Higher-order differential cryptanalysis

Higher-order derivatives

A block cipher which maps n-bit strings to n-bit strings can, for a fixed key, be thought of as a function f:\mathbb{F}^n_2\to\mathbb{F}^n_2. In standard differential cryptanalysis, one is interested in finding a pair of an input difference \alpha and an output difference \beta such that two input texts with difference \alpha are likely to result in output texts with a difference \beta i.e., that f(m\oplus\alpha)\oplus f(m) = \beta is true for many m\in\mathbb{F}^n_2. Note that the difference used here is the XOR which is the usual case, though other definitions of difference are possible.

This motivates defining the derivative of a function f:\mathbb{F}^n_2\to\mathbb{F}^n_2 at a point \alpha as

Using this definition, the i-th derivative at (\alpha_1,\alpha_2,\dots,\alpha_i) can recursively be defined as

f(x)\right).}}

Thus for example \Delta^{(2)}_{\alpha_1,\alpha_2} f(x) = f(x)\oplus f(x\oplus\alpha_1)\oplus f(x\oplus\alpha_2)\oplus f(x\oplus\alpha_1\oplus\alpha_2).

Higher order derivatives as defined here have many properties in common with ordinary derivative such as the sum rule and the product rule. Importantly also, taking the derivative reduces the algebraic degree of the function.

Higher-order differential attacks

To implement an attack using higher order derivatives, knowledge about the probability distribution of the derivative of the cipher is needed. Calculating or estimating this distribution is generally a hard problem but if the cipher in question is known to have a low algebraic degree, the fact that derivatives reduce this degree can be used. For example, if a cipher (or the S-box function under analysis) is known to only have an algebraic degree of 8, any 9th order derivative must be 0.

Therefore, it is important for any cipher or S-box function in specific to have a maximal (or close to maximal) degree to defy this attack.

Cube attacks have been considered a variant of higher-order differential attacks.

Resistance against Higher-order differential attacks

Limitations of Higher-order differential attacks

Works for small or low algebraic degree S-boxes or small S-boxes. In addition to AND and XOR operations.

References

References

1. [[Xuejia Lai. (1994). "Communications and Cryptography". Springer US.
2. [[Lars Knudsen. (1994). "Truncated and Higher Order Differentials". [[Springer-Verlag]].
3. [[Thomas Jakobsen. (1997). "Fast Software Encryption". Springer Berlin Heidelberg.
4. [[Daniel J. Bernstein]]. (2009-01-14). "Why haven't cube attacks broken anything?".