Higher order differential cryptanalysis

In cryptography, higher-order differential cryptanalysis is a generalization of differential cryptanalysis, an attack used against block ciphers. While in standard differential cryptanalysis the difference between only two texts is used, higher-order differential cryptanalysis studies the propagation of a set of differences between a larger set of texts. Xuejia Lai, in 1994, laid the groundwork by showing that differentials are a special case of the more general case of higher order derivates.Lars Knudsen, in the same year, was able to show how the concept of higher order derivatives can be used to mount attacks on block ciphers. These attacks can be superior to standard differential cryptanalysis. Higher-order differential cryptanalysis has notably been used to break the KN-Cipher, a cipher which had previously been proved to be immune against standard differential cryptanalysis.

A block cipher which maps ${\displaystyle n}$-bit strings to ${\displaystyle n}$-bit strings can, for a fixed key, be thought of as a function ${\displaystyle f:\mathbb {F} _{2}^{n}\to \mathbb {F} _{2}^{n}}$. In standard differential cryptanalysis, one is interested in finding a pair of an input difference ${\displaystyle \alpha }$ and an output difference ${\displaystyle \beta }$ such that two input texts with difference ${\displaystyle \alpha }$ are likely to result in output texts with a difference ${\displaystyle \beta }$ i.e., that ${\displaystyle f(m\oplus \alpha )\oplus f(m)=\beta }$ is true for many ${\displaystyle m\in \mathbb {F} _{2}^{n}}$. Note that the difference used here is the XOR which is the usual case, though other definitions of difference are possible.

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