Note: All this is only for integer (mod n) discrete log, not rationals or anything else.
Adapt from
http://www2.math.umd.edu/~lcw/MathematicaCode/crypto6.nb
In[1]:= addell[p1_, p2_, a_, b_, n_] := Module[{z, m, x3, y3, p3, z1},
z = 0; z1 = 1;
If[p1 == {"infinity", "infinity"}, p3 = p2; z = 1, " "];
If[z == 1, " ", If[p2 == {"infinity", "infinity"}, p3 = p1; z = 1, " "]];
If[z == 1, " ",
If[p1[[1]] == p2[[1]] && p1[[2]] == p2[[2]] == 0,
p3 = {"infinity", "infinity"}; z = 1, " "]];
If[z == 1, " ",
If[p1[[1]] == p2[[1]] && p1[[2]] != p2[[2]],
p3 = {"infinity", "infinity"}; z = 1, " "]];
If[z == 1, " ",
If[p1 == p2 && GCD[p1[[2]], n] != 1 && GCD[p1[[2]], n] != n, z = 1;
z1 = GCD[p1[[2]], n], " "]];
If[z == 1, " ",
If[p1 == p2, m = Mod[(3*p1[[1]]^2 + a)*PowerMod[2*p1[[2]], -1, n], n];
z = 1; x3 = m^2 - p1[[1]] - p2[[1]]; y3 = m*(p1[[1]] - x3) - p1[[2]];
p3 = Mod[{x3, y3}, n], " "]];
If[z == 1, " ", If[GCD[p2[[1]] - p1[[1]], n] != 1,
z = 1; z1 = GCD[p2[[1]] - p1[[1]], n], " "]];
If[z == 1, " ",
m = Mod[(p2[[2]] - p1[[2]])*PowerMod[p2[[1]] - p1[[1]], -1, n], n];
x3 = m^2 - p1[[1]] - p2[[1]]; y3 = m*(p1[[1]] - x3) - p1[[2]];
p3 = Mod[{x3, y3}, n]]; If[z1 == 1, p3, {"factor=", z1}]];
Now use that to solve the discrete log problem.
dlog[{px,py},{qx,qy},a,b,n] finds the discrete log k of p*k==q on the curve y^2==x^3+ax+b mod n
In[2]:= dlog[p_, q_, a_, b_, n_] := Module[{r},
r = p;
For[k = 1, k < 2*n, k++,
r = addell[r, p, a, b, n];
If[r == q, Break[]]];
k + 1];
Test this on
http://www.certicom.com/index.php/52-the-elliptic-curve-discrete-logarithm-problem
In[3]:= dlog[{16, 5}, {4, 5}, 9, 17, 23]
Out[3]= 9
Test this on http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node89.html
In[4]:= dlog[{2, 2}, {0, 6}, 1, 1, 7]
Out[4]= 3
Test this on https://www.crm.cat/Paginas/Conferences/0607/ContentSecurity/Slides/Leprevost.pdf
In[5]:= dlog[{1, 237}, {190, 271}, 71, 602, 1009]
Out[5]= 419
Please test this on lots more examples, and possibly even reverse engineer the code to try to prove correctness, before you even begin to think about trusting it. And this is brute force sequential searching for the discrete log, so if you are thinking of trying anything more than 24 bit coefficients you should expect to wait a very long time.
And on the comment
"Seems like the standard Java "write once, debug everywhere" scenario..."
Rule 1: EVERY internet service will be abused.
Rule 2: NOBODY effectively plans for Rule 1.
Java seemed like it was very good, at least until every petty criminal and moron on the planet tried hijacking every computer they could find and web browser authors started shoveling in partly baked, broken and very late attempts at fixing their security cess pool products.