elliptic curves and finite fields in Matlab

Hi,

How to work over finite fields in matlab? Finding inverses also algebraic closure of finite fields.

Also how to work with elliptic curves over finite fields in matlab specifically point addition.

To clarify I'm looking for software/built in functions that can do this not to do this myself.

[David Hill]

You may want to look at my file exchanges: secp256k1, curve448, curve25519

If you inspect the functions you will be able to see how I did point addition, and inverses.

Answer 1

[Truman]

Short answer is Matlab is not the best tool to analyze finite fields, field extensions of finite fields, elliptic curves over finite fields (or even the rationals). Matlat excells for "engineering" applications but not for general mathematical applications.

For what you want, Mathematica with its build in function over finite fields and handing of symbolic mathematics is a better choice.

Answer 2

[John D'Errico]

An inverse is trivial in MATLAB. Just use gcd. That is, if you want to solve the problem

a*x = 1, mod P

where a and P are given and relatively prime, then the inverse comes directly from

[G,C,D] = gcd(a,P).

The inverse has no solution if G ~= 1. For example...

a = 12;
P = nextprime(sym('1e12'))

P = 

1000000000039

[G,C,D] = gcd(a,P)

G = 

1

C = 

416666666683

D = 

Again, as long as they are relatively prime so that G == 1, then we have

G = C*a + D*P

Modulo P, we know that C*a == 1, and so C is the multiplicative inverse of a in the corresponding field.

If you want, you can find my modInv on the File Exchange, which does exactly this. And, GCD is a built-in part of MATLAB.

Is Mathematica better at these things? Probably so.