Q1) A randomized algorithm A taking k inputs x1, . . . , xk can be viewed as a deterministic algorithm taking k + 1 inputs x1, . . . , xk, r, where the (randomized) output of A(x1, . . . , xk) is determined by choosing a sufficiently-long random string r and then outputting the (deterministic) value A(x1, . . . , xk; r). (In this case, r is called the random coins used by A, and we distinguish it from other inputs by using a semicolon instead of a comma.)

Prove that, in the context of private-key encryption, we can assume without loss of generality that keys are chosen uniformly at random (and so Gen is trivial). I.e., show that for any encryption scheme (Gen′, Enc′,Dec′) (over any message space), there is a functionally equivalent encryption scheme (Gen, Enc,Dec) where Gen simply outputs its random coins. (I am looking for formal specifications of Enc and Dec.)