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

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

block cipher In cryptography, a block cipher is a deterministic algorithm that operates on fixed-length groups of bits, called ''blocks''. Block ciphers are the elementary building blocks of many cryptographic protocols. They are ubiquitous in the storage a ...

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. It was originally designed by

Lars Knudsen Lars Ramkilde Knudsen (born 21 February 1962) is a Denmark, Danish researcher in cryptography, particularly interested in the design and cryptanalysis, analysis of block ciphers, cryptographic hash function, hash functions and message authentic ...

as a dedicated attack against

Square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

, 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 submitted as a candidate for the Advanced Encryption Standard (AES). It is very efficient in hardware implementations and was designed by Chae Hoon Lim of Future Systems Inc. The CRYPTON alg ...

Rijndael The Advanced Encryption Standard (AES), also known by its original name Rijndael (), is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001. AES is a variant ...

, and

SHARK Sharks are a group of elasmobranch cartilaginous fish characterized by a ribless endoskeleton, dermal denticles, five to seven gill slits on each side, and pectoral fins that are not fused to the head. Modern sharks are classified within the ...

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 generalisi ...

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 researc ...

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 gover ...

IDEA In philosophy and in common usage, an idea (from the Greek word: ἰδέα (idea), meaning 'a form, or a pattern') is the results of thought. Also in philosophy, ideas can also be mental representational images of some object. Many philosophe ...

Camellia ''Camellia'' (pronounced or ) is a genus of flowering plants in the family Theaceae. They are found in tropical and subtropical areas in East Asia, eastern and South Asia, southern Asia, from the Himalayas east to Japan and Indonesia. There are ...

, 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 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 ...

, 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 the ...

, 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 wa ...

). 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 a ...

, which uses pairs of chosen plaintexts with a fixed

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 is true if and only if the inputs differ (one ...

difference, integral cryptanalysis uses

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

s 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 ...

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 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

References

* * * * * * * * * * * * Cryptographic attacks {{crypto-stub