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Assume $g$ is generator of multiplicative group modulo prime $p=2q+1$ where $q$ is prime.

Assume we know $g^{2t}\bmod p$ and $g^{2}\bmod p$ and assume we can have access to a Diffie-Hellman oracle.

Can we find $g^t\bmod p$ in polynomial time?

Note if we can do that we can break discrete log with access to a DH oracle when generator order is even.

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Can we find $g^t \bmod p$ in polynomial time?

We can find either $g^t$ or $-g^{t} = g^{t + (p-1)/2}$; obviously, we can't tell which one was the correct one with the information we were given.

Because $p \equiv 3 \pmod 4$ (because $(p-1)/2$ is assumed to be prime, and taking $p=5$ off the table - that can be handled as a special case), then [1] we can compute modular square-roots with the simple computation $\sqrt{x} = \pm x^{(p+1)/4}$.

So, we have $g^t \in\{ -(g^{2t})^{(p+1)/4},+(g^{2t})^{(p+1)/4}\}$, easily doable in polytime.

[1]: If $p \equiv 1 \bmod 4$, then it is still practical to compute modular squareroots, it's a bit more involved.

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  • $\begingroup$ True.. but can the oracle resolve the ambiguity in the sign? $\endgroup$
    – Turbo
    Mar 27 at 18:25
  • $\begingroup$ @Turbo: no, it can't, because both values are possible solutions for $g^t$ $\endgroup$
    – poncho
    Mar 27 at 19:31

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