Z superscript confusion

Ricky

1/23/24, 4:51 PM

I was practicing some questions on cryptography (newbie) and came across this question:

I know that Z₂₆ means modulo-n arithmetic is used, but what does the superscript (3) denote? My guess is that the superscript represents the dimension of the square, key matrix. But, I would like it if someone confirms it for me or corrects my understanding.

1 + 0

hill-cipher

modular-arithmetic

Score:3

Crypto

Amit

1/23/24, 5:00 PM

Usually $\mathbb{Z}^3$ denotes the triple cartesian product (triplets) of the set of integers, so it is a set consisting of all triplets $(a,b,c)$ where $a,b,c \in \mathbb{Z}$

Which makes sense, because Hill is a polyalphabetic cipher where you encrypt by applying a square matrix of some dimension (3 I guess in this case) to a column vector of this same dimension.

+ 1

fgrieu

1/25/24, 3:46 PM

Yes. My reading is that the authors write $Z_{26}$ for $\mathbb Z/26\mathbb Z$ (also noted $\mathbb Z_{26}$ in some circles), and $Z_{26}^3$ when they could have written ${Z_{26}}^3$ or $Z_{26}\times Z_{26}\times Z_{26}$, meaning the set of triplets of elements of $Z_{26}$.

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