CFB (Cipher FeedBack) as a Stream Cipher

This section describes how DES CFB (Cipher FeedBack) operation mode can be modified as a 1-bit stream cipher or a 1-byte stream cipher.

CFB (Cipher FeedBack) operation mode as a block cipher can be described with notations defined earlier as the following formula and diagram:

```C[i] = P[i] XOR E(C[i-1])
C = P XOR E(IV)

IV
|        -----|        -----|
E()     /     E()     /     E()
|     /       |     /       |
|    /        |    /        |
P--XOR  /  P--XOR  /  P--XOR
|  /          |  /          |
| /           | /           |
C          C          C
```

As you can see from the formula and the diagram, in CBC mode, each block of plaintext is XORed with the encrypted version of the previous ciphertext to generate the current ciphertext block. In this way, each ciphertext block is depending on all plaintext blocks up to that point. Note that for the first block, the Initial Vector (IV) is used as the previous ciphertext block.

In order to run the CFB operation mode as a stream cipher, FIPS 81 defines CFB variations where plaintext blocks can have any size less than 64 bits. To describe CFB variations, we need the following additional notations:

k - Representing the size plaintext blocks. k can have a value between 1 and 64.

Fk() - Representing a filter function that take the first k bits of a 64-bit block.

I = I, I, I, ..., I[i], ... - Representing the input block, I, used as input to the DES encryption process.

Sk() - Representing a shifting function that shifts k bits out of the input block from the left side. The missing k bits are taken from the ciphertext block.

With the above notations, a k-bit CFB operation mode variation can be described as:

```C[i] = P[i] XOR Fk(E(I[i]))
I[i] = Sk(I[i-1], C[i-1])
I = IV

IV
|
I------Sk()--I------Sk()--I
|       /       |       /       |
E()     /       E()     /       E()
|     /         |     /         |
Fk()   /        Fk()   /        Fk()
|   /           |   /           |
P--XOR /     P--XOR /     P--XOR
| /             | /             |
C            C            C
```

Note that there are some special cases of CFB variations:

• The 64-bit CFB variation is the original CFB operation mode.
• The 8-bit CFB variation is a stream cipher that takes 1 byte (8 bits) from the plaintext message at a time.
• The 1-bit CFB variation is a stream cipher that takes 1 bit from the plaintext message at a time.