Discrete Mathematics
Hans Georg Schaathun
4th January 2016

# Exercises Week 11 Advanced Encryption Standard

### Wednesday 28 October 2015

Please watch the videos once before class. It may be helpful to peek at the exercises between the videos. You should not worry if you do not understand them after the first run. We will discuss the exercises in class, and if that does help you start to understand, then you are allowed to worry.

#### 1.1 Matrix calculations over finite sets

Exercise 1.1 Write the following numbers in Hexadecimal:

1. 1024
Decimal 1024 $\to$ Hexadecimal $400$
2. 129
Decimal 129 $\to$ Hexadecimal $81$
3. 721

Exercise 1.2 Write the following numbers in Octal:

1. 1024
2. 129
Decimal 129 $\to$ Octal $201$

Exercise 1.3 Write the following hexadecimal numbers in decimal:

1. FF
Hexadecimal FF $\to$ Decimal $255$
2. 17A
3. 111
Hexadecimal 111 $\to$ Decimal $273$

Exercise 1.4 Rewrite hexadecimal A9 in binary.

Hexadecimal A9 $\to$ Binary $10101001$

Exercise 1.5 Write 0155 (octal) in decimal.

Exercise 1.6 Take the word ‘Try’, find the ASCII number for each letter, write it in binary form, and encode it using the parity check code.

1. Translating to ASCII, we get 54, 72, 79 in Hexadecimal.
2. In binary, this becomes $\left(1010100\right),\left(1110010\right),\left(1111001\right)$
3. Encoding, with a parity check bit, we get $\left(11010100\right),\left(01110010\right),\left(11111001\right)$

#### 1.2 Theory of finite fields

Exercise 1.7 Calculate the following over ${ℤ}_{2}$:

Exercise 1.8 Calculate the following over ${ℤ}_{2}$:

Exercise 1.9 Calculate the following over ${ℤ}_{3}$:

1. $\left[121\right]+\left[210\right]=$
$\left[121\right]+\left[210\right]=\left[1+2,2+1,1+0\right]=\left[0,0,1\right]$

2. $\left[121\right]\cdot \left[210\right]=$

$\begin{array}{llll}\hfill \left[121\right]\cdot {\left[210\right]}^{T}& =1\cdot 2+2\cdot 1+1\cdot 0=4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Hans Georg Schaathun / hasc@hials.no