Abstract
Symmetric block ciphers, such as the Advanced Encryption Standard (AES), are deterministic algorithms which transform plaintexts to ciphertexts using a secret key. These ciphers are designed such that it is computationally very difficult to recover the secret key if only pairs of plaintexts and ciphertexts are provided to the attacker. Constraint solvers have recently been suggested as a way of recovering the secret keys of symmetric block ciphers. To carry out such an attack, the attacker provides the solver with a set of equations describing the mathematical relationship between a known plaintext and a known ciphertext, and then attempts to solve for the unknown secret key. This approach is known to be intractable against AES unless side-channel data – information leaked from the cryptographic device due to its internal physical structure – is introduced into the equation set. A significant challenge in writing equations representing side-channel data is measurement noise. In this work we show how casting the problem as a pseudo-Boolean optimization instance provides an efficient and effective way of tolerating this noise. We describe a theoretical analysis, connecting the measurement signal-to-noise ratio and the tolerable set size of a non-optimizing solver with the success probability. We then conduct an extensive performance evaluation, comparing two optimizing variants for dealing with measurement noise to a non-optimizing method. Our best optimizing method provides a successful attack on the AES cipher which requires surprisingly little side-channel data and works in reasonable computation time. We also make available a set of AES cryptanalysis instances and provide some practical feedback on our experience of using open-source constraint solvers.
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Notes
It is believed that this form of attack was well known to the signals intelligence community from as early as WWII.
This is not a sufficient condition – even if all leaks are recovered correctly the problem may still be under-defined or computationally intractable.
It was already established in [13] that the Hamming weights leaked from an 8-bit micro-controller implementation of AES during key expansion are sufficient for full key recovery, even without any additional state information.
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Acknowledgements
The authors thank the anonymous reviewers for their detailed and helpful suggestions. The authors wish to acknowledge Mario Kirschbaum, Thomas Popp, Mathieu Renauld and Francois-Xavier Standaert for their contributions to this research.
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Appendix: A sample TASCA instance
Appendix: A sample TASCA instance
The appendix demonstrates some of the equations used in a TASCA attack on AES with set size k = 3, following the notation introduced in Subsection 5.1. We show a sample of the goal function, the plaintext assignment, the round functions and the measurement equations. The equations are given in the OPB format supported by the SCIP solver [4]. Full instances can be downloaded from our website at [26].
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Oren, Y., Wool, A. Side-channel cryptographic attacks using pseudo-boolean optimization. Constraints 21, 616–645 (2016). https://doi.org/10.1007/s10601-015-9237-3
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DOI: https://doi.org/10.1007/s10601-015-9237-3