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42-609: Many modern symmetric block ciphers are based on Feistel networks. Feistel networks were first seen commercially in IBM's Lucifer cipher, designed by Horst Feistel and Don Coppersmith in 1973. Feistel networks gained respectability when the U.S. Federal Government adopted the DES (a cipher based on Lucifer, with changes made by the NSA ) in 1976. Like other components of the DES, the iterative nature of

84-437: A 32-bit block, using one addition mod 4 and a singular 4-bit S-box. The construction is designed to operate on 4 bits per clock cycle. This may be one of the smallest block-cipher implementations known. Feistel later described a stronger variant that uses a 128-bit key and operates on 128-bit blocks. Sorkin (1984) described a later Lucifer as a 16-round Feistel network , also on 128-bit blocks and 128-bit keys. This version

126-404: A 48-bit key and operates on 48-bit blocks. The cipher is a substitution–permutation network and uses two 4-bit S-boxes . The key selects which S-boxes are used. The patent describes the execution of the cipher operating on 24 bits at a time, and also a sequential version operating on 8 bits at a time. Another variant by John L. Smith from the same year uses a 64-bit key operating on

168-532: A 64-bit half-block of data, together with a 64-bit subkey and 8 " interchange control bits " (ICBs). The ICBs control a swapping operation. The 64-bit data block is considered as a series of eight 8-bit bytes, and if the ICB corresponding to a particular byte is zero, the left and right 4-bit halves ( nibbles ) are swapped. If the ICB is one, the byte is left unchanged. Each byte is then operated on by two 4×4-bit S-boxes, denoted S 0 and S 1 — S 0 operates on

210-472: A random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source. XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes 10011100 2 and 01101100 2 from two (or more) hard drives by XORing

252-486: A simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output. On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) than to load and store the value zero. In cryptography , XOR is sometimes used as a simple, self-inverse mixing function, such as in one-time pad or Feistel network systems. XOR

294-484: A substitution box that could cause timing side-channels in software implementations. The structure and properties of Feistel ciphers have been extensively analyzed by cryptographers . Michael Luby and Charles Rackoff analyzed the Feistel cipher construction and proved that if the round function is a cryptographically secure pseudorandom function , with K i used as the seed, then 3 rounds are sufficient to make

336-873: Is symbolized by the prefix operator J {\displaystyle J} and by the infix operators XOR ( / ˌ ɛ k s ˈ ɔː r / , / ˌ ɛ k s ˈ ɔː / , / ˈ k s ɔː r / or / ˈ k s ɔː / ), EOR , EXOR , ∨ ˙ {\displaystyle {\dot {\vee }}} , ∨ ¯ {\displaystyle {\overline {\vee }}} , ∨ _ {\displaystyle {\underline {\vee }}} , ⩛ , ⊕ {\displaystyle \oplus } , ↮ {\displaystyle \nleftrightarrow } , and ≢ {\displaystyle \not \equiv } . The truth table of A ⊕ B {\displaystyle A\oplus B} shows that it outputs true whenever

378-557: Is a group . This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring . However, the system using exclusive or ( { T , F } , ⊕ ) {\displaystyle (\{T,F\},\oplus )} is an abelian group . The combination of operators ∧ {\displaystyle \wedge } and ⊕ {\displaystyle \oplus } over elements { T , F } {\displaystyle \{T,F\}} produce

420-557: Is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French soit... soit . The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen: If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2. Exclusive disjunction

462-497: Is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR). In simple threshold-activated artificial neural networks , modeling the XOR function requires a second layer because XOR is not a linearly separable function. Similarly, XOR can be used in generating entropy pools for hardware random number generators . The XOR operation preserves randomness, meaning that

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504-408: 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. A generalized Feistel algorithm can be used to create strong permutations on small domains of size not a power of two (see format-preserving encryption ). Whether

546-406: Is an example of such a cipher. The Texas Instruments digital signature transponder uses a proprietary unbalanced Feistel cipher to perform challenge–response authentication . The Thorp shuffle is an extreme case of an unbalanced Feistel cipher in which one side is a single bit. This has better provable security than a balanced Feistel cipher but requires more rounds. The Feistel construction

588-478: Is called the function's algebraic normal form . Disjunction is often understood exclusively in natural languages . In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet. However, disjunction can also be understood inclusively, even in combination with "either". For instance,

630-550: Is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence . In summary, we have, in mathematical and in engineering notation: By applying the spirit of De Morgan's laws , we get: ¬ ( p ↮ q ) ⇔ ¬ p ↮ q ⇔ p ↮ ¬ q . {\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.} Although

672-469: Is often used for bitwise operations. Examples: As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two n -bit strings is identical to the standard vector of addition in the vector space ( Z / 2 Z ) n {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}} . In computer science, exclusive disjunction has several uses: In logical circuits,

714-398: Is sometimes useful to write p ↮ q {\displaystyle p\nleftrightarrow q} in the following way: or: This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof. The exclusive or is also equivalent to the negation of a logical biconditional , by the rules of material implication (a material conditional

756-619: Is susceptible to differential cryptanalysis ; for about half the keys, the cipher can be broken with 2 chosen plaintexts and 2 time complexity. IBM submitted the Feistel-network version of Lucifer as a candidate for the Data Encryption Standard (compare the more recent AES process ). It became the DES after the National Security Agency reduced the cipher's key size to 56 bits, reduced

798-436: Is the plaintext again. 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 {\displaystyle L_{0}} and R 0 {\displaystyle R_{0}} are not of equal lengths. The Skipjack cipher

840-425: Is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd . It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR excludes that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B". It

882-423: The logical conjunction ("logical and", ∧ {\displaystyle \wedge } ), the disjunction ("logical or", ∨ {\displaystyle \lor } ), and the negation ( ¬ {\displaystyle \lnot } ) as follows: The exclusive disjunction p ↮ q {\displaystyle p\nleftrightarrow q} can also be expressed in

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924-529: The operators ∧ {\displaystyle \wedge } ( conjunction ) and ∨ {\displaystyle \lor } ( disjunction ) are very useful in logic systems, they fail a more generalizable structure in the following way: The systems ( { T , F } , ∧ ) {\displaystyle (\{T,F\},\wedge )} and ( { T , F } , ∨ ) {\displaystyle (\{T,F\},\lor )} are monoids , but neither

966-442: The 128 key bits are loaded into a shift register . Each round, the left 64 bits of the register form the subkey, and right eight bits form the ICB bits. After each round, the register is rotated 56 bits to the left. XOR Exclusive or , exclusive disjunction , exclusive alternation , logical non-equivalence , or logical inequality is a logical operator whose negation is the logical biconditional . With two inputs, XOR

1008-401: The Feistel construction makes implementing the cryptosystem in hardware easier (particularly on the hardware available at the time of DES's design). A Feistel network uses a round function , a function which takes two inputs – a data block and a subkey – and returns one output of the same size as the data block. In each round, the round function is run on half of

1050-536: The above have motivated analyses of the exclusivity inference as pragmatic conversational implicatures calculated on the basis of an inclusive semantics . Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity . However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it. This behavior of English "or"

1092-434: The basic operation is as follows: Split the plaintext block into two equal pieces: ( L 0 {\displaystyle L_{0}} , R 0 {\displaystyle R_{0}} ). For each round i = 0 , 1 , … , n {\displaystyle i=0,1,\dots ,n} , compute where ⊕ {\displaystyle \oplus } means XOR . Then

1134-407: The block cipher a pseudorandom permutation , while 4 rounds are 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). Because of this very important result of Luby and Rackoff, Feistel ciphers are sometimes called Luby–Rackoff block ciphers. Further theoretical work has generalized

1176-611: The block size to 64 bits, and made the cipher resistant against differential cryptanalysis , which was at the time known only to IBM and the NSA. The name "Lucifer" was apparently a pun on "Demon". This was in turn a truncation of "Demonstration", the name for a privacy system Feistel was working on. The operating system used could not handle the longer name . The variant described by Sorkin (1984) has 16 Feistel rounds , like DES, but no initial or final permutations. The key and block sizes are both 128 bits. The Feistel function operates on

1218-567: The ciphertext is ( R n + 1 , L n + 1 ) {\displaystyle (R_{n+1},L_{n+1})} . Decryption of a ciphertext ( R n + 1 , L n + 1 ) {\displaystyle (R_{n+1},L_{n+1})} is accomplished by computing for i = n , n − 1 , … , 0 {\displaystyle i=n,n-1,\ldots ,0} Then ( L 0 , R 0 ) {\displaystyle (L_{0},R_{0})}

1260-461: The construction somewhat and given more precise bounds for security. Let F {\displaystyle \mathrm {F} } be the round function and let K 0 , K 1 , … , K n {\displaystyle K_{0},K_{1},\ldots ,K_{n}} be the sub-keys for the rounds 0 , 1 , … , n {\displaystyle 0,1,\ldots ,n} respectively. Then

1302-400: The data to be encrypted, and its output is XORed with the other half of the data. This is repeated a fixed number of times, and the final output is the encrypted data. An important advantage of Feistel networks compared to other cipher designs such as substitution–permutation networks is that the entire operation is guaranteed to be invertible (that is, encrypted data can be decrypted), even if

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1344-525: The entire cipher is a Feistel cipher or not, Feistel-like networks can be used as a component of a cipher's design. For example, MISTY1 is a Feistel cipher using a three-round Feistel network in its round function, Skipjack is a modified Feistel cipher using a Feistel network in its G permutation, and Threefish (part of Skein ) is a non-Feistel block cipher that uses a Feistel-like MIX function. Feistel or modified Feistel: Generalised Feistel: Lucifer (cipher) In cryptography , Lucifer

1386-406: The first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans. Examples such as

1428-399: The following way: This representation of XOR may be found useful when constructing a circuit or network, because it has only one ¬ {\displaystyle \lnot } operation and small number of ∧ {\displaystyle \land } and ∨ {\displaystyle \lor } operations. A proof of this identity is given below: It

1470-596: The inputs differ: Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction p ↮ q {\displaystyle p\nleftrightarrow q} , also denoted by p ? ⁡ q {\displaystyle p\operatorname {?} q} or J p q {\displaystyle Jpq} , can be expressed in terms of

1512-406: The just mentioned bytes, resulting in ( 11110000 2 ) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing 01101100 2 is lost, 10011100 2 and 11110000 2 can be XORed to recover the lost byte. XOR is also used to detect an overflow in

1554-422: The left 4-bit nibble and S 1 operates on the right. The resultant outputs are concatenated and then combined with the subkey using exclusive or (XOR); this is termed " key interruption ". This is followed by a permutation operation in two stages; the first permutes each byte under a fixed permutation. The second stage mixes bits between the bytes. The key-scheduling algorithm is relatively simple. Initially,

1596-404: The logical "AND" operation as multiplication on F 2 {\displaystyle \mathbb {F} _{2}} and the "XOR" operation as addition on F 2 {\displaystyle \mathbb {F} _{2}} : The description of a Boolean function as a polynomial in F 2 {\displaystyle \mathbb {F} _{2}} , using this basis,

1638-777: The result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow. XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm ; however this is regarded as more of a curiosity and not encouraged in practice. XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures. In computer graphics , XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes. It

1680-492: The round function is not itself invertible. The round function can be made arbitrarily complicated, since it does not need to be designed to be invertible. Furthermore, the 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. Unlike substitution-permutation networks, Feistel networks also do not depend on

1722-520: The well-known two-element field F 2 {\displaystyle \mathbb {F} _{2}} . This field can represent any logic obtainable with the system ( ∧ , ∨ ) {\displaystyle (\land ,\lor )} and has the added benefit of the arsenal of algebraic analysis tools for fields. More specifically, if one associates F {\displaystyle F} with 0 and T {\displaystyle T} with 1, one can interpret

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1764-461: Was the name given to several of the earliest civilian block ciphers , developed by Horst Feistel and his colleagues at IBM . Lucifer was a direct precursor to the Data Encryption Standard . One version, alternatively named DTD-1 , saw commercial use in the 1970s for electronic banking . Lucifer uses a combination of transposition and substitution crypting as a starting point in decoding ciphers. One variant, described by Feistel in 1971, uses

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