# Integral cryptanalysis

In cryptography, integral cryptanalysis is a cryptanalytic attack that is particularly applicable to block ciphers based on substitution–permutation networks. It was originally designed by Lars Knudsen as a dedicated attack against Square, so it is commonly known as the Square attack. It was also extended to a few other ciphers related to Square: CRYPTON, Rijndael, and SHARK. Stefan Lucks generalized the attack to what he called a saturation attack and used it to attack Twofish, which is not at all similar to Square, having a radically different Feistel network structure. Forms of integral cryptanalysis have since been applied to a variety of ciphers, including Hierocrypt, IDEA, Camellia, Skipjack, MISTY1, MISTY2, SAFER++, KHAZAD, and FOX (now called IDEA NXT).

Unlike differential cryptanalysis, which uses pairs of chosen plaintexts with a fixed XOR difference, integral cryptanalysis uses sets or even multisets 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.

## References

• Joan Daemen, Lars Knudsen, Vincent Rijmen (January 1997). The Block Cipher Square (PDF). 4th International Workshop on Fast Software Encryption (FSE '97), Volume 1267 of Lecture Notes in Computer Science. Haifa: Springer-Verlag. pp. 149–165. Retrieved 2007-02-15.CS1 maint: multiple names: authors list (link)
• Carl D'Halluin, Gert Bijnens, Vincent Rijmen, Bart Preneel (March 1999). Attack on Six Rounds of Crypton (PDF/PostScript). 6th International Workshop on Fast Software Encryption (FSE '99). Rome: Springer-Verlag. pp. 46–59. Retrieved 2007-03-03.CS1 maint: multiple names: authors list (link)
• N. Ferguson, J. Kelsey, S. Lucks, B. Schneier, M. Stay, D. Wagner, D. Whiting (April 2000). Improved Cryptanalysis of Rijndael (PDF/PostScript). 7th International Workshop on Fast Software Encryption (FSE 2000). New York City: Springer-Verlag. pp. 213–230. Retrieved 2007-03-06.CS1 maint: multiple names: authors list (link)
• Stefan Lucks (September 14, 2000). The Saturation Attack - a Bait for Twofish (PDF/PostScript). 8th International Workshop on Fast Software Encryption (FSE '01). Yokohama: Springer-Verlag. pp. 1–15. Retrieved 2006-11-30.
• Paulo S. L. M. Barreto, Vincent Rijmen, Jorge Nakahara, Jr., Bart Preneel, Joos Vandewalle, Hae Yong Kim (April 2001). Improved SQUARE Attacks against Reduced-Round HIEROCRYPT (PDF). 8th International Workshop on Fast Software Encryption (FSE '01). Yokohama: Springer-Verlag. pp. 165–173. Retrieved 2007-03-03.CS1 maint: multiple names: authors list (link)
• Jorge Nakahara, Jr.; Paulo S.L.M. Barreto; Bart Preneel; Joos Vandewalle; Hae Y. Kim (2001). "SQUARE Attacks on Reduced-Round PES and IDEA Block Ciphers" (PDF/PostScript). Retrieved 2007-03-03. Cite journal requires `|journal=` (help)
• Yongjin Yeom; Sangwoo Park; Iljun Kim (February 2002). On the Security of CAMELLIA against the Square Attack (PDF). 9th International Workshop on Fast Software Encryption (FSE '02). Leuven: Springer-Verlag. pp. 89–99. Retrieved 2007-03-03.
• Kyungdeok Hwang; Wonil Lee; Sungjae Lee; Sangjin Lee; Jongin Lim (February 2002). Saturation Attacks on Reduced Round Skipjack. 9th International Workshop on Fast Software Encryption (FSE '02). Leuven: Springer-Verlag. pp. 100–111. Archived from the original (PDF) on 2007-05-25. Retrieved 2007-03-03.
• Lars Knudsen; David Wagner (December 11, 2001). Integral cryptanalysis (PDF/PostScript). 9th International Workshop on Fast Software Encryption (FSE '02). Leuven: Springer-Verlag. pp. 112–127. Retrieved 2006-11-30.
• Gilles Piret, Jean-Jacques Quisquater (February 16, 2003). "Integral Cryptanalysis on reduced-round Safer++" (PDF/PostScript). Retrieved 2007-03-03. Cite journal requires `|journal=` (help)
• Frédéric Muller (December 2003). A New Attack against Khazad. Advances in Cryptology - ASIACRYPT 2003. Taipei: Springer-Verlag. pp. 347–358. Archived from the original (PDF) on 2007-07-23. Retrieved 2007-03-03.
• Wu Wenling; Zhang Wentao; Feng Dengguo (August 25, 2005). "Improved Integral Cryptanalysis of FOX Block Cipher" (PDF). Retrieved 2007-03-03. Cite journal requires `|journal=` (help)