# how to solve discrete logarithm problem information requested

I have seen your example about ecdlp solver :The Problem is as follows:

$$E\backslash GF(p):y^2=x^3+17230x+22699$$

where $$p=23981$$, point $$G$$ with prime order $$|G| = 109$$

Alice creates a public key by selecting a private key $$d, public key $$Q=[d]G = (3141,12767)$$

Therefore;

• public information : $$a,b,p,G,q,Q$$
• private key : $$d$$

However, this curve has the following characteristic:

$$\Delta= −16( 4a^3+27b^2) \bmod p =0$$ That is, the discriminant is 0. and embedding degree is 2

I, however, don't understand where comes out the value (23796,0) i.e how can I calculate it? and then the following formula equation: after which we obtain the corresponding curve $$y^2=x^3+23426x^2$$

• is it possible for a further explanation of how you determine the point $$(27396,0)$$?
• Is there still anything missing from the answer that keeps you from accepting it? – Maarten Bodewes Dec 12 '19 at 20:07

Let $$f(x,y) = -y^2 + x^3+17230x+22699$$ over $$\Bbb F_p$$ with $$p=23981$$. A point on the curve is a singular point if and only if the partial derivatives are vanishes at that point. The partial derivatives are;

• $$\frac{\partial f}{\partial x} = 3x^2 + 17230 =0 \pmod p$$ and vanishes at $$x=\{185,23796\}$$ found by WolframAlpha or it can be found by Tonelli-Shanks.

• $$\frac{\partial f}{\partial y} = -2y = 0 \pmod p$$ and vanishes at $$y=0$$

The vanishing point $$(185,0)$$ is not on the curve, however, $$(23796,0)$$ is.

Therefore $$(23796,0)$$ is a singular point for the Curve.

Then we then translate the origin to this singular point $$(23796,0)$$, that is replace $$(x,y)$$ by $$(x+23796,y+0$$) in the equation of $$E$$, yielding the equation of the curve in the shifted referential: $$y^2 = x^3 + 23426x^2$$ The rest is in the answer which the OP had trouble with.

Note: credit goes to @kelalaka for the better of the present answer, in particular introducing $$f(x,y)$$ and its partial derivatives.

• We begin with the singular curve y2=x3+17230x+22699. This curve is singular, as can be immediately determined by its 0 discriminant. Furthermore, it has a singular point (23796,0), where both partial derivatives vanish. We translate the curve to have this singular point at (0,0) by changing thanks for your explaination.variables (x,y)↦(x−23796,y−0), after which we obtain the corresponding curve y2=x3+23426x2, which can be rewritten as y2=x2(x+23426). – fabio Dec 9 '19 at 20:25
• thanks for your explaination, after i have obtained the point of the curve, (23796,0) how can i obtain the value 23426 put in the script? how can i obtain the new equation: y2=x^3+23426^x2? – fabio Dec 9 '19 at 20:27
• @fabio did you apply the change of variables? – kelalaka Dec 9 '19 at 20:28
• @kelalaka..can you write me as i can do the change of variables please? – fabio Dec 9 '19 at 20:35
• Thanks @kelalaka – fabio Dec 9 '19 at 20:40