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Since you say the input strings are outputs of a TRNG, I shall take each of the terms as uniformly distributed random vectors. You define $$f(x, y)=\min\{H(x \oplus y),H(x \oplus R(y,1)),\ldots,H(x \oplus R(y,\ell-1)\} $$ which I shall model as the minimum Hamming weight of a set of $k$ randomly uniformly independently chosen binary $\ell-$tuples. Each ...

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