Beyond Lip Service: Sustaining Modelling in Curricula and Coursework

Appendix

Codes

Coding and decoding messages can be carried out using matrices with the aid of modular arithmetic. In modular arithmetic we say, for example, that $18 \equiv 4(\text{mod}\ 7)$ because the remainder is 4 when 18 is divided by 7. Modulo 26 is useful for representing the letters of the alphabet. In modulo 26 we can say, for example, $85 \equiv 7(\text{mod}\ 26)$. A positive remainder can be found for negative numbers. For example, $-15 \equiv 11(\text{mod}\ 26)$ since $26 \times -1 +11 = -15$. The multiplicative inverse can sometimes, but not always, be found. For example, $9^{-1}(\text{mod}\ 26)=3$ because $9\times 3 (\text{mod}\ 26)$ results in the identity element, 1.

Using 2 × 2 matrices, a message can be coded as shown in this example:

Starting with the message LEAVE NOW, the message is divided into pairs of letters. Each letter is represented by a number in the range from 0 to 25 according to its position in the alphabet, where the letters A–Y are numbered 1–25 and Z is numbered 0. In this case the message becomes,

$$ 12\ 5\quad 1\ 22\quad 5\ 14\quad 15\ 23 $$

The pairs of numbers are then entered as columns in a matrix, which is then pre‐multiplied by a secret 2 × 2 matrix to produce the code.

If the secret 2 × 2 matrix is $A=\begin{bmatrix} 3&1 \\ 5&4 \end{bmatrix}$, then the code would become:

$$ A=\begin{bmatrix} 3&1 \\ 5&4 \end{bmatrix}=\begin{bmatrix} 12& 1 & 5& 15 \\ 5 & 22 & 14 & 23 \end{bmatrix}=\begin{bmatrix} 41 & 25 & 29 & 68 \\ 80 & 93 & 81 & 167 \end{bmatrix}$$

which is $\begin{bmatrix} 15 & 25 & 3 & 16 \\ 2 & 15 & 3 & 5 \end{bmatrix}$ mod 26 and which represents: OB YO CC PE.

If the secret 2 × 2 matrix is known, then the message can be decoded by pre‐multiplying it by A ⁻¹ where $A^{-1}=(ad - bc)^{-1}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}(\text{mod}\ 26)$ and $(ad - bc)^{-1}$ is the multiplicative inverse of $(ad - bc)$ in modulo 26.

• a) Imagine you are a spy. Compose a message comprising of at least 8 letters and code the message using a suitable secret 2 × 2 matrix. Use MATLAB to perform the matrix operations and submit a print‐out of your working.

• b) Show how the coded message you created in part (a) can be decoded using the secret 2 × 2 matrix. Use MATLAB to perform the matrix operations and submit a print‐out of your working.

Now compose a message with at least 50 letters. Using a suitable 4 × 4 matrix, code the message and show how it can be decoded. Use MATLAB to perform the matrix operations and submit a print‐out of your working.

Describe one strength and one limitation of the methods you used in this investigation to create codes (included two additional tasks including an open problem).

• (a) Identify one significant assumption that was made in this investigation regarding language and describe, using examples, how you could adapt your method to allow for this assumption.

• (b) Comment briefly on the reasonableness of your results with reference to real world data.