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Title: Counting and characterising functions with “fast points” for differential attacks
Authors: Salagean, A.M.
Mandache-Salagean, Matei
Keywords: Higher order differential attacks
Higher order derivative
Polynomials over finite fields
Issue Date: 2015
Publisher: Springer / © The Authors
Citation: SALAGEAN, A.M. and MANDACHE-SALAGEAN, M., 2015. Counting and characterising functions with “fast points” for differential attacks. Cryptography and Communications, 9 (2), pp. 217-239.
Abstract: Higher 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.
Description: This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Version: Published
DOI: 10.1007/s12095-015-0166-1
URI: https://dspace.lboro.ac.uk/2134/19814
Publisher Link: http://dx.doi.org/10.1007/s12095-015-0166-1
ISSN: 1936-2447
Appears in Collections:Published Articles (Computer Science)

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