How to apply Pollard's Rho Method on elliptic curves to solve discrete logarithm problem in finite field?

Question

I have ElGamal signature scheme implemented in finite field $\mathbb{F}_p$. The thing is that I need to apply Pollard's Rho Method on elliptic curve $E(\mathbb{F}_p)$ to this scheme, solve discrete logarithm problem and find private key $x$. In ElGamal scheme I have $a^x\equiv b\ (mod\ p)$ where $a,\ b\in \mathbb{F}_p$ and in Pollard's Rho Method I have $Q=dP$ where $P,Q\in E(\mathbb{F}_p)$.
So how can I get $P,\ Q$ from $a,\ b$ and, after I get $d$, how can I get $x$ from it?
Is it even possible or am I totally wrong?

Answer 1

In the question you said that $a^x\equiv b\ (mod\ p)$ and $P,Q\in E(\mathbb{F}_p)$. In general case, the number of elliptic curve points $\#E(\mathbb{F}_p)$ is not equal to $p$. So these two groups are not isomorphic and your question is wrong.

If $\#E(\mathbb{F}_p)=p$ the curve is called anomalous and we can find the map $\psi : E(\mathbb{F}_p) \to \mathbb{Z}_p$ in polynomial time. In this state, $a=\psi(P),b=\psi(Q)$ and $x=d$.

The main idea behind Pollard’s rho algorithm is to find distinct pairs $(c' , d' )$ and $(c'' , d'' )$ of integers modulo $n$ such that $c' P + d' Q = c'' P + d'' Q$. So $(c' − c'' ) = (d'' − d' )d \pmod n$. Hence $d$ can be obtained by computing $$d = (c' − c'' )(d'' − d' )^{ −1} \pmod n.$$

For more details you can see page $156,168$ of "Guide to elliptic curve cryptography", Hankerson, ....

Answer 2

Pollard's Rho Method is a general method for finding discrete logarithms in a finite group. It doesn't use any of the properties of Elliptic Curves.

So just use the same algorithm, over the multiplicative group of $\mathbb F_p$ instead of $E(\mathbb F_p)$.