# Block Cipher

Block cipher is an encryption method which the encryption algorithm operates on a plaintext block of n bits and produces a block of n bits ciphertext.

Block Cipher has \({2}^{n}\) possible different plaintext block to encrypt.

Nonsingular Transformation mean the encryption algorithm must be
reversible (Nonsingular) to decrypt the ciphertext into the original
plaintext.

The ciphertext must be unique for each plaintext block.

Example:
Encryption block cipher with a plaintext block length n=4 bits and ciphertext, with \({2}^{4}=16\) possible input states

plaintext | Ciphertext

0000 0101

0001 0111

0010 1001

0011 0000

0100 1100

0101 0110

0110 1010

0111 0011

1000 1011

1001 1000

1010 0010

1011 1111

1100 1110

1101 0001

1110 0100

1111 1101

This example has 16 possible input states, with a Nonsingular transformation as you can see in the next table

Ciphertext | Plaintext

0000 0011

0001 1101

0010 1010

0011 0111

0100 1110

0101 0000

0110 0101

0111 0001

1000 1001

1001 0010

1010 0110

1011 1000

1100 0100

1101 1111

1110 1100

1111 1011

As we can see this algorithm is reversible (Nonsingular).

There is a problem with a small block size n that this system is vulnerable to statistical analysis, but if n is large enough so the statistical analysis will be much harder.

Also it is not practical to establish a large enough reversible substitution cipher (The ideal block cipher), the solution of this problem is Feistel Cipher.

Feistel proposed that we can approximate the ideal block cipher system for large n ,built up by components that are easily realizable, which is the execution of 2 or more simple cipher in sequence and the result is a strong encryption.

The required form this approach is to develop a block cipher with a key length = k bits, and a block length = n bits with a total possible transformation = \({2}^{k}\) rather than \({2}^{n}\).

Feistel Cipher consists of a number of identical rounds of processing, and in each round a substitution is preformed on one half of data being processed followed by a permutation that make interchange the two halves.

The original key is expanded so that a different key is used for each round.

Feistel encryption algorithm(at the left hand side), the inputs to encryption algorithm are plaintext block of length 2w bits, and key K.

The plaintext is divided into two halves \({L}_{o}\) and \({R}_{o}\), passes through n rounds of processing, each round i has
inputs \({L}_{i}-1\) and \({R}_{i}-1\), and then combined to produce the ciphertext block.

The subkey \({K}_{i}\) is derived from the overall \(K\), \({K}_{i}\) is different from \(K\) and from each other.

Feistel Cipher actually performs two operations

**1- Substitution** is performed on the
left half of data by applying a round function F to the right half of data, then by doing XOR the output of round function F with the left
half of data.

The round function F has the same structure every round but there is a change in parameter subkey \({K}_{i}\) for each round.

**2- Permutation** is performed that consists of interchange on the two halves of data.

Feistel Cipher structure is form of substitution permutation network (SPN).

Feistel Decryption Algorithm(at the right hand side), the inputs to decryption algorithm are ciphertext and subkey \({K}_{i}\) but in
reverse order, start with \({K}_{n}\) then \({K}_{n}-1\) and so on until \({K}_{1}\) in the last round.

Note 1:At every round the value of encryption algorithm is equal to the corresponding value(n-ith) of decryption algorithm, but the two halves are swapped.

For example: For a system with 16 rounds, the value of 5th round in encryption algorithm is equal to the value of 12th round of decryption algorithm, but L and R are swapped.

Note 2:The round function F does not required to be a reversible function.

- Block size: Large block size mean greater security but reduced encryption and decryption speed.
- Key size: Large key size mean greater security but may also reduced the encryption and decryption speed.
- Number of rounds: Increasing security can be achieved by increasing the number of rounds.
- Round Function F: Increasing the complexity of round function F increasing the resistance to cryptanalysis.
- Subkey generation algorithm: Increasing of complexity here also increasing the resistance to cryptanalysis.