1-to-1 cryptographically secure bit shuffling

Given an input item (N bytes), I'm looking for a function that will map this to an output (still N bytes). The function should have the following qualities:

• It should be 1-to-1 so that all inputs map to some output and so that no two inputs map to the same output.

• Given an output element it should be difficult to guess the input that lead to that output, even when the mapping is completely known.

Does such a function exist? Where can I learn more?

This is known as a one-way permutation. The "permutation" refers to the first of your two requirements; the "one-way" refers to the second of your two requirements. There are various candidate constructions for one-way permutations, e.g., based on raising to the third power modulo an RSA modulus or other schemes.

StackExchange Computer Science Q#50952, answer score: 10