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Counting and characterising functions with “fast points” for differential attacks

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journal contribution
posted on 10.12.2015, 12:16 by Ana SalageanAna Salagean, Matei Mandache-Salagean
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.

History

School

  • Science

Department

  • Computer Science

Published in

Cryptography and Communications

Volume

9

Issue

2

Pages

217 - 239

Citation

SALAGEAN, A.M. and MANDACHE-SALAGEAN, M., 2017. Counting and characterising functions with “fast points” for differential attacks. Cryptography and Communications, 9 (2), pp. 217-239.

Publisher

Springer / © The Authors

Version

VoR (Version of Record)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) licence. Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/

Acceptance date

20/10/2015

Publication date

2015-11-26

Notes

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.

ISSN

1936-2447

eISSN

1936-2455

Language

en