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For a scheme to be information-theoretically secure, you need that $$\Pr[M=m\mid C=c]=\Pr[M=m\mid C=c^\prime]$$ for all $c,c^\prime$ (that is, any ciphertext has the same probability $M=m$, so the ciphertext doesn't change the probability $M=m$). Let's suppose we have a $c$ and a $c^\prime$. Both of them have the same number of ones and zeroes, because both ...