# Meaning of the $m$ power of a modular world

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?

• The obvious meaning of that notation is probably a sequence of m characters, where each character is chosen from an alphabet of 26 letters. – 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.)