Given $n$-bit block cipher $E$ (and its inverse $E^{-1}$), define block cipher $E^\prime_k(m) = E_k(E_{f(k)}^{-1}(m))$ where $k,f(k) \in \{0,1\}^n$ and $\forall k:f(k) \ne k$. Under the ideal block cipher model, there exists no function $f$ which would give an attacker an advantage against $E^\prime$. Are there any real block ciphers for which any $E^\prime$ would be weaker than $E$ (excluding those with equivalent keys, of course)?
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