Why wasn't the length of key mentioned in this algebraic notation of Vigenere Cipher?

Question

Let, $M=m_ 1 m_ 2 m_ 3 ... m_ n$

and, $K=k_ 1 k_ 2 k_ 3 ... k_ m$

Then how algebraic notations of Vigenere Cipher should be?

In the following pages key-length and message-length are shown same.

This page.

This page.

This page.

Answer 1

In your notation, if $m<n$, then the key phrase is repeated in whole or in part as often as is necessary to make a key of length $n$. Thus, $k_{m+1}=k_1,k_{m+2}=k_2$, and son on.

This is explained at your second link (where I’ve corrected a few typos):

$K_i$ – $i$-th character of the key phrase (if the key phrase is shorter than the open text, which is usual, than the key phrase is repeated to match the length of the open text)

You can also see this in the example in the PDF at your first link, in which the plaintext CRYPTOGRAPHY is $12$ characters long, while the key phrase LUCK is only $4$ characters long and is therefore repeated twice to form the key LUCKLUCKLUCK of length $12$.