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Incryptography,confusion anddiffusion are two properties of a securecipher identified byClaude Shannon in his 1945 classified reportA Mathematical Theory of Cryptography.[1] These properties, when present, work together to thwart the application ofstatistics, and other methods ofcryptanalysis.
Confusion in asymmetric cipher is obscuring the local correlation between the input (plaintext), and output (ciphertext) by varying the application of thekey to the data, while diffusion is hiding the plaintext statistics by spreading it over a larger area of ciphertext.[2] Although ciphers can be confusion-only (substitution cipher,one-time pad) or diffusion-only (transposition cipher), any "reasonable"block cipher uses both confusion and diffusion.[2] These concepts are also important in the design ofcryptographic hash functions, andpseudorandom number generators, where decorrelation of the generated values is the main feature. Diffusion (and itsavalanche effect) is also applicable tonon-cryptographic hash functions.
Confusion means that each binary digit (bit) of the ciphertext should depend on several parts of the key, obscuring the connections between the two.[3]
The property of confusion hides the relationship between the ciphertext and the key.
This property makes it difficult to find the key from the ciphertext and if a single bit in a key is changed, the calculation of most or all of the bits in the ciphertext will be affected.
Confusion increases the ambiguity of ciphertext and it is used by both block and stream ciphers.
Insubstitution–permutation networks, confusion is provided bysubstitution boxes.[4]
Diffusion means that if we change a single bit of the plaintext, then about half of the bits in the ciphertext should change, and similarly, if we change one bit of the ciphertext, then about half of the plaintext bits should change.[5] This is equivalent to the expectation that encryption schemes exhibit anavalanche effect.
The purpose of diffusion is to hide the statistical relationship between the ciphertext and the plain text. For example, diffusion ensures that any patterns in the plaintext, such as redundant bits, are not apparent in the ciphertext.[3] Block ciphers achieve this by "diffusing" the information about the plaintext's structure across the rows and columns of the cipher.
In substitution–permutation networks, diffusion is provided bypermutation boxes (a.k.a. permutation layer[4]). In the beginning of the 21st century a consensus had appeared where the designers preferred the permutation layer to consist oflinear Boolean functions, although nonlinear functions can be used, too.[4]
In Shannon's original definitions,confusion refers to making the relationship between theciphertext and thesymmetric key as complex and involved as possible;diffusion refers to dissipating the statistical structure ofplaintext over the bulk ofciphertext. This complexity is generally implemented through a well-defined and repeatable series ofsubstitutions andpermutations. Substitution refers to the replacement of certain components (usually bits) with other components, following certain rules. Permutation refers to manipulation of the order of bits according to some algorithm. To be effective, any non-uniformity of plaintext bits needs to be redistributed across much larger structures in the ciphertext, making that non-uniformity much harder to detect.
In particular, for a randomly chosen input, if one flips thei-th bit, then the probability that thej-th output bit will change should be one half, for anyi andj—this is termed thestrict avalanche criterion. More generally, one may require that flipping a fixed set of bits should change each output bit with probability one half.
One aim of confusion is to make it very hard to find the key even if one has a large number of plaintext-ciphertext pairs produced with the same key. Therefore, each bit of the ciphertext should depend on the entire key, and in different ways on different bits of the key. In particular, changing one bit of the key should change the ciphertext completely.
Design of a modernblock cipher uses both confusion and diffusion,[2]with confusion changing data between the input and the output by applying a key-dependent non-linear transformation (linear calculations are easier to reverse and thus are easier to break).
Confusion inevitably involves some diffusion,[6] so a design with a very wide-inputS-box can provide the necessary diffusion properties,[citation needed] but will be very costly in implementation. Therefore, the practical ciphers utilize relatively small S-boxes, operating on small groups of bits ("bundles"[7]). For example, the design of AES has 8-bit S-boxes,Serpent − 4-bit,BaseKing and3-way − 3-bit.[8] Small S-boxes provide almost no diffusion, so the resources are spent on simpler diffusion transformations.[6] For example, thewide trail strategy popularized by theRijndael design, involves a linear mixing transformation that provides high diffusion,[9] although the security proofs do not depend on the diffusion layer being linear.[10]
One of the most researched cipher structures uses thesubstitution-permutation network (SPN) where eachround includes a layer of local nonlinear permutations (S-boxes) for confusion and alinear diffusion transformation (usually a multiplication by a matrix over afinite field).[11] Modern block ciphers mostly follow the confusion layer/diffusion layer model, with the efficiency of the diffusion layer estimated using the so-calledbranch number, a numerical parameter that can reach the value fors input bundles for the perfect diffusion transformation.[12] Since the transformations that have high branch numbers (and thus require a lot of bundles as inputs) are costly in implementation, the diffusion layer is sometimes (for example, in the AES) composed from two sublayers, "local diffusion" that processes subsets of the bundles in abricklayer fashion (each subset is transformed independently) and "dispersion" that makes the bits that were "close" (within one subset of bundles) to become "distant" (spread to different subsets and thus be locally diffused within these new subsets on the next round).[13]
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TheAdvanced Encryption Standard (AES) has both excellent confusion and diffusion. Its confusion look-up tables are very non-linear and good at destroying patterns.[14] Its diffusion stage spreads every part of the input to every part of the output: changing one bit of input changes half the output bits on average. Both confusion and diffusion are repeated multiple times for each input to increase the amount of scrambling. The secret key is mixed in at every stage so that an attacker cannot precalculate what the cipher does.
None of this happens when a simple one-stage scramble is based on a key. Input patterns would flow straight through to the output. It might look random to the eye but analysis would find obvious patterns and the cipher could be broken.