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Information Theoretic Comparison of Side-Channel Distinguishers: Inter-class Distance, Confusion, and Success

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Trusted Computing for Embedded Systems

Abstract

Different side-channel distinguishers have different efficiencies. Their fair comparison is a difficult task because many factors come into play—in particular, their intrinsic statistical properties and the quality of their estimation.

In this work, we first evaluate two related information-theoretic distinguishers: mutual information analysis and inter-class information analysis. The latter requires the same underlying probability distributions but uses a different comparing strategy. These distinguishers are not only interesting on their own and suitable for a mathematical study, but they also exhibit an example where the theoretical and empirical evaluation framework agree. The IIA was found to distinguish better than MIA in theory as well as in practice.

Moreover, we develop a new metric, called success metric, capturing the relevant parameters of the success rate, while providing more feedback about the distinguishing power. We additionally state closed-form expressions of the theoretical success metric for additive distinguisher like CPA and DPA and highlight that these expressions are much more convenient than for the theoretical success rate. In the case of a low signal-to-noise ratio (realistic practical condition), we derive the conditions on the cipher’s substitution boxes (sboxes) to minimize the success metric (hence the success rate). This result supersedes a previous characterization on sboxes known as transparency order, which is derived from a metric on a distinguisher, and not from a success metric/rate. Moreover, we are also able to formulate a closed-form expression for MIA, which has not been shown before.

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Notes

  1. 1.

    Another reason is that differential entropy is not “coordinate-free” – it depends on the underlying coordinate system.

  2. 2.

    Note that, unlike in our previous definitions, the random variable Y is also continuous in this example. Thus sums have to be replaced by integrals.

  3. 3.

    Note that this equivalence of metrics is not the same as the equivalence between distinguishers stated in [8].

  4. 4.

    A well-known information-theoretic property commonly referred to as “mixing increases entropy”.

  5. 5.

    Interestingly, it is not true that II(X; Y ) ≥ I(X; Y ) for general random variables X and Y. For example, we can find a counterexample when X, Y are binary variables with small p(x | y) for all x, y ≠ 0.

  6. 6.

    In [27] the term exact instead of theoretical is used.

  7. 7.

    Note that, in some publications, the relative distinguishing margin is also called nearest-rival distinguishing score.

  8. 8.

    Note that, a similar model was also implicitly used in [9, 37].

  9. 9.

    Namely \(\left [\kappa (k^{{\ast}},i,j)\right ]_{(i,j)\in \mathcal{K}\setminus \{0\}}\) and \(\left [\kappa (k^{{\ast}},i) \times \kappa (k^{{\ast}},j)\right ]_{(i,j)\in \mathcal{K}\setminus \{0\}}\).

  10. 10.

    One can easily extend the calculation also for the Hamming distance model.

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Acknowledgements

Annelie Heuser is partly founded by the Google Doctoral European Fellowship in the field of privacy.

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Authors and Affiliations

  1. Department Comelec, Telecom ParisTech, Institut Mines-Telecom, CNRS LTCI, 46 rue Barrault, 75 634, Paris Cedex 13, France

    Annelie Heuser, Olivier Rioul, Sylvain Guilley & Jean-Luc Danger

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  1. Annelie Heuser
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Correspondence to Annelie Heuser .

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  1. Thales Communications, Colombes, France

    Bernard Candaele

  2. Department of Computer Science, National Technical University of Athens, Athens, Greece

    Dimitrios Soudris

  3. Department of Computer Science, National Technical University of Athens, Athens, Greece

    Iraklis Anagnostopoulos

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Heuser, A., Rioul, O., Guilley, S., Danger, JL. (2015). Information Theoretic Comparison of Side-Channel Distinguishers: Inter-class Distance, Confusion, and Success. In: Candaele, B., Soudris, D., Anagnostopoulos, I. (eds) Trusted Computing for Embedded Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-09420-5_10

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