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14

The example is using a shorthand notation for the rotors that somewhat obscures the way they actually work. For example, the first rotor in your example, BDFHJLCPRTXVZNYEIWGAKMUSQO, actually applies the following permutation of the alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ BDFHJLCPRTXVZNYEIWGAKMUSQO Applying this rotor in the ...


6

This example is correct. The inversed versions are the inverse permutation; that is, if the forward direction is the permutation $P$, then the inverse permutation $P^{-1}$ has the property that $P^{-1}(P(X)) = X$ for all $X$. That is, if $X$ is a plaintext letter, and we run it through in the forward direction (giving us $P(X)$), and then run it through in ...


4

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 $n$ ...


3

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 ...


2

It just means vectors of dimension $n$, where each entry is in $\mathbb{Z}_q$. I.e., $(x_1, \ldots, x_n) \in \mathbb{Z}_q^n$ means that $x_i \in \mathbb{Z}_q$ for all $i = 1, \ldots, n$.


1

It contains the equivalence classes represented by $\{1,\ldots,6\}$. To elaborate on Nova's comment below, $\mathbb{Z}_p$ is the collection of congruence classes when you divide by $p$. When $p=7$, two integers represent the same congruence class when they have the same remainder when divided by $7$. As you probably know, the only possible remainders are ...


1

It means $Q$ divides $P-1$. In other words, $P-1$ is a multiple of $Q$.


1

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.



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