Feistel cipher

Feistel cipher

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 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.

Feistel networks and similar constructions are product ciphers, and so combine multiple rounds of repeated operations, such as:
* Bit-shuffling (often called permutation boxes or P-boxes)
* Simple non-linear functions (often called substitution boxes or S-boxes)
* Linear mixing (in the sense of modular algebra) using XORto 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.


Feistel networks were first seen commercially in IBM's Lucifer cipher, designed by Feistel and Don Coppersmith. Feistel networks gained respectability when the US 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). Things have changed through the decades as hardware has become more capable.

Theoretical Work

Many modern symmetric block ciphers are based on 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).M. Luby and C. Rackoff. "How to Construct Pseudorandom Permutations and Pseudorandom Functions." In "SIAM J. Comput.," vol. 17, 1988, pp. 373-386.] 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.Jacques Patarin, Luby-Rackoff: 7 Rounds Are Enough for Security, Lecture Notes in Computer Science, Volume 2729, Oct 2003, Pages 513 - 529]

Construction Details

Let { m F} be the round function and letK_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, (L_0, R_0)

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

:L_{i+1} = R_i,:R_{i+1}= L_i oplus { m F}(R_i, K_i).

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

Decryption of a ciphertext (R_n, L_n) is accomplished by computing for i=n,n-1,ldots,0

:R_{i} = L_{i+1},:L_{i} = R_{i+1} oplus { m F}(L_{i+1}, K_{i}).

Then (L_0,R_0) is the plaintext again.

One advantage of this model is that the round function { m 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 ciphers use a modified structure where L_0 and R_0 are not of equal lengths. 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.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. [http://www.usenix.org/events/sec05/tech/bono/bono.pdf (pdf)] ]

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:
Triple DES,

Generalised Feistel:


ee also

* Cryptography
* Stream cipher
* Substitution-permutation network

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