Tag Archive for 'encryption'

Output Feedback Mode

Published on March 18, 2009 in Miscel. 0 Comments

[ Sometime the LATEX does not render properly. Just refresh the page and it should do. ]

OFB Encryption

OFB Decryption

Output feedback mode is similar to CFB mode except that the quantity XORed with each plain text block is generated independently of both the plain text and cipher text. An initialization vector $s_0$ is used as a seed for a sequence of data blocks $s_i$ , and each data block $s_i$ is derived from the encryption of the previous data block $s_{i-1}$ . The encryption of a plain text block is derived by taking the XOR of the plain text block with the relevant data block.

It is essential for security that the initial value is chosen randomly and independently from the previous ones. This prevents almost with certainty that the same initial value $s_0$ is used for more than one encryption.

A transmission bit error in block $c_i$ only affects the decryption of that block. The block recovered from $c_i$ has bit errors precisely where $c_i$ did. However, the output feedback mode will not recover from a lost cipher text block – all following cipher text blocks will be decrypted incorrectly.

The speed of encryption is identical to that of the block cipher. Even though the process cannot easily be parallelized, time can be saved by generating the key stream before the data is available for encryption.

The output feedback mode is implemented by the following algorithm

Algorithm:

bitStream ofbEncrypt(bitStream $m$ , $s_0$ )

divide $m$ into $m_0m_1...m_l$

for $i \gets 1$ to $l$ do

$c_i \gets m_i \oplus {msb_r}{({E_K}({s_i}))}$

$x_i \gets {E_k}({s_i})$

return $c_1c_2...c_l$

[ This is a part of a series of post on Modes Of Encryption. I had to scribe a lecture as a requirement of a course on the Foundations Of Cryptology at the Indian Institute Of Technology. The scribe has been broken into smaller chunks so that it is easily readable. ]

Tags: cryptography, encryption, iit, kgp, scribe.

Cipher Block Chaining

Published on March 18, 2009 in Miscel. 0 Comments

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In the cipher block chaining mode, each block of plain text is XORed with the previous cipher text block before being encrypted. This was, each cipher text block is dependent on all the plain text blocks that have been processed up to that point. Also to make each message unique, an initialization vector must be used in the first block.

CBC Encryption

CBC Decryption

CBC mode is as secure as the underlying block cipher against standard attacks. In addition any patterns in the plain text are concealed by the XORing of the previous cipher text blocks with the plain text block. Note also that the plain text cannot be directly manipulated except by removal of blocks from the beginning or the end of the cipher text. The initialization vector should be different for any two messages encrypted with the same key and is preferably randomly chosen. It does not have to be encrypted and it can be transmitted with (or considered as the first part of) the cipher text.

If the first block has index $1$ , the mathematical formula for CBC encryption is

$C_i = E_K(P_i \oplus C_{i-1}), C_0 = IV$ while the mathematical formula for CBC decryption is $P_i = D_K(C_i) \oplus C_{i-1}, C_0 = IV$

Choosing the initial value $c_0$ at random prevents almost with certainty that the same initial value $c_0$ is used for more than one encryption. This is important for security. Suppose for a moment that the same $c_0$ is used for two messages $m$ and $m^`$ . Then, an eavesdropper can immediately detect whether the first $l$ blocks of $m$ and $m^`$ coincide, because in this case the first $l$ ciphertext blocks are the same.

The speed of encryption is identical to that of the block cipher, but the encryption process cannot be easily parallelized, although the decryption process can be.

In this mode, we have $r = n$ . Encryption in the cipher-block chaining mode is implemented by the following algorithm

Algorithm:

bitStream cbcEncrypt(bitStream m)

select $c_0 \in \{0, 1\}^n$ at random

divide $m$ into $m_1m_2...m_l$

for $i$ $\gets$ $1$ to $l$ do

$c_i \gets E_k(m_i \oplus c_{i-1})$

return $c_0c_1c_2..c_l$

Decryption in cipher-block chaining mode is implemented by the following algorithm

Algorithm:

bitStream cbcDecrypt(bitStream c)

divide $c$ into $c_0c_1c_2...c_l$

for $i$ $\gets$ $1$ to $l$ do

$m_i \gets {E_k}^{-1}(c_i) \oplus c_{i-1}$

return $m_1m_2...m_l$

A transmission bit error in block $c_i$ affects the decryption of the blocks $c_i$ and $c_{i+1}$ . The block recovered from $c_i$ will appear random (here we assume that even a small change in the input of a block cipher will produce a random looking output), while the plaintext recovered from $c_{i+1}$ has bit errors precisely where $c_i$ did. The block $c_{i+2}$ is decrypted correctly. The cipher block chaining mode is self synchronizing, even if one or more entire blocks are lost. A lost ciphertext block results in the loss of the corresponding plaintext block and errors in the next plaintext block.

In both the electronic codebook mode and cipher block chaining mode, ${E_k}^{-1}$ is applied for decryption. Hence, both modes are also applicable with public key encryption methods, where the computation of ${E_k}^{-1}$ requires the recipient’s secret, while $E_k$ can be easily computed by everyone.

Tags: cryptography, encryption, iit, kgp, scribe.

Electronic Code Book

Published on March 18, 2009 in Miscel. 1 Comment

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The simplest of all the encryption modes is the electronic codebook mode. The message is divided into blocks and each block is encrypted separately.

ECB Encryption

ECB Decryption

ECB is as secure as the underlying block cipher. However, plaintext patterns are not concealed. Each identical block of plaintext gives an identical block of ciphertext. The plaintext can be easily manipulated by removing, repeating or interchanging blocks. As the encryption is deterministic, it is not CPA secure.

The speed of each encryption is identical to that of the block cipher. ECB allows easy parallelization to yield higher performance. Unfortunately, no processing is possible before a block is seen.

In this mode we have $r = n$ . The ECB is implemented by the following algorithm

Algorithm:

bitStream ecbEncrypt(bitStream m)

divide $m$ into $m_1m_2...m_l$

for $i$ $\gets$ $1$ to $l$ do

$c_i = E_k(m_i)$

return $c_1c_2...c_l$

For decryption the same algorithm can be used with the decryption function ${E_k}^{-1}$ instead of $E_k$ .

If we encrypt many blocks, partial information about the plain text is revealed. Therefor other modes are preferable.

Tags: cryptography, encryption, iit, kgp, scribe.

Modes Of Encryption

Published on March 18, 2009 in Miscel. 0 Comments

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Often the length of message exceeds the block length. So, the block ciphers need some extension. Consider a block cipher of length $n$ . We fix a key $k$ , and denote the encryption function with this key as

$E_k\colon \{0, 1\}^n \to \{0, 1\}^n$
To encrypt a message $m$ that is longer than $n$ , the message is decomposed into blocks of fixed size $r, m = m_1m_2...m_l$ . The individual blocks are encrypted iteratively.

The message block size $r$ need not equal $n$ . In few modes of encryption, $r$ is smaller than $n$ .

Also if the length of message $m$ is not an integral multiple of $r$ , then we have to complete the last block. The last block of the message can be padded out with some bits and encrypted. After decryption, the receiver must remove the padding. Therefore he must know how many bits were padded. This can be achieved, for exmple, by storing the number of padded bits in the last byte of the last block.

Tags: cryptography, encryption, iit, kgp, scribe.

Tag Archive for 'encryption'

Output Feedback Mode

Cipher Block Chaining

Electronic Code Book

Modes Of Encryption

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