32-bit or 16-bits elliptic curves

Question

I would like test vectors for 32-bit or 16-bits elliptic curves like $[p, a, b, G, n, h]$ , to test the Pohlig-Hellman algorithm in order to attack ECDLP over a finite prime field $F_p$.

Does anybody know a method to generate small $F_p$ parameters or can anybody supply the test vectors?

Answer 1

Here is an example curve with smooth order $E/\mathbb{F}_p:y^2=x^3+ax+b$, generated with Magma.

\begin{align*} p &= 2^{31}-1 \\ a &= 1456400922 \\ b &= 2005615003 \\ n &= 2^5\cdot 3^7 \cdot 5\cdot 17\cdot 19^2. \end{align*}

I'd expect that if you are able to run PH, you should also be able to generate some test vectors yourself. The code is

p := 2^31-1;
Fp := GF(p);

repeat
    ct := 0;
    repeat
        a := Random(Fp);
        b := Random(Fp);
        D := 4*a^3+27*b^2;
    until D ne 0;
    E := EllipticCurve([Fp|a,b]);
    F := Factorization(#E);
    for f in F do
        if f[1] gt 25 then
            ct := 1;
        end if;
    end for;
until ct eq 0;