Simplified International Data Encryption Algorithm (IDEA)

22 Oct, 2021
In cryptography, block ciphers are very important in the designing of many cryptographic algorithms and are widely used to encrypt the bulk of data in chunks. By chunks, it means that the cipher takes a fixed size of the plaintext in the encryption process and generates a fixed size ciphertext using a fixed-length key. An algorithm’s strength is determined by its key length. 
The Simplified International Data Encryption Algorithm (IDEA) is a symmetric key block cipher that:  
uses a fixed-length plaintext of 16 bits and
encrypts them in 4 chunks of 4 bits each
to produce 16 bits ciphertext.
The length of the key used is 32 bits.
The key is also divided into 8 blocks of 4 bits each.
This algorithm involves a series of 4 identical complete rounds and 1 half-round. Each complete round involves a series of 14 steps that includes operations like:  
Bitwise XOR
Addition modulo 
Multiplication modulo +1
After 4 complete rounds, the final “half-round” consists of only the first 4 out of the 14 steps previously used in the full rounds. To perform these rounds, each binary notation must be converted to its equivalent decimal notation, perform the operation and the result obtained should be converted back to the binary representation for the final result of that particular step. 
Key Schedule: 6 subkeys of 4 bits out of the 8 subkeys are used in each complete round, while 4 are used in the half-round. So, 4.5 rounds require 28 subkeys. The given key, ‘K’, directly gives the first 8 subkeys. By rotating the main key left by 6 bits between each group of 8, further groups of 8 subkeys are created, implying less than one rotation per round for the key (3 rotations). 
 
 
* denotes a shift of bits 
Notations used in the 14 steps: 
 
The 16-bit plaintext can be represented as X1 || X2 || X3 || X4, each of size 4 bits. The 32-bit key is broken into 8 subkeys denoted as K1 || K2 || K3 || K4 || K5 || K6 || K7 || K8, again of size 4 bits each. Each round of 14 steps uses the three algebraic operation-Addition modulo (2^4), Multiplication modulo (2^4)+1 and Bitwise XOR. The steps involved are as follows:  
X1 * K1X2 + K2X3 + K3X4 * K4Step 1 ^ Step 3Step 2 ^ Step 4Step 5 * K5Step 6 + Step 7Step 8 * K6Step 7 + Step 9Step 1 ^ Step 9Step 3 ^ Step 9Step 2 ^ Step 10Step 4 ^ Step 10
X1 * K1
X2 + K2
X3 + K3
X4 * K4
Step 1 ^ Step 3
Step 2 ^ Step 4
Step 5 * K5
Step 6 + Step 7
Step 8 * K6
Step 7 + Step 9
Step 1 ^ Step 9
Step 3 ^ Step 9
Step 2 ^ Step 10
Step 4 ^ Step 10
The input to the next round is Step 11 || Step 13 || Step 12 || Step 14, which becomes X1 || X2 || X3 || X4. This swap between 12 and 13 takes place after each complete round, except the last complete round (4th round), where the input to the final half round is Step 11 || Step 12 || Step 13 || Step 14. 
After last complete round, the half-round is as follows:  
X1 * K1X2 + K2X3 + K3X4 * K4
X1 * K1
X2 + K2
X3 + K3
X4 * K4
The final output is obtained by concatenating the blocks. 
Example:  
Key: 1101 1100 0110 1111 0011 1111 0101 1001 
Plaintext: 1001 1100 1010 1100
Ciphertext: 1011 1011 0100 1011
Explanation: The explanation is only for 1st complete round (the remaining can be implemented similarly) and the last half-round.  
Round 1: From the plaintext: X1 – 1001, X2 – 1100, X3 – 1010, X4 – 1100  From the table above: K1 – 1101, K2 – 1100, K3 – 0110, K4 – 1111, K5 – 0011, K6 – 1111   
From the plaintext: X1 – 1001, X2 – 1100, X3 – 1010, X4 – 1100  
From the table above: K1 – 1101, K2 – 1100, K3 – 0110, K4 – 1111, K5 – 0011, K6 – 1111  
 
(1001(9) * 1101(13))(mod 17) = 1111(15)
(1100(12) + 1100(12))(mod 16) = 1000(8)
(1010(10) + 0110(6))(mod 16) = 0000(0)
(1100(12) * 1111(15))(mod 17) = 1010(10)
(1111(15) ^ 0000(0)) = 1111(15)
(1000(8) ^ 1010(10)) = 0010(2)
(1111(15) * 0011(3))(mod 17) = 1011(11)
(0010(2) + 1011(11))(mod 16) = 1101(13)
(1101(13) * 1111(15))(mod 17) = 1000(8)
(1011(11) + 1000(8))(mod 16) = 0011(3)
(1000(8) ^ 1111(15)) = 0111(7)
(1000(8) ^ 0000(0)) = 1000(8)
(0011(3) ^ 1000(8)) = 1011(11)
(0011(3) ^ 1010(10)) = 1001(9)
 
Round 1 Output: 0111 1011 1000 1001 (Step 12 and Step 13 results are interchanged)
Round 2: From Round 1 output: X1 – 0111, X2 – 1011, X3 – 1000, X4 – 1001  From the table above: K1 – 0101, K2 – 1001, K3 – 0001, K4 – 1011, K5 – 1100, K6 – 1111  Round 2 Output: 0110 0110 1110 1100 (Step 12 and Step 13 results are interchanged)
From Round 1 output: X1 – 0111, X2 – 1011, X3 – 1000, X4 – 1001  
From the table above: K1 – 0101, K2 – 1001, K3 – 0001, K4 – 1011, K5 – 1100, K6 – 1111  
Round 2 Output: 0110 0110 1110 1100 (Step 12 and Step 13 results are interchanged)
Round 3: From Round 2 Output: X1 – 0110, X2 – 0110, X3 – 1110, X4 – 1100  From the table above: K1 – 1101, K2 – 0110, K3 – 0111, K4 – 0111, K5 – 1111, K6 – 0011  Round 3 Output: 0100 1110 1011 0010 (Step 12 and Step 13 results are interchanged)
From Round 2 Output: X1 – 0110, X2 – 0110, X3 – 1110, X4 – 1100  
From the table above: K1 – 1101, K2 – 0110, K3 – 0111, K4 – 0111, K5 – 1111, K6 – 0011  
Round 3 Output: 0100 1110 1011 0010 (Step 12 and Step 13 results are interchanged)
Round 4: From Round 3 Output: X1 – 0100, X2 – 1110, X3 – 1011, X4 – 0010  From the table above: K1 – 1111, K2 – 0101, K3 – 1001, K4 – 1101, K5 – 1100, K6 – 0110  Round 4 Output: 0011 1110 1110 0100 (Step 12 and Step 13 results are interchanged)
From Round 3 Output: X1 – 0100, X2 – 1110, X3 – 1011, X4 – 0010  
From the table above: K1 – 1111, K2 – 0101, K3 – 1001, K4 – 1101, K5 – 1100, K6 – 0110  
Round 4 Output: 0011 1110 1110 0100 (Step 12 and Step 13 results are interchanged)
Round 4.5: From Round 4 Output: X1 – 0011, X2 – 1110, X3 – 1110, X4 – 0100  From the table above: K1 – 1111, K2 – 1101, K3 – 0110, K4 – 0111  Round 4.5 Output: 1011 1011 0100 1011 (Step 2 and  Step 3 results are not interchanged) 
From Round 4 Output: X1 – 0011, X2 – 1110, X3 – 1110, X4 – 0100  
From the table above: K1 – 1111, K2 – 1101, K3 – 0110, K4 – 0111  
Round 4.5 Output: 1011 1011 0100 1011 (Step 2 and  Step 3 results are not interchanged)
 
 
(0011(3) * 1111(15))(mod 17) = 1011(11)
(1110(14) + 1101(13))(mod 16) = 1011(11)
(1110(14) + 0110(6))(mod 16) = 0100(4)
(0100(4) * 0111(7))(mod 17) = 1011(11) 
 
Final Ciphertext is 1011 1011 0100 1011
NOTE: For every round except the final transformation, a swap occurs, and the input is given to the next round 
 
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