X = {0, 1}

E: K x X ---> Y

E(k, x) = x xor k

E is a secure PRP

How do we understand it?

**Permutation.**

A function f: X ---> Y is a permutation means that it is an one-to-one and onto function, and the domain X is equal to the codomain Y.

**Pseudo Random Permutation (PRP)**

The function E(k, .) is a PRP means that E(k, .) is a permutation and k is a random variable that defines the permutation.

**Secure PRP.**

The function E(k, .) is a Secure PRP means that it is indistinguishable from a random function in the set of all permutations.

**The above 1-bit PRP is secure.**

We can imagine that there are two worlds as the below picture, World 0 and Word 1. World 0 is for 1-bit PRP and World 1 is for truly random permutation for the 1-bit.

There are two permutations in World 0 and they are same in World 1. We can say that World 0 is equals World 1. An adversary cannot distinguish both worlds. Therefore the 1-bit PRP is secure.

-Count

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