# I don't know how to find the four square roots of an elliptic curve modulo a prime

1. Jul 19, 2012

### SneakyG

So there are four square roots for an elliptic curve represented by an equation something like this: y^2 = x^3 + x + 6 (mod 5)

How would one go about calculating these?

2. Jul 19, 2012

### DonAntonio

Re: I don't know how to find the four square roots of an elliptic curve modulo a prim

To begin with, why not write the equation in modulo 5?
$$y^2=x^3+x+1$$

Let's now check the cubes and squares modulo 5:

$$0^2=0\,\,,\,1^2=1\,\,,\,2^2=4\,\,,\,3^2=4\,\,,\,4^2=1$$
$$0^3=0\,\,,\,1^3=1\,\,,\,2^3=3\,\,,\,3^3=2\,\,,\,4^3=4$$

We get at once the solutions
$$(0,1)\,\,,\,(0,4)\,\,,\,(2,1)\,\,,\,(2,4)\,\,,\,(3,1)\,\,,\,(3,4)\,\,,\,(4,2)\,\,,\,(4,3)$$

DonAntonio

3. Jul 20, 2012

### SneakyG

Re: I don't know how to find the four square roots of an elliptic curve modulo a prim

Thanks. How do you calculate the orders?

4. Jul 20, 2012

### DonAntonio

Re: I don't know how to find the four square roots of an elliptic curve modulo a prim

Apply the group law to the points...you know it, right? Otherwise it'll be impossible for you to understand what's

going on. You can read this in Silverman's "The Arithmetic of Elliptic Curves", for example. Let's do one of them, say:

$$(0,1)+(0,1)=(4,2)\,\,,\,\,(0,1)+(4,2)=(1,0)=0=\,\,\text{the group's zero}\,$$

So the element $\,(0,1)\in\Bbb E(\Bbb F_5)\,$ has order $\,3\,$ ...

DonAntonio