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I am looking for a trapdoor one-to-one function $f : \{0,1\}^n \rightarrow \{0,1\}^n$ (that is, mapping the set of $n$ bit numbers 1-1 onto itself [any $n$ is OK]).

I was considering the use of something like AES-256 but unfortunately AES is symmetric.

I don't think plain RSA will work since I want powers of two.

Basically, I want something like RSA except instead of working with a mod a semi-prime $N$, I want to work mod $2^n$.

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