Using Pari for elliptic curves
I recently tried Pari, a piece of software that’s designed for doing number-theoretical calculations – like those you need to do in cryptography. After using it for a few hours, I have to say that I’m very impressed by it. It has lots of built-in functions for doing calculations involving elliptic curves, and because lots of the technology that we use at Voltage involves elliptic curves, I found that very useful.
On the other hand, Pari seems to assume that you already know a lot about things before you start using it. Here’s an example of defining the elliptic curve y2 = x3 + 1 and finding all of the points of finite order on the curve:
The ellinit() function initializes a data structure for an elliptic curve, but what it tells you when it does this probably isn’t useful to most people. Here's how the Pari User's Guide explains the output of ellinit():
a1–a6,b2–b8,c4–c6: coefficients of the elliptic curve
area: volume of the complex lattice defining E
disc: discriminant of the curve
j: j-invariant of the curve
omega: [ω1,ω2], periods forming the basis of the complex lattice defining E (ω1 is the real period and ω2 belongs to Poincare's half-plane).
eta: quasi-periods [η1,η2] such that η1ω2 – η2ω1 = 2πi.
roots: roots of the associated Weierstrass equation
tate: [u2,u,v] in the notation of Tate
w: Mestre's w (this is technical)
The elltors() function finds all of the points of finite order. Its output is slightly more user-friendly, but it’s probably not obvious to most people what it’s telling you.
So overall, I’d have to say that I’m very impressed with Pari, and I’ll probably be using it a lot in the future. On the other hand, I can’t really recommend it for most people. If you feel comfortable reading Silverman’s The Arithmetic of Elliptic Curves, you’ll probably find it very useful. You'll also understand how to interpret the output of ellinit(). Otherwise, you might find its output a bit cryptic and tricky to interpret.