# Feistel cipher: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

In cryptography, a Feistel cipher is a symmetric structure used in the construction of block ciphers, named after the German-born physicist and cryptographer Horst Feistel who did pioneering research while working for IBM (USA); it is also commonly known as a Feistel network. A large proportion of block ciphers use the scheme, including the Data Encryption Standard (DES). The Feistel structure has the advantage that encryption and decryption operations are very similar, even identical in some cases, requiring only a reversal of the key schedule. Therefore the size of the code or circuitry required to implement such a cipher is nearly halved.

A Feistel network is an iterated cipher with an internal function called a round function.[1]

## Historical

Feistel networks were first seen commercially in IBM's Lucifer cipher, designed by Horst Feistel and Don Coppersmith. Feistel networks gained respectability when the U.S. Federal Government adopted the DES (a cipher based on Lucifer, with changes made by the NSA). Like other components of the DES, the iterative nature of the Feistel construction makes implementing the cryptosystem in hardware easier (particularly on the hardware available at the time of DES' design).

## Theoretical Work

Many modern and also some old symmetric block ciphers are based on Feistel networks (eg GOST_28147-89 block cipher), and the structure and properties of Feistel ciphers have been extensively explored by cryptographers. Specifically, Michael Luby and Charles Rackoff analyzed the Feistel cipher construction, and proved that if the round function is a cryptographically secure pseudorandom function, with Ki used as the seed, then 3 rounds is sufficient to make the block cipher a pseudorandom permutation, while 4 rounds is sufficient to make it a "strong" pseudorandom permutation (which means that it remains pseudorandom even to an adversary who gets oracle access to its inverse permutation).[2]

Because of this very important result of Luby and Rackoff, Feistel ciphers are sometimes called Luby-Rackoff block ciphers. Further theoretical work has generalized the construction somewhat, and given more precise bounds for security.[3]

## Construction Details

Let F be the round function and let $K_0,K_1,\ldots,K_{n}$ be the sub-keys for the rounds $0,1,\ldots,n$ respectively.

Then the basic operation is as follows:

Split the plaintext block into two equal pieces, (L0, R0)

For each round $i =0,1,\dots,n$, compute

$L_{i+1} = R_i\,$
$R_{i+1}= L_i \oplus {\rm F}(R_i, K_i)$.

Then the ciphertext is (Rn + 1,Ln + 1).

Decryption of a ciphertext (Rn + 1,Ln + 1) is accomplished by computing for $i=n,n-1,\ldots,0$

$R_{i} = L_{i+1}\,$
$L_{i} = R_{i+1} \oplus {\rm F}(L_{i+1}, K_{i})$.

Then (L0,R0) is the plaintext again.

One advantage of the Feistel model compared to a substitution-permutation network is that the round function F does not have to be invertible, and can be very complex.

The diagram illustrates both encryption and decryption. Note the reversal of the subkey order for decryption; this is the only difference between encryption and decryption:

### Unbalanced Feistel Cipher

Unbalanced Feistel ciphers use a modified structure where L0 and R0 are not of equal lengths.[4] The Skipjack encryption algorithm is an example of such a cipher. The Texas Instruments Digital Signature Transponder uses a proprietary unbalanced Feistel cipher to perform challenge-response authentication.[5]

The Feistel construction is also used in cryptographic algorithms other than block ciphers. For example, the Optimal Asymmetric Encryption Padding (OAEP) scheme uses a simple Feistel network to randomize ciphertexts in certain asymmetric key encryption schemes.

## List of Feistel ciphers

Feistel or modified Feistel: Blowfish, Camellia, CAST-128, DES, FEAL, ICE, KASUMI, LOKI97, Lucifer, MARS, MAGENTA, MISTY1, RC5, TEA, Triple DES, Twofish, XTEA, GOST_28147-89

Generalised Feistel: CAST-256, MacGuffin, RC2, RC6, Skipjack, SMS4

## References

1. ^ Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Fifth Printing (August 2001) page 251.
2. ^ Luby, Michael; Rackoff, Charles (April 1988), "How to Construct Pseudorandom Permutations from Pseudorandom Functions", SIAM Journal on Computing 17 (2): 373–386, doi:10.1137/0217022, ISSN 0097-5397
3. ^ Patarin, Jacques (October 2003), "Luby-Rackoff: 7 Rounds Are Enough for 2n(1−ε) Security", Advances in Cryptology—CRYPTO 2003, Lecture Notes in Computer Science 2729: 513–529, doi:10.1007/b11817, retrieved 2009-07-27
4. ^ http://www.schneier.com/paper-unbalanced-feistel.html
5. ^ S. Bono, M. Green, A. Stubblefield, A. Rubin, A. Juels, M. Szydlo. "Security Analysis of a Cryptographically-Enabled RFID Device". In Proceedings of the USENIX Security Symposium, August 2005. (pdf)

# Simple English

In cryptography, a Feistel cipher is a symmetric structure used in the construction of block ciphers, named after the German IBM cryptographer Horst Feistel; it is also commonly known as a Feistel network. A large set of block ciphers use the scheme, including the Data Encryption Standard (DES).

The Feistel structure has the advantage that encryption and decryption operations are very similar, even identical in some cases, requiring only a reversal of the key schedule. Therefore the size of the code or circuitry required to implement such a cipher is nearly halved.

Feistel construction is iterative in nature which makes implementing the cryptosystem in hardware easier.

Feistel networks and similar constructions are product ciphers, and so combine multiple rounds of repeated operations, such as:

to produce a function with large amounts of what Claude Shannon described as "confusion and diffusion".

Bit shuffling creates the diffusion effect, while substitution is used for confusion.

## Theoretical Work

Many modern symmetric block ciphers uses Feistel networks, and the structure and properties of Feistel ciphers have been extensively explored by cryptographers. Specifically, Michael Luby and Charles Rackoff analyzed the Feistel block cipher construction, and proved that if the round function is a cryptographically secure pseudorandom function, with Ki used as the seed, then 3 rounds is sufficient to make the block cipher a pseudorandom permutation, while 4 rounds is sufficient to make it a "strong" pseudorandom permutation (which means that it remains pseudorandom even to an adversary who gets oracle access to its inverse permutation).[1] Because of this very important result of Luby and Rackoff, Feistel ciphers are sometimes called Luby-Rackoff block ciphers. Further theoretical studies generalized the construction, and defined more precise limits for security.[2]

## Construction Details

[[File:|thumb|right|Feistel network operation on block cipher, where P is the plaintext and C is the ciphertext]]

Let $\left\{\rm F\right\}$ be the round function and let $K_1,K_2,\ldots,K_\left\{n\right\}$ be the sub-keys for the rounds $1,2,\ldots,n$ respectively.

Then the basic operation is as follows:

Split the plaintext block into two equal pieces, ($L_1$, $R_1$)

For each round $i =1,2,\dots,n$, compute (calculate)

$L_\left\{i+1\right\} = R_i\,$
$R_\left\{i+1\right\}= L_i \oplus \left\{\rm F\right\}\left(R_i, K_i\right)$.

Then the ciphertext is $\left(R_n, L_n\right)$. (Commonly the two pieces $R_n$ and $L_n$ are not switched after the last round.)

Decryption of a ciphertext $\left(R_n, L_n\right)$ is accomplished by computing for $i=n,n-1,\ldots,1$

$R_\left\{i\right\} = L_\left\{i+1\right\}\,$
$L_\left\{i\right\} = R_\left\{i+1\right\} \oplus \left\{\rm F\right\}\left(L_\left\{i+1\right\}, K_\left\{i\right\}\right)$.

Then $\left(L_1,R_1\right)$ is the plaintext again.

One advantage of this model is that the round function $\left\{\rm F\right\}$ does not have to be invertible, and can be very complex.

The diagram illustrates the encryption process. Decryption requires only reversing the order of the subkey $K_\left\{n\right\},K_\left\{n-1\right\},\ldots,K_1$ using the same process; this is the only difference between encryption and decryption:

Unbalanced Feistel ciphers use a modified structure where $L_1$ and $R_1$ are not of equal lengths. The MacGuffin cipher is an experimental example of such a cipher.

The Feistel construction is also used in cryptographic algorithms other than block ciphers. For example, the Optimal Asymmetric Encryption Padding (OAEP) scheme uses a simple Feistel network to randomize ciphertexts in certain asymmetric-key encryption schemes.

## List of Feistel ciphers

Feistel or modified Feistel: Blowfish, Camellia, CAST-128, DES, FEAL, ICE, KASUMI, LOKI97, Lucifer, MARS, MAGENTA, MISTY1, RC5, TEA, Triple DES, Twofish, XTEA, GOST 28147-89

Generalised Feistel: CAST-256, MacGuffin, RC2, RC6, Skipjack

## References

1. M. Luby and C. Rackoff. "How to Construct Pseudorandom Permutations and Pseudorandom Functions." In SIAM J. Comput., vol. 17, 1988, pp. 373-386.
2. Jacques Patarin, Luby-Rackoff: 7 Rounds Are Enough for Security, Lecture Notes in Computer Science, Volume 2729, Oct 2003, Pages 513 - 529