Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 3 Solutions - Ex3.1
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- Exercise 3.1
- Exercise 3.2
- Exercise 3.3
- Exercise 3.4
- Exercise 3.5
- Exercise 3.6
- Exercise 3.7
- Exercise 3.8
- Exercise 3.9
- Exercise 3.10
- Exercise 3.11
- Exercise 3.12
- Exercise 3.13
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:
- \(x_1 = 000000_2, x_2 = 000001_2\)
- \(x_1 = 111111_2, x_2 = 100000_2\)
- \(x_1 = 101010_2, x_2 = 010101_2\)
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.
\[S_1(101010_2) \oplus S_1(010101_2) \neq S_1(101010_2 \oplus 010101_2)\] \[0110_2 \oplus 1100_2 \neq S_1(111111_2)\] \[1010_2 \neq 1101_2\]
Thomas Busby
I write about computing stuff