The first cryptographic protocol of this book, “Coin Flipping Over Telephone” (Prot 1.1), is specified using a “magic function” *f*. Let us recap two properties of this function (Property 1.1):

For every integer

*x*, it is easy to compute*f*(*x*) from*x*while given any value*f*(*x*) it is*impossible*to find any information about a pre-image*x*, e.g., whether*x*is an odd or even number.It

*impossible*to find a pair of integers (*x, y*) satisfying*x ≠ y*and*f*(*x*) =*f*(*y*).

So far, this “magic function” remains magic. No supporting evidence for the two uses of the word “impossible” has been provided, let alone the provision of a concrete realization of the function (and hence of Prot 1.1).

In fact, in ...

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