# Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 3 Solutions - Ex3.1

- 1 min

## Exercise 3.1

As stated in Sect. 3.5.2, one important property which makes DES secure is that the S-boxes are nonlinear. In this problem we verify this property by computing the output of S1 for several pairs of inputs.

Show that $S_1(x_1) \oplus S_1(x_2) \neq S_1(x_1 \oplus x_2)$, where “$\oplus$” denotes bitwise XOR, for:

1.   $x_1 = 000000_2, x_2 = 000001_2$
2.   $x_1 = 111111_2, x_2 = 100000_2$
3.   $x_1 = 101010_2, x_2 = 010101_2$

### Solution

This solution is verified as correct by the official Solutions for Odd-Numbered Questions manual.

1.

$S_1(000000_2) \oplus S_1(000001_2) \neq S_1(000000_2 \oplus 000001_2)$ $1110_2 \oplus 0000_2 \neq S_1(000001_2)$ $1110_2 \neq 0000_2$

2.

$S_1(111111_2) \oplus S_1(100000_2) \neq S_1(111111_2 \oplus 100000_2)$ $1101_2 \oplus 0100_2 \neq S_1(011111_2)$ $1001_2 \neq 1000_2$

3.

$S_1(101010_2) \oplus S_1(010101_2) \neq S_1(101010_2 \oplus 010101_2)$ $0110_2 \oplus 1100_2 \neq S_1(111111_2)$ $1010_2 \neq 1101_2$