Hil 2-cipher with 26 letter alphabet

Question

A Hil 2-cipher with a 26-letter alphabet $A=1, B=2, \dots, Y=25, Z=0$ has enciphering matrix $A = \begin{bmatrix}19 & 13 \\ 6 & 3\end{bmatrix}$

Questions

1. Verify that $A$ is suitable for use as an enciphering matrix.

(Can someone please show me how to do this?)

$~~$

2. Enchipher 'REEVA'

First split the letters into groups of two: $$RE ~~ EV ~~ A$$

We can then encipher each block separately as $$\bigg( \begin{bmatrix}19 & 23 \\ 6 & 3\end{bmatrix}\begin{bmatrix}18 \\ 5\end{bmatrix} \bigg)mod \ 26 = \begin{bmatrix}15 \\ 19\end{bmatrix} = OS$$

$$\bigg( \begin{bmatrix}19 & 23 \\ 6 & 3\end{bmatrix}\begin{bmatrix}5 \\ 22\end{bmatrix} \bigg)mod \ 26 = \begin{bmatrix}3 \\ 18\end{bmatrix} = CR$$

I am unsure how I will encipher the last block, since this only consists out of one entry. How will I do this?

Answer 1

For 1, you need to verify that A is invertible. The easiest way is to check the determinant. That will mean you can decrypt the message. Do you see why?

For 2, you just pad the message with something to make enough characters. For example purposes, you often use X, so the last block would be AX. If you imagine adding one X (or, for some other codes, several X's) to the end of real English text, the receiver can easily see that they are spurious.