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Let's approach the question from this direction. PAKE tries to solve this problem: Alice has a password $PW_{a}$, and Bob has a password $PW_{b}$; consider a protocol where Alice and Bob exchange messages, and generate encryption keys $E_{a}, E_{b}$ where $E_{a} = E_{b}$ if $PW_a = PW_b$ (and unrelated if $PW_a \ne PW_b$); if they're the same, Alice and Bob ...

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