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Imagine that Alice generates a number $a$ and Bob generates a number $b$. Is there a way that they can check that $a \not = b$ without revealing any information on $a$ and $b$ (Alice must not know $b$ and Bob must not know $a$) ?

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Yes, this is a standard secure two computation problem. There was a classic paper discussing this from many years ago called Comparing Information Without Leaking It. There are more efficient methods today (in fact, the oblivious transfer method today is very efficient using oblivious transfer extensions).

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