cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...

, integral cryptanalysis is a cryptanalytic attack that is particularly applicable to

block cipher In cryptography, a block cipher is a deterministic algorithm operating on fixed-length groups of bits, called ''blocks''. Block ciphers are specified cryptographic primitive, elementary components in the design of many cryptographic protocols and ...

s based on

substitution–permutation network In cryptography, an SP-network, or substitution–permutation network (SPN), is a series of linked mathematical operations used in block cipher algorithms such as AES (Rijndael), 3-Way, Kalyna, Kuznyechik, PRESENT, SAFER, SHARK, and Square. S ...

s. It was originally designed by

Lars Knudsen Lars Ramkilde Knudsen (born 21 February 1962) is a Danish researcher in cryptography, particularly interested in the design and analysis of block ciphers, hash functions and message authentication codes (MACs). Academic After some early work i ...

as a dedicated attack against

Square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, so it is commonly known as the Square attack. It was also extended to a few other ciphers related to Square:

CRYPTON In cryptography, CRYPTON is a symmetric block cipher In cryptography, a block cipher is a deterministic algorithm operating on fixed-length groups of bits, called ''blocks''. Block ciphers are specified cryptographic primitive, elementary c ...

, Rijndael, and

SHARK Sharks are a group of elasmobranch fish characterized by a cartilaginous skeleton, five to seven gill slits on the sides of the head, and pectoral fins that are not fused to the head. Modern sharks are classified within the clade Selachi ...

Stefan Lucks Stefan Lucks is a researcher in the fields of communications security and cryptography. Lucks is known for his attack on Triple DES, and for extending Lars Knudsen's Square attack to Twofish, a cipher outside the Square family, thus generalis ...

generalized the attack to what he called a ''saturation attack'' and used it to attack

Twofish In cryptography, Twofish is a symmetric key block cipher with a block size of 128 bits and key sizes up to 256 bits. It was one of the five finalists of the Advanced Encryption Standard contest, but it was not selected for standardization. Two ...

, which is not at all similar to Square, having a radically different

Feistel network In cryptography, a Feistel cipher (also known as Luby–Rackoff block 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 whi ...

structure. Forms of integral cryptanalysis have since been applied to a variety of ciphers, including

Hierocrypt In cryptography, Hierocrypt-L1 and Hierocrypt-3 are block ciphers created by Toshiba in 2000. They were submitted to the NESSIE project, but were not selected. Both algorithms were among the cryptographic techniques recommended for Japanese gove ...

IDEA In common usage and in philosophy, ideas are the results of thought. Also in philosophy, ideas can also be mental representational images of some object. Many philosophers have considered ideas to be a fundamental ontological category of be ...

Camellia ''Camellia'' (pronounced or ) is a genus of flowering plants in the family Theaceae. They are found in eastern and southern Asia, from the Himalayas east to Japan and Indonesia. There are more than 220 described species, with some controvers ...

, Skipjack,

MISTY1 In cryptography, MISTY1 (or MISTY-1) is a block cipher designed in 1995 by Mitsuru Matsui and others for Mitsubishi Electric. MISTY1 is one of the selected algorithms in the European NESSIE project, and has been among the cryptographic techniq ...

MISTY2 Misty may refer to: Music * ''Misty'' (Ray Stevens album), an album by Ray Stevens featuring the above song * ''Misty'' (Richard "Groove" Holmes album), an album by Richard "Groove" Holmes featuring the above song * ''Misty'' (Eddie "Lockjaw" ...

, SAFER++,

KHAZAD In cryptography, KHAZAD is a block cipher designed by Paulo S. L. M. Barreto together with Vincent Rijmen, one of the designers of the Advanced Encryption Standard ( Rijndael). KHAZAD is named after Khazad-dûm, the fictional dwarven realm in ...

, and ''FOX'' (now called

IDEA NXT In cryptography, the IDEA NXT algorithm (previously known as FOX) is a block cipher designed by Pascal Junod and Serge Vaudenay of EPFL (Lausanne, Switzerland). It was conceived between 2001 and 2003. The project was originally named FOX and ...

). Unlike

differential cryptanalysis Differential cryptanalysis is a general form of cryptanalysis applicable primarily to block ciphers, but also to stream ciphers and cryptographic hash functions. In the broadest sense, it is the study of how differences in information input can af ...

, which uses pairs of chosen plaintexts with a fixed

XOR Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , , ...

difference, integral cryptanalysis uses sets or even

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...

s of chosen plaintexts of which part is held constant and another part varies through all possibilities. For example, an attack might use 256 chosen plaintexts that have all but 8 of their bits the same, but all differ in those 8 bits. Such a set necessarily has an XOR sum of 0, and the XOR sums of the corresponding sets of ciphertexts provide information about the cipher's operation. This contrast between the differences of pairs of texts and the sums of larger sets of texts inspired the name "integral cryptanalysis", borrowing the terminology of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

References

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