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The original question was: Alice knows: $a,b$ and $x$ such that $a^{(x\cdot x)} = b$ Bob knows: $a,b$ and DrLecter referenced this paper (fixed the link), which covers the question. Now, the question was changed to Alice knows: $b$ and $x$ such that $x^x=b$; Bob knows: $b$. The given structure was: ... multiplicative group $G$ ...


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Since Blum-Micali is cryptographically secure, you know that there won't be a short cycle (not one short enough to detect in polynomial time), because a short cycle would violate cryptographic security. Therefore, you don't need to worry about this. It's not worth your time worrying about it: it's very unlikely you'll run into a short cycle. That said, ...



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