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I am having a bit of difficulty understanding what this notation is saying exactly and hoping someone could work it out.
Let F: {0,1}^k -> {{0,1}^k -> {0,1}^k}, k-> F_k be a pseudo random function .
From what I gather, F is a function that takes in a string of 0,1 of length k. it then transforms that string but its that transformation I am confused by
It looks like it takes in a $k$-bit integer (the seed, presumably), and returns a function that maps $k$-bit integer inputs to $k$-bit integer outputs. (So it's more a pseudo-random function generator than the pseudo-random function itself; given the same seed, it produces the same pseudo-random function.) Without more context, it's a bit difficult to say for sure.