it is still “1-dimensional”, so the affine solutions $(x,y)$ can be depicted on a 2-dimensional diagram like the following:

Fortunately, the familiar geometric description of the group operation on elliptic curves in terms of line intersections still works (why?). That is, any two points can be added by drawing a line through them, finding the third point of intersection, and reflecting through the line $y=0$, and the point $\mathcal{O}$ corresponds to the vertical direction and is the identity element of the group.

For example, it is immediate from this rule that $A+B=C$. Remembering that lines wrap-around our diagram (which is actually a torus), what do you think $C+D$ is equal to? (Hint: it’s the next letter of the alphabet).

As in the case over $\mathbb{R}$:

  • If a vertical line passes through two distinct affine points such as $(0,2)$ and $(0,3)$, then (since it also intersects with $\mathcal{O}$ in the projective plane) these points are inverses of one another w.r.t. the group operation. (We’ve labelled $-D, -E$ to reflect this.)
  • If a vertical line hits a single affine point (e.g. the line $x=1$) then this point is its own inverse.

Thus $A, B, C$ are all group elements of order 2.

Amusingly, the geometric rule for point doubling using tangents still works, as well. The slope of the tangent at a point $(x,y)$ on our elliptic curve can be calculated in the usual way.
$$ s = \frac{\frac{\partial}{\partial x} RHS}{\frac{\partial}{\partial y} LHS} = \frac{3x^2 – 2x + 1}{2y}.$$
These slopes are depicted on our diagram with dashed blue lines. Following these tangents, you can immediately verify that
$$ \pm E + \pm E = B, \qquad \pm D + \pm D = B,$$
and so $\pm D, \pm E$ have order $4$ as group elements.

The orders of our group elements are enough to conclude that our group (call it $\mathbb{G}$) is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_4$. Indeed, since $A+B=C$ and $A+D=E$ (to check, just follow the lines!) we have that
\mathcal{O} & \mapsto & (0,0) \\
A & \mapsto & (1,0) \\
B & \mapsto & (0,2) \\
C & \mapsto & (1,2) \\
\pm D & \mapsto & (0,\pm 1) \\
\pm E & \mapsto & (1,\pm 3)\\
is an isomorphism of groups $\mathbb{G} \rightarrow \mathbb{Z}_2 \times \mathbb{Z}_4.$