Introduction to Cryptography with Coding Theory, 3rd Edition

D.10 Computations for Chapter 21

Elliptic curves

Let’s set up the elliptic curve $y^2\equiv x^3+2x+3$ mod 7:

E=EllipticCurve(IntegerModRing(7),[2,3])

The entry [2,3] gives the coefficients $a\text{,}b$ of the polynomial $x^3+ax+b$. More generally, we could use the vector [a,b,c,d,e] to specify the coefficients of the general form $y^2+axy+cy\equiv x^3+b{x}^2+dx+e$. We could also use GF(7) instead of IntegerModRing(7) to specify that we are working mod 7. We could replace the 7 in IntegerModRing(7) with a composite modulus, but this will sometimes result in error messages when adding points (this is the basis of the elliptic curve factorization method).

We can list the points on $E$:

E.points()
[(0:1:0), (2:1:1), (2:6:1), (3:1:1), (3:6:1), (6:0:1)]

These are given in projective ...