Sălăgean-Mandache-Sălăgean2017_Article_CountingAndCharacterisingFunct.pdf (508.99 kB)
Download fileCounting and characterising functions with “fast points” for differential attacks
journal contribution
posted on 2015-12-10, 12:16 authored by Ana SalageanAna Salagean, Matei Mandache-SalageanHigher order derivatives have been introduced by Lai in a cryptographic context.
A number of attacks such as differential cryptanalysis, the cube and the AIDA attack
have been reformulated using higher order derivatives. Duan and Lai have introduced the
notion of “fast points” of a polynomial function f as being vectors a so that computing the
derivative with respect to a decreases the total degree of f by more than one. This notion
is motivated by the fact that most of the attacks become more efficient if they use fast
points. Duan and Lai gave a characterisation of fast points and Duan et al. gave some results
regarding the number of functions with fast points in some particular cases. We firstly give
an alternative characterisation of fast points and secondly give an explicit formula for the
number of functions with fast points for any given degree and number of variables, thus
covering all the cases left open in Duan et al. Our main tool is an invertible linear change of
coordinates which transforms the higher order derivative with respect to an arbitrary set of
linearly independent vectors into the higher order derivative with respect to a set of vectors
in the canonical basis. Finally we discuss the cryptographic significance of our results.
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