Abstract
We generalise the cube attack of Dinur and Shamir (and the similar AIDA attack of Vielhaber) to a more general higher order differentiation attack, by summing over an arbitrary subspace of the space of initialisation vectors. The Moebius transform can be used for efficiently examining all the subspaces of a big space, similar to the method used by Fouque and Vannet for the usual cube attack.
Secondly we propose replacing the Generalised Linearity Test proposed by Dinur and Shamir with a test based on higher order differentiation/Moebius transform. We show that the proposed test provides all the information provided by the Generalised Linearity Test, at the same computational cost. In addition, for functions that do not pass the linearity test it also provides, at no extra cost, an estimate of the degree of the function. This is useful for guiding the heuristics for the cube/AIDA attacks.
Finally we implement our ideas and test them on the stream cipher Trivium.
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Acknowledgements
The authors would like to thank the referees for useful comments. One of the referees brought to our attention the recent paper [2] that we had not been aware of, and in which higher order derivatives with respect to an arbitrary vector space (as explored in Sect. 3) were used for statistical attacks on the NORX cipher.
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Winter, R., Salagean, A., Phan, R.CW. (2015). Comparison of Cube Attacks Over Different Vector Spaces. In: Groth, J. (eds) Cryptography and Coding. IMACC 2015. Lecture Notes in Computer Science(), vol 9496. Springer, Cham. https://doi.org/10.1007/978-3-319-27239-9_14
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DOI: https://doi.org/10.1007/978-3-319-27239-9_14
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