Any encription algorithm y = E(x) takes a string x to a new string y,
which is easy to compute.  The inverse function x = E^-1(y) is a
mathematical function that is "extremely hard to compute (if the
encription algorithm is any good)  but very easy to verify."  I.e. if
you give me x for a given y, I'll just compute E(x) and verify that
it's equal to y.

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