quickstart.magma

load "utils.magma";

// Define the parameters: R_k = F_{p^e}[X]/<X^k>
p := 7; k := 3; e := 3; 

// Construct R_k
q := p^e;
Fq<gamma> := GF(q); // gamma is the primitive element of Fq
F<eps> := PolynomialRing(Fq);
I := ideal<F | eps^k>;
Rk<eps> := F/I;
// printf "R_k: %o\n", Rk;

// Set the curve, with 2 or 5 coefficients in R_k (casting is important!)
E := [Rk | 6, 2];
//printf "E: %o\n", E;
// printf "Is a valid EC: %o\n", IsEC(E); // Check determinant

// Get a random curve over Rk - automatically checks det
E := RandomCurve(Rk); 
// printf "Random E: %o\n", E;

// Pick two points and check if they are on the curve
P := [Rk | 0, 1, 0];
Q := [Rk | gamma^34*eps^2 + 1, gamma^21*eps+2, 0];
Q := SF(Q); // Standard Form

// printf "P: %o -> Is on E: %o\n", P, IsOn(P, E);
// printf "Q: %o -> Is on E: %o\n", Q, IsOn(Q, E);


// Pick random Q on the curve
Q := RandomEPoint(E, k); //k is used to lift the point - automatic detection shoul be added
// printf "Random Q: %o\n", Q;

// Point addition
printf "P+Q: %o\n", add(P, Q, E);
printf "Q+Q: %o\n", add(Q, Q, E);

// Point multiplication
printf "5Q: %o\n", mul(Q, 5, E);

// Points over O
P := [Rk | 0, 1, 0];
P := RandomLift(P, E, k);

// Take a generic curve on F_q with random points
E := EllipticCurve([GF(q) | 2, 6]);

P := Random(E);
Q := Random(E);
P; Q; P+Q;

// Lift the curve
E := [Rk | i: i in Coefficients(E)];
E;

// Lift the specific points
P := RandomLift(P, E, k);
Q := RandomLift(Q, E, k);

// Point addition again - we get a lift of P+Q since projection is hom
P; Q; add(P, Q, E);