Abstract
We consider a game-theoretic setting to model the interplay between attacker and defender in the context of information flow, and to reason about their optimal strategies. In contrast with standard game theory, in our games the utility of a mixed strategy is a convex function of the distribution on the defender’s pure actions, rather than the expected value of their utilities. Nevertheless, the important properties of game theory, notably the existence of a Nash equilibrium, still hold for our (zero-sum) leakage games, and we provide algorithms to compute the corresponding optimal strategies. As typical in (simultaneous) game theory, the optimal strategy is usually mixed, i.e., probabilistic, for both the attacker and the defender. From the point of view of information flow, this was to be expected in the case of the defender, since it is well known that randomization at the level of the system design may help to reduce information leaks. Regarding the attacker, however, this seems the first work (w.r.t. the literature in information flow) proving formally that in certain cases the optimal attack strategy is necessarily probabilistic.
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Notes
- 1.
Following the convention of security games, we set the first player to be the defender.
- 2.
Conventionally in game theory the utility u is set to be that of the first player, but we prefer to look at the utility from the point of view of the attacker to be in line with the definition of utility as vulnerability, as we will introduce in Sect. 2.3.
- 3.
More precisely, if posterior vulnerability is defined as the expectation of the vulnerability of posterior distributions, the measure respects the data-processing inequality and yields non-negative leakage iff vulnerability is convex.
- 4.
The reason to involve Jeeves is that Alice may not want to reveal a to Don, either.
- 5.
Note that d should not be revealed to the attacker: although d is not sensitive information in itself, knowing it would help the attacker figure out the value of x.
- 6.
Note that two channel matrices with different column indices can always be made compatible by adding appropriate columns with 0-valued cells in each of them.
- 7.
Note that this is true only for \(\delta \), the \(\alpha \)-solution of the minimax problem is not necessarily part of an equilibrium; we need to solve the maximin problem for this.
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Acknowledgments
The authors are thankful to Arman Khouzani and Pedro O. S. Vaz de Melo for valuable discussions. This work was supported by JSPS and Inria under the project LOGIS of the Japan-France AYAME Program, and by the project Epistemic Interactive Concurrency (EPIC) from the STIC AmSud Program. Mário S. Alvim was supported by CNPq, CAPES, and FAPEMIG. Yusuke Kawamoto was supported by JSPS KAKENHI Grant Number JP17K12667.
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Alvim, M.S., Chatzikokolakis, K., Kawamoto, Y., Palamidessi, C. (2017). Information Leakage Games. In: Rass, S., An, B., Kiekintveld, C., Fang, F., Schauer, S. (eds) Decision and Game Theory for Security. GameSec 2017. Lecture Notes in Computer Science(), vol 10575. Springer, Cham. https://doi.org/10.1007/978-3-319-68711-7_23
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