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This construction is well-known as XORP = "XOR of independent permutations". In your case the permutations are obtained by invoking a block cipher $E$ with different keys. If $E$ is an ideal cipher, and $k_1, k_2$ are distinct (fixed, public) keys, then $E_{k_1}(\cdot)$ and $E_{k_2}(\cdot)$ are independent, ideal permutations. Ideal permutation ...


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Function $H$ must be defined as $H(x_1, x_2) = E_{k_1}(x_1) \oplus E_{k_2}(x_2)$ where $x_1, x_2$ are separate inputs for it to be compressive. That's not to say that $x_1 = x_2$ cannot happen by chance, but they must be from separate streams if you will. Compression in this case means a loss of information whereby less bits are output by the 'compressor' ...


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