A cryptosystem is defined by a 5 tuple $(P,C,K,E,D)$=(plain text, cipher text, key, encryption function, decryption function)

So, my question is with the definition given for permutation cipher, which is stated
"If $m$ be a +ve integer, and let $P=C=(Z_{26})^m$ and let K consists of all the permutation of $\{1,\ldots,m\}$, for a key, $\pi$, we define: $$e_π(x_1,\ldots,x_m)=(x_{π(1)},\ldots,x_{π(m)})$$ $$d_π(y_1,\ldots,y_m)=(y_{π(1)}^{-1},\ldots,y_{π(m)}^{-1})$$

What I don't understand is, what does $(Z_{26})^m$ mean?

  • 2
    $\begingroup$ The obvious meaning of that notation is probably a sequence of m characters, where each character is chosen from an alphabet of 26 letters. $\endgroup$ – DannyNiu Mar 17 at 11:28

The notation $(Z_{26})^m$ means the $m$-fold cartesian product of $Z_{26}$. This is a set whose elements are all tuples of length $m$ with components in $Z_{26}$. For example, $(Z_2)^2$ means the set of tuples (0,0), (0,1), (1,0), (1,1), where $Z_2$ alone is taken to mean the set of integers modulo 2, i.e. 0 or 1. In this context, you can read $(Z_{26})^m$ as the set of strings that are $m$ English letters long.

(Beware: Sometimes $Z$ means the center of a group, not the integers, so I prefer to write the unambiguous $\mathbb Z$. Sometimes $Z_p$ or $\mathbb Z_p$ means the $p$-adic integers, so I prefer to write $\mathbb Z/p\mathbb Z$ which is the unambiguous quotient of $\mathbb Z$ by the ideal $p\mathbb Z$, i.e. the integers modulo $p$. But in this case context suggests that $Z_{26}$ means the set of integers modulo 26, corresponding to the English alphabet.)


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