This note is in support of elliptic curve crypto which is presumably the main current application of elliptic curve arithmetic but first some simple math concerning elliptic curves.
For computational elliptic curve stuff we first choose a finite field and then parameters a and b, from that field, for the equation:
y2 = x3 + ax + b . Pairs of field values (x, y) that satisfy this equation are called ‘points on the curve’. I show here how to compute with finite fields but we need little or none of that code here. All of this logic works when the field is the reals or rationals in which case your intuition leads you to some useful insights. Recall, however that finite fields seem to have no useful graphics to lean on and you must ground your results in the logic of the situation.
In analogy to ordinary Cartesian geometry a straight line is a set of points (x, y) that satisfy an equation cx + dy = e for some fixed field values for c, d and e, not all zero. If p and q are distinct points on the curve there will be a unique line thru them and generally this line will intersect the curve at some other unique point r. In this case we write r = U(p, q). Sometimes a line intersects the curve at just two points and then we add the ‘point at infinity’ which we must allow for computationally. Also there is a concept of tangent to the curve at a point p on the curve and then U(p, p)