# Cryptography/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 an input can affect the resultant difference at the output. In the case of a block cipher, it refers to a set of techniques for tracing differences through the network of transformations, discovering where the cipher exhibits non-random behaviour, and exploiting such properties to recover the secret key.

## History

The discovery of differential cryptanalysis is generally attributed to Eli Biham and Adi Shamir in the late 1980s, who published a number of attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard (DES). It was noted by Bamford in The Puzzle Palace that DES is surprisingly resilient to differential cryptanalysis, in the sense that even small modifications to the algorithm would make it much more susceptible.
In 1994, a member of the original IBM DES team, Don Coppersmith, published a paper stating that differential cryptanalysis was known to IBM as early as 1974, and that defending against differential cryptanalysis had been a design goal. According to author Steven Levy, IBM had discovered differential cryptanalysis on its own, and the NSA was apparently well aware of the technique. IBM kept some secrets, as Coppersmith explains: "After discussions with NSA, it was decided that disclosure of the design considerations would reveal the technique of differential cryptanalysis, a powerful technique that could be used against many ciphers. This in turn would weaken the competitive advantage the United States enjoyed over other countries in the field of cryptography." Within IBM, differential cryptanalysis was known as the "T-attack", or "Tickle attack".
While DES was designed with resistance to differential cryptanalysis in mind, other contemporary ciphers proved to be vulnerable. An early target for the attack was the FEAL block cipher. The original proposed version with four rounds (FEAL-4) can be broken using only eight chosen plaintexts, and even a 31-round version of FEAL is susceptible to the attack.

## Attack mechanics

Differential cryptanalysis is usually a chosen plaintext attack, meaning that the attacker must be able to obtain encrypted ciphertexts for some set of plaintexts of his choosing. The scheme can successfully cryptanalyze DES with an effort on the order 2$^{47}$ chosen plaintexts. There are, however, extensions that would allow a known plaintext or even a ciphertext-only attack. The basic method uses pairs of plaintext related by a constant difference; difference can be defined in several ways, but the eXclusive OR (XOR) operation is usual. The attacker then computes the differences of the corresponding ciphertexts, hoping to detect statistical patterns in their distribution. The resulting pair of differences is called a differential. Their statistical properties depend upon the nature of the S-boxes used for encryption, so the attacker analyses differentials $(\Delta _{X},\Delta _{Y})$, where $\Delta _{Y}=S(X)\oplus S(X\oplus \Delta _{X})$ (and $\oplus$ denotes exclusive or) for each such S-box $S$. In the basic attack, one particular ciphertext difference is expected to be especially frequent; in this way, the cipher can be distinguished from randomness. More sophisticated variations allow the key to be recovered faster than exhaustive search.
In the most basic form of key recovery through differential cryptanalysis, an attacker requests the ciphertexts for a large number of plaintext pairs, then assumes that the differential holds for at least r-1 rounds, where r is the total number of rounds. The attacker then deduces which round keys (for the final round) are possible assuming the difference between the blocks before the final round is fixed. When round keys are short, this can be achieved by simply exhaustively decrypting the ciphertext pairs one round with each possible round key. When one round key has been deemed a potential round key considerably more often than any other key, it is assumed to be the correct round key.
For any particular cipher, the input difference must be carefully selected if the attack is to be successful. An analysis of the algorithm's internals is undertaken; the standard method is to trace a path of highly probable differences through the various stages of encryption, termed a differential characteristic.
Since differential cryptanalysis became public knowledge, it has become a basic concern of cipher designers. New designs are expected to be accompanied by evidence that the algorithm is resistant to this attack, and many, including the Advanced Encryption Standard, have been proven secure against the attack.

## References

* 1 2 3 Coppersmith, Don (1994). "The Data Encryption Standard (DES) and its strength against attacks" (PDF). IBM Journal of Research and Development. 38 (3): 243. {{cite journal}}: Unknown parameter |month= ignored (help) (subscription required)
* ↑ Levy, Steven (2001). "Crypto: How the Code Rebels Beat the Government — Saving Privacy in the Digital Age. Penguin Books. pp. 55–56. ISBN 0-14-024432-8.
* ↑ Matt Blaze, sci.crypt, 15 August 1996, Re: Reverse engineering and the Clipper chip"
* Eli Biham, Adi Shamir, Differential Cryptanalysis of the Data Encryption Standard, Springer Verlag, 1993. ISBN 0-387-97930-1, ISBN 3-540-97930-1.
* Biham, E. and A. Shamir. (1990). Differential Cryptanalysis of DES-like Cryptosystems. Advances in Cryptology — CRYPTO '90. Springer-Verlag. 2–21.
* Eli Biham, Adi Shamir,"Differential Cryptanalysis of the Full 16-Round DES," CS 708, Proceedings of CRYPTO '92, Volume 740 of Lecture Notes in Computer Science, December 1991. (Postscript)
* Eli Biham, slides from "How to Make a Difference: Early History of Differential Cryptanalysis"PDF (850 KB), March 16, 2006, FSE 2006, Graz, Austria