What is the real formula for Hill Cipher? (Book Correction)

Question

My book "Cryptography Theory and Practice" by Douglas R. Stinson states that the formulas for encrypting and decrypting on page 19 are follow:

For a key $K$, we define

$$e_K(x)=xK$$

and

$$d_K(y)=yK^{-1}$$

Where all operations are performed in $\mathbb{z}_{26}$

but in all sources like wikipedia, journal, etc state that the formula is:

$$e_K(x)=Kx$$

and

$$d_K(y)=K^{-1}y$$

Which one is correct? $K$ is a matrix, and i know matrices aren't commutative. Thus, those formulas are different.

I've tried encrypted and decrypted with the first formula, i got different result. But when i tried the second formula, i got the same result.

Is that mean the book is wrong or what?

Please help me to clear this. Thanks.

score 2 · Answer 1 · answered Mar 14 '20 at 02:01

2

Your book has adopted row vector convention. That is your $x$ is a row vector.

Wikipedia has adopted column vector convention. That is $x$ is a column vector.

To change one convention to the other convention, you just have to transpose everything, note that $$(xK)^T=K^Tx^T$$ where $x$ is a row vector here.

You have to fix and state your convention clearly.

answered Mar 14 '20 at 02:01

Siong Thye Goh

Hi Siong thanks for the answer. Btw is an italic T stands for Transpose? But there is no transpose on my book – user516076 Mar 14 '20 at 02:33
yes, that is because the book is consistent that they just use the row convention throughout. – Siong Thye Goh Mar 14 '20 at 02:35
Here is 1 example on page 19: encrypt "ju" with the key $$K=\begin{pmatrix} 11 & 8\ 3 & 7 \end{pmatrix}$$ so $$(9,20)\begin{pmatrix} 11 & 8\ 3 & 7 \end{pmatrix}=(99+60,72+140)=(3,4)$$ and the term $(a,b)$ is a column vector i guess? – user516076 Mar 14 '20 at 02:36
$(a,b)$ is a row vector since it is a row. $\begin{bmatrix} a\ b\end{bmatrix}$ is a column vector. – Siong Thye Goh Mar 14 '20 at 02:38
And if we want to encrypt "ju" with the formula that stated by wikipedia and another journals, it supposed to be $$\begin{pmatrix} 11 & 8\ 3 & 7 \end{pmatrix}(9,20)$$ which gives me the answer $(25,11)$ – user516076 Mar 14 '20 at 02:40
ah, sorry. Yes it's a row vector. – user516076 Mar 14 '20 at 02:42
Let $K_B$ be the matrix in your book, then the corresponding matrix in the wikipedia should be $K_W=K_B^T$. Also remember that the matrix size must be compatible to perform marix multiplication. – Siong Thye Goh Mar 14 '20 at 02:42
But still gives me different result – user516076 Mar 14 '20 at 02:42
Let us continue this discussion in chat. – Siong Thye Goh Mar 14 '20 at 02:43

1 Answers1