IMT4531 Introduction to Cryptology
Exercise 3 – Block ciphers
1. Suppose that we have a 3-bit block cipher with 2 bits of key and that, given a fixed key, the
permutation is (5,2,7,1,8,6,4,3). Determine the vectorial Boolean functions for encipherment
and decipherment. Check encipherment of the plaintext 001 and decipherment of the
ciphertext originating from this plaintext.
1
𝑃=(
5
2 3 4 5
2 7 1 8
6 7 8
)
6 4 3
This means that the binary value of 1-1 (upper row) goes to the address 5-1, binary value of
2-1 goes to the address 2-1, binary value of 3-1 goes to the address 7-1, etc.
Vectorial Boolean function for encipherment:
Address (decimal)
Address (binary)
Value
0
000
011
1
001
001
2
010
111
3
011
110
4
100
000
5
101
101
6
110
010
7
111
100
For decipherment, we need the inverse permutation P-1.
1 2 3 4 5 6 7 8
𝑃−1 = (
)
4 2 8 7 1 6 3 5
This means that the binary value of 1-1 (upper row) goes to the address 4-1, binary value of
2-1 goes to the address 2-1, binary value of 3-1 goes to the address 8-1, etc.
Vectorial Boolean function for decipherment:
Address (decimal)
Address (binary)
Value
0
000
100
1
001
001
2
010
110
3
011
000
4
100
111
5
101
101
6
110
011
7
111
010
Encipherment of the plaintext 001:
E(001)=001
Decipherment of the ciphertext that corresponds to enciphering of the plaintext 001
E-1(001)=001
Enciphering and deciphering of some other inputs:
E(110)=010, E-1(010)=110
E(000)=011, E-1(011)=000
2. We are given a 4-bit block cipher with 5 bits of key. Suppose that the permutation that
corresponds to a fixed key is (16,12,8,11,1,9,15,6,5,2,13,4,3,7,10,14). Construct the vectorial
Boolean functions for encipherment and decipherment.
1
𝑃=(
16
2 3 4 5
12 8 11 1
6 7 8 9 10 11 12 13 14 15 16
)
3
7 10 14
9 15 6 5 2 13 4
This means that the binary value of 1-1 (upper row) goes to the address 16-1, binary value of
2-1 goes to the address 12-1, binary value of 3-1 goes to the address 8-1, etc.
Vectorial Boolean function for encipherment:
Address (decimal)
Address (binary)
Value
0
0000
0100
1
0001
1001
2
0010
1100
3
0011
1011
4
0100
1000
5
0101
0111
6
0110
1101
7
0111
0010
8
1000
0101
9
1001
1110
10
1010
0011
11
1011
0001
12
1100
1010
13
1101
1111
14
1110
0110
15
1111
0000
For decipherment, we need the inverse permutation P-1.
1 2
3 4 5 6 7 8 9 10 11 12 13 14 15 16
𝑃−1 = (
)
5 10 13 12 9 8 14 3 6 15 4 2 11 16 7
1
This means that the binary value of 1-1 (upper row) goes to the address 5-1, binary value of
2-1 goes to the address 10-1, binary value of 3-1 goes to the address 13-1, etc.
Vectorial Boolean function for decipherment:
Address (decimal)
Address (binary)
Value
0
0000
1111
1
0001
1011
2
0010
0111
3
0011
1010
4
0100
0000
5
0101
1000
6
0110
1110
7
0111
0101
8
1000
0100
9
1001
0001
10
1010
1100
11
1011
0011
12
1100
0010
13
1101
0110
14
1110
1001
15
1111
1101
3. A Feistel 8-bit block cipher has the 1st round function (for a fixed round key K1) f1(x) given by
the following table:
x
f1(x)
0000
0001
0001
0010
0010
0011
0011
0100
0100
0101
0101
0110
0110
0111
0111
1000
1000
1001
1001
1010
1010
1011
1011
1100
1100
1101
1101
1110
1110
1111
1111
0000
Given the 8-bit input to the first round 11010001, determine the output of the first round.
The Feistel structure of a round (example with 64-bit input from the lecture):
Since the 8-bit input in our case is 11010001=(L,R), we have L=1101 and R=0001. Then
L’=R=0001 and R’=Lf1(R)=1101f1(0001)=11010010=1111. So the 8-bit output of the first
round is (L’,R’)=00011111.