AES – Advanced Encryption Standard

AES is based on the substitution-permutation network principle and is similar to Rijndael cipher. It’s block length is fixed to be 128 bits and key length can vary like 128, 192 or 256 bits where the encryption requires 10, 12 and 14 rounds respectively to get the cipher text. In this every input and output of an operation is termed as a state which is an array of 4X4 bytes.

Eg. Plain Text : AES USES A MATRIX ZZ

Hexadecimal : 00 04 12 14 12 04 12 00 0C 00 13 11 08 23 19 19

State representation

AES encryption involves four transformations which transforms the state given as the input to a different state. The four transformations are:

1. Sub Bytes
2. Shift Rows
3. Mix Columns
4. Add Round key

Sub Bytes Transformation

The values in the state is substituted by another values according to a lookup table called S-box.

S-Box

Shift Rows Transformation

ShiftRows method operates on the rows of the state, the nth row is shifted left circular by n-1 bytes.

Mix Columns Transformation

In this step four bytes of each column is replaced by doing multiplication with a fixed matrix shown below,

Matrix multiplication with hexadecimal is a combination of multiplication and addition similar to the normal matrix multiplication. The multiplication part follow the following mentioned rules:

Multiplication with 1: No change in the value

Multiplication with 2: Left shift with no carry. In addition to that XOR with 0X1B if the shifted bit is 1.

Multiplication with 3: Perform multiplication with 2 then XOR with the initial value.

While the addition part is simply XOR.

Add Round key Transformation

Here each byte of the input state is combined with corresponding byte of the subkey by doing bitwise XOR. The subkey for each round is generated by Rijndael’s key expansion method.

Key expansion in AES

Key expansion is done using three operations described below:

1. Rotate : A 32 bit word is rotated eight bits to the left.
   • 08 23 19 0C   =>   23 19 0C 08
2. S-box : 
   

   S-Box

   The value is replaced by using the lookup table given above as follows :

   •   C3   =>   2E
3. Rcon : The value is XORed with the round constant depending on the round number.

Rcon constants for different rounds

Key used for encryption is of 128 bits arranged as four words,

(i) If i is not a multiple of 4, w_i is formed as,

w_i = w_i-1 ⊕ w_i-4

(ii) If i is a multiple of 4, w_i is formed as,

w_i = temp ⊕ w_i-4

temp is a 32 bit word formed from w_i-1 in the following manner :

w_i-1  =>  Left shift 8 bits  =>  S-Box  =>  ⊕ Rcon_i/4  =>  temp_i

Key expansion in AES

Now, the AES encryption can be done in 11 rounds, where the initial round, Round 0 involves an Add Round Key transformation with the initial key comprising of w₀, w₁, w₂ and w₃. Following 9 rounds comprises of Sub Bytes, Shift Rows, Mix Columns followed by Add Round key with the subkey generated for that round. The final round involves all the above rounds except Mix Columns. After that the cipher text is ready.

AES encryption and decryption

 

DES – Data Encryption Standard

DES is a block cipher which is based on a Fiestel Cipher structure where encryption is carried out in multiple rounds, each of which uses a different subkey derived from the original key. The encryption/decryption is done on blocks of 64 bit with a 56 bit long key. The encryption/decryption completes in 16 rounds, where each round uses a 48 bit subkey.

Each round follows the following steps:

• Partition of the input block into two equal halves
• DES function of the right half and the subkey
• XOR of the left half with the output of the function

Now let me explain each steps in detail,

DES Encryption:

1. The bits of the plain text is permuted by the initial permutation table which is shown in the following figure.

Eg. Plain Text, M = 0123456789ABCDEF

M (in binary) = 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

M (after permutation) = 1100 1100 0000 0000 1100 1100 1111 1111 1111 0000 1010 1010 1111 0000 1010 1010

IP Table

2. The permuted bits is then divided into two equal halves L₀ and R₀

L₀  = 1100 1100 0000 0000 1100 1100 1111 1111

R₀ = 1111 0000 1010 1010 1111 0000 1010 1010

Division of the bits

3. L_i = R_i-1

    R_i = L_i-1 ⊕ f(R_i-1, K_i)

Finding f(R_i-1, K_i)

DES function

Let the key given be K = 133457799BBCDFF1

K (in binary) = 0001 0011 0011 0100 0101 0111 0111 1001 1001 1011 1011 1100 1101 1111 1111 0001

The 64 bit key is permuted by the key permutation table which is given below, to obtain a 56 bit key after discarding the parity bits.

K (after permutation) = 1111 0000 1100 1100 1010 1010 1111 0101 0101 0110 0110 0111 1000 1111

Key permutation table

Now divide K into two equal parts, Co and Do containing 28 bits each

Co = 1111 0000 1100 1100 1010 1010 1111

Do = 0101 0101 0110 0110 0111 1000 1111

From this we have to obtain C_i and D_i for 16 rounds from which the subkeys are created. The C_i and D_i for the first round is obtained by left shifting Co and Do by 1,

C₁ = 1110 0001 1001 1001 0101 0101 1111

D₁ = 1010 1010 1100 1100 1111 0001 1110

By permuting C₁ and D₁ by the following table the subkey K₁ is obtained which is of 48 bit long.

K₁ = 0001 1011 0000 0010 1110 1111 1111 1100 0111 0000 0111 0010

Key – PC-2

Likewise C₂ and D₂ is obtained from C₁ and D₁ by doing left shift by 1 and K₂ is obtained by passing C₂ and D₂ to PC-2. Similarly C_i and D_i is obtained from C_i-1 and D_i-1 by doing left shift according to the following table,

____________________________________________________________________________________
|   Round   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|Left-Shifts| 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |  2 |  2 |  2 |  2 |  2 |  2 |  1 |
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 Now, the R₀ has to be passed to an extension table which makes it 48 bits from 32 bits,

Expansion permutation table

Thus, E(R₀) = 0111 1010 0001 0101 0101 0101 0111 1010 0001 0101 0101 0111

E(R₀) ⊕ K₁ = 0110 0001 0001 0111 1011 1010 1000 0110 0110 0101 0010 0101

The result of E(R₀) ⊕ K₁ is written in form of 8 rows with 6 bits each as follows,

B₁ = 011000              B₂ = 010001              B₃ = 011110             B₄ = 111010

B₅ = 100001              B₆ = 100110              B₇ = 010100            B₈ = 100101

There are 8 S-boxes which gives a 4 bit output when a 6 bit input is given, each B_i is given as input to S-boxes of corresponding i value.

S-Box 1

S-Box 2

S-Box 3

S-Box 4

S-Box 5

S-Box 6

S-Box 7

S-Box 8

How to lookup in a S-Box?

It’s simple, the first and last bit determines the row and the other four determines the column and the value inside that box is the required 4 bit output.

Output from 8 S-boxes

The result from the S boxes => 0101 1100 1000 0010 1011 0101 1001 1110 is then permuted by the following permutation table,

After doing the permutation what we finally get is the value of f(R₀, K₁). It is then XOR with L₀ to get R₁. Similarly the process is repeated 16 times. After the 16th round L₁₆ and R₁₆ is reversed ie,

 L₁₆ = 0100 0011 0100 0010 0011 0010 0011 0100
R₁₆ = 0000 1010 0100 1100 1101 1001 1001 0101

becomes, R₁₆L₁₆ => 0000 1010 0100 1100 1101 1001 1001 0101 0100 0011 0100 0010 0011 0010 0011 0100 which is permuted with the final permutation table IP^-1 to get the cipher text.

DES Decryption:

Decryption is also done in the same manner in the reverse order.

Encryption and Decryption