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how does $A$ know $g^y$ to include in the formula above? Actually, what they mean here is, in fact, $g^y \bmod p$; that is, the value that $A$ received. It wouldn't work to insert the literal values $g^x$ and $g^y$; apart from the fact that $A$ doesn't know the second one, there's also the practical difficulty that since $x$ and $y$ are (perhaps) 256 ...

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