Checking if point is on EC twist

Question

Given a short Weierstrass elliptic curve $C$ over $F_p$ and a point $(x, y)$, it is easy to verify that $(x, y)$ either satisfies the curve equation (on-curve) or does not (off-curve).

In the case that it is off-curve, is there an efficient way to test if $(x, y)$ is on a quadratic twist of the curve? I can easily generate domain parameters $a, b$ of a twisted curve $C'$, but unless I have the correct coefficients, $(x, y)$ might not satisfy that particular curve equation $C'$ even though $(x, y)$ are on a twist of $C$.

Answer 1

Let $E$ be an elliptic curve over $\mathbf F_p$ given by the equation $y^2 = x^3 + Ax + B$. Then a quadratic twist $E'$ of $E$ is given by the equation $\beta y^2 = x^3 + Ax + B$ where $\beta$ is not a square.

Now you have a point $(x_0, y_0)$ you suppose is not on $E'$. Then to make it on a twist, you need to find $\beta$ such that $\beta {y_0}^2 = {x_0}^3 + Ax_0 + B$. If $\beta$ is not a square, then you found a quadratic twist of the curve with $(x_0, y_0)$ belonging on it.

Now, if you want $E'$ to have a short Weierstrass form, you can instead look for a non-square $\beta$ such that the following equation is satisfied: \begin{equation} \beta^3({y_0}^2 - {x_0}^3) - \beta Ax - B = 0. \end{equation} Then, $E'$ is given by the equation $y^2 = x^3 + A'x + B'$ where $A'=\frac{A}{\beta^2}$ and $B'=\frac{B}{\beta^3}$.

Example

Let $E$ the elliptic curve of equation $y^2 = x^3 - 3x + 60$ over the field $\mathbf F_{101}$ and the point $P=(5,17)$ that is not on the curve.

We will try to find the short Weierstrass equation of the twist $E'$ where $P$ lies on it.

We take the above equation and solve for $\beta$: \begin{equation} \beta^3 ({y_0}^2 - {x_0}^3) + 3 \beta x_0 - 60 = 0. \end{equation} With our favourite computer algebra software, we find $\beta = 67$. Then the equation of $E'$ is \begin{equation} y^2 = x^3 + 74x + 97. \end{equation}

Let's check everything:

sage: p = 101
sage: E = EllipticCurve(GF(p), [-3, 60])
sage: q = E.cardinality()
sage: q
113
sage: x_0, y_0 = 5, 17
sage: E.is_on_curve(x_0,y_0)
False
sage: R.<X> = GF(p)[]
sage: f = X^3*(y_0^2 - x_0^3) + 3*X*x_0 - 60
sage: f.roots()
[(67, 1)]
sage: F(67).is_square()
False
sage: EE = EllipticCurve(GF(p), [-3/beta^2,60/beta^3])
sage: qq = EE.cardinality() 
sage: qq
91
sage: sage: EE.is_on_curve(x_0,y_0)
True
sage: q + qq == 2*(p + 1)
True

The last instructions check the relation between the cardinality of $E$ and its quadratic twist.

Final remark.

This shows how to construct a specific quadratic twist having a specific point, but if you do this with another random point, not on $E$, then it might not be on the previously constructed twist $E'$.