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The explanation of why the security parameter $k$ is given in the unary form of $1^k$ as mentioned at the second footnote at page 366 of Foundations of Cryptography Volume 2 is to allow a smooth transition to fully non-uniform formulations...Specifically $1^n$ indicates that the $n^{th}$ circuit is to be used.


That definition is a standard definition which defines encryption as a function $E$. That function takes two inputs, a $\kappa$ bit key and a $n$ bit message. Hence it is defined over the cartesian product - denoted as $\times$ - over these two sets, i.e. all bitstring of length $\kappa$ and $n$ respectively. It maps - denoted as $\rightarrow$ - to an $n$ ...


As one of the authors of the paper, let me give you an answer. The operation $F$ is indeed applied to both $x$ and $x'$. By stating that $\oplus$ is invariant under rotation, we mean that if you first rotate $x$ and $x'$ and take the difference with $\oplus$, you get the same result as if you first take the difference with $\oplus$ and then rotate the ...

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