1 Introduction
A fundamental problem in computer security is the leakage of sensitive information due to correlation of secret values with observables
—i.e., any information accessible to the attacker, such as, for instance, the system’s outputs or execution time. The typical defense consists in reducing this correlation, which can be done in, essentially, two ways. The first, applicable when the correspondence secretobservable is deterministic, consists in coarsening the equivalence classes of secrets that give rise to the same observables. This can be achieved with postprocessing, i.e., sequentially composing the original system with a program that removes information from observables. For example, a typical attack on encrypted web traffic consists in the analysis of the packets’ length, and a typical defense consists in padding extra bits so to diminish the length variety
Sun et al. (2002).The second kind of defense, on which we focus in this work, consists in adding controlled noise to the observables produced by the system. This can be usually seen as a composition of different protocols via probabilistic choice. [Differential privacy] Consider a counting query , namely a function that, applied to a dataset , returns the number of individuals in that satisfy a given property. A way to implement differential privacy Dwork et al. (2006) is to add geometrical noise to the result of
, so to obtain a probability distribution
on integers of the form , where is a normalization factor. The resulting mechanism can be interpreted as a probabilistic choice on protocols of the form , where the probability assigned to and to decreases exponentially with . [Dining cryptographers] Consider two agents running the dining cryptographers protocol Chaum (1988), which consists in tossing a fair binary coin and then declaring the exclusive or of their secret value and the result of the coin. The protocol can be thought as the fair probabilistic choice of two protocols, one consisting simply of declaring , and the other declaring .Most of the work in the literature of quantitative information flow (QIF) considers passive attacks, in which the attacker only observes the system. Notable exceptions are the works Boreale and Pampaloni (2015); Mardziel et al. (2014); Alvim et al. (2017), which consider attackers who interact with and influence the system, possibly in an adaptive way, with the purpose of maximizing the leakage of information.
[CRIME attack] Compression Ratio Infoleak Made Easy (CRIME) Rizzo and Duong (2012) is a security exploit against secret web cookies over connections using the HTTPS and SPDY protocols and data compression. The idea is that the attacker can inject some content in the communication of the secret from the target site to the server. The server then compresses and encrypts the data, including both and , and sends back the result. By observing the length of the result, the attacker can then infer information about . To mitigate the leakage, one possible defense would consist in transmitting, along with , also an encryption method selected randomly from a set . Again, the resulting protocol can be seen as a composition, using probabilistic choice, of the protocols in the set .
[Timing sidechannels] Consider a passwordchecker, or any similar system in which the user authenticates himself by entering a secret that is checked by the system. An adversary does not know the real secret, of course, but a timing sidechannel could reveal the part (eg. which bit) of the secret in which the adversary’s input fails. By repeating the process with different inputs, the adversary might be able to fully retrieve the secret. A possible counter measure is to make the side channel noisy, by randomizing the order in which the secret’s bit are checked against the user input. This example is studied in detaul in Section 7.
In all examples above the main use of the probabilistic choice is to obfuscate the relation between secrets and observables, thus reducing their correlation—and, hence, the information leakage. To achieve this goal, it is essential that the attacker never comes to know the result of the choice. In the CRIME example, however, if and are chosen independently, then (in general) it is still better to choose probabilistically, even if the attacker will come to know, afterwards, the choice of . In fact, this is true also for the attacker: his best strategies (in general) are to choose according to some probability distribution. Indeed, suppose that are the defender’s choices and are the attacker’s, and that leaks more than , while leaks less than . This is a scenario like the matching pennies in game theory: if one player selects an action deterministically, the other player may exploit this choice and get an advantage. For each player the optimal strategy is to play probabilistically, using a distribution that maximizes his own gain for all possible actions of the attacker. In zerosum games, in which the gain of one player coincides with the loss of the other, the optimal pair of distributions always exists, and it is called saddle point. It also coincides with the Nash equilibrium, which is defined as the point in which neither of the two players gets any advantage in changing unilaterally his strategy.
Motivated by these examples, this paper investigates the two kinds of choice, visible and hidden (to the attacker), in a gametheoretic setting. Looking at them as language operators, we study their algebraic properties, which will help reason about their behavior in games. We consider zerosum games, in which the gain (for the attacker) is represented by the leakage. While for visible choice it is appropriate to use the “classic” gametheoretic framework, for hidden choice we need to adopt the more general framework of the information leakage games proposed in Alvim et al. (2017). This happens because, in contrast with standard game theory, in games with hidden choice the payoff of a mixed strategy is a convex function of the distribution on the defender’s pure actions, rather than simply the expected value of their utilities. We will consider both simultaneous games—in which each player chooses independently—and sequential games— in which one player chooses his action first. We aim at comparing all these situations, and at identifying the precise advantage of the hidden choice over the visible one.
To measure leakage we use the wellknown informationtheoretic model.
A central notion in this model is that of entropy, but here
we use its converse, vulnerability, which
represents the magnitude of the threat.
In order to derive results as general as possible, we adopt
the very comprehensive notion of vulnerability as any convex and continuous
function, as used in Boreale and Pampaloni (2015) and
Alvim et al. (2016).
This notion has been shown Alvim et al. (2016)
to be, in a precise sense, the most general
information measures w.r.t. a set of fundamental
informationtheoretic axioms.
Our results, hence, apply to all information measures that respect such
fundamental principles, including the widely adopted measures of Bayes vulnerability (aka minvulnerability, aka (the converse of)
Bayes risk) Smith (2009); Chatzikokolakis et al. (2008),
Shannon entropy Shannon (1948),
guessing entropy Massey (1994), and
vulnerability Alvim et al. (2012).
The main contributions of this paper are:

We present a general framework for reasoning about information leakage in a gametheoretic setting, extending the notion of information leakage games proposed in Alvim et al. (2017) to both simultaneous and sequential games, with either hidden or visible choice.

We present a rigorous compositional way, using visible and hidden choice operators, for representing attacker’s and defender’s actions in information leakage games. In particular, we study the algebraic properties of visible and hidden choice on channels, and compare the two kinds of choice with respect to the capability of reducing leakage, in presence of an adaptive attacker.

We provide a taxonomy of the various scenarios (simultaneous and sequential) showing when randomization is necessary, for either attacker or defender, to achieve optimality. Although it is wellknown in information flow that the defender’s best strategy is usually randomized, only recently it has been shown that when defender and attacker act simultaneously, the attacker’s optimal strategy also requires randomization Alvim et al. (2017).

We compare the vulnerability of the leakage games for these various scenarios and establish a hierarchy of leakage games based on the order between the value of the leakage in the Nash equilibrium. Furthermore, we show that when the attacker moves first in a sequential game with hidden choice, the behavioral strategies (where the defender chooses his probabilistic distribution after he has seen the choice of the attacker) are more advantageous for the defender than the mixed strategies (where the defender chooses the probabilistic distribution over his possible functional dependency on the choice of the attacker). This contrast with the standard game theory, where the two types of strategies are equivalent. Another difference is that in our attackerfirst sequential games there may not exist Nash equilibria with deterministic strategies for the defender (although the defender has full visibility of the attacker’s moves).

We use our framework in a detailed case study of a passwordchecking protocol. A naive program, which checks the password bit by bit and stops when it finds a mismatch, is clearly very insecure, because it reveals at each attempt (via a timing sidechannel), the maximum correct prefix. On the other hand, if we continue checking until the end of the string (time padding), the program becomes very inefficient. We show that, by using probabilistic choice instead, we can obtain a good tradeoff between security and efficiency.
Plan of the paper.
The remaining of the paper is organized as follows. In Section 2 we review some basic notions of game theory and quantitative information flow. In Section 3 we introduce our running example. In Section 4 we define the visible and hidden choice operators and demonstrate their algebraic properties. In Section 5, the core of the paper, we examine various scenarios for leakage games. In Section 6 we compare the vulnerability of the various leakage games, and establish a hierarchy among those games. In Section 7 we show an application of our framework to a password checker. In Section 8 we discuss related work and, finally, in Section 9 we conclude.
A preliminary version of this paper appeared in Alvim et al. (2018). One difference with respect to Alvim et al. (2018) is that in the present paper we consider both behavioral and mixed strategies in the sequential games, while in Alvim et al. (2018) we only considered the latter. We also show that the two kinds of strategies are not equivalent in our context (Example 5.1.6: the optimal strategy profile yields a different payoff depending on whether the defender adopts mixed strategies or behavioral ones). In light of this difference, we provide new results that concern behavioral strategies, and in particular:

Theorem 5.1.2, which concerns the defender’s behavioral strategies in the defenderfirst game with visible choice (Game II) ,

the second half of Theorem 5.1.6, which deals with the adversary’s behavioral strategies in the attackerfirst game with hidden choice (Game VI).
Furthermore, in this paper we define formally all concepts, and provide all the proofs. In particular, we provide a precise formulation of the comparison among games with visible/hidden choices (Propositions 6.2 and 6.2, Corollaries 6.2, 6.2, and 6.2) in Section 6. Finally, in Section 7 we provide a new result, expressed by Theorem 7.2, regarding the optimal strategies for the defender in the presence of a uniform prior on passwords.
2 Preliminaries
In this section we review some basic notions from game theory and quantitative information flow. We use the following notation: Given a set , we denote by the set of all probability distributions over . Given , its support is the set of its elements with positive probability. We use to indicate that a value is sampled from a distribution on . A set is convex if for all and . For such a set, a function is convex if for all , and concave if is convex.
2.1 Basic concepts from game theory
2.1.1 Twoplayer games
Twoplayer games are a model for reasoning about the behavior of two players. In a game, each player has at its disposal a set of actions that he can perform, and he obtains some gain or loss depending on the actions chosen by both players. Gains and losses are defined using a realvalued payoff function. Each player is assumed to be rational, i.e., his choice is driven by the attempt to maximize his own expected payoff. We also assume that the set of possible actions and the payoff functions of both players are common knowledge.
In this paper we only consider finite games, in which the set of actions available to the players are finite, and which are also zerosum games, so the payoff of one player is the loss of the other. Next we introduce an important distinction between simultaneous and sequential games. In the following, we will call the two players defender and attacker.
2.1.2 Simultaneous games
In a simultaneous game, each player chooses his action without knowing the action chosen by the other. The term “simultaneous” here does not mean that the players’ actions are chosen at the same time, but only that they are chosen independently. Formally, such a game is defined as a tuple^{1}^{1}1Following the convention of security games, we set the first player to be the defender. , where is a nonempty set of defender’s actions, is a nonempty set of attacker’s actions, is the defender’s payoff function, and is the attacker’s payoff function.
Each player may choose an action deterministically or probabilistically. A pure strategy of the defender (resp. attacker) is a deterministic choice of an action, i.e., an element (resp. ). A pair is called pure strategy profile, and , represent the defender’s and the attacker’s payoffs, respectively. A mixed strategy of the defender (resp. attacker) is a probabilistic choice of an action, defined as a probability distribution (resp. ). A pair is called mixed strategy profile. The defender’s and the attacker’s expected payoff functions for mixed strategies are defined, respectively, as:
A defender’s mixed strategy is a best response to an attacker’s mixed strategy if . Symmetrically, is a best response to if . A mixedstrategy Nash equilibrium is a profile such that is a best response to and vice versa. This means that in a Nash equilibrium, no unilateral deviation by any single player provides better payoff to that player. If and are point distributions concentrated on some and respectively, then is a purestrategy Nash equilibrium, and will be denoted by . While not all games have a pure strategy Nash equilibrium, every finite game has a mixed strategy Nash equilibrium.
2.1.3 Sequential games
In a sequential game players may take turns in choosing their actions. In this paper, we only consider the case in which each player moves only once, in such a way that one of the players (the leader) chooses his action first, and commits to it, before the other player (the follower) makes his choice. The follower may have total knowledge of the choice made by the leader, or only partial. We refer to the two scenarios by the terms perfect and imperfect information, respectively. Another distinction is the kind of randomization used by the players, namely whether the follower chooses probabilistically his action after he knows (partially or totally) the move of the leader, or whether he chooses at the beginning of the game a probabilistic distribution on (deterministic) strategies that depend on the (partial or total) knowledge of the move of the leader. In the first case the strategies are called behavioral, in the second case mixed.
We now give the precise definitions assuming that the leader is the defender. The definitions for the case in which the leader is the attacker are analogous.
A defenderfirst sequential game with perfect information is a tuple where , , and are defined as in simultaneous games: The choice of an action represents a pure strategy of the defender. As for the attacker, his choice depends functionally on the prior choice of the defender, and for this reason the pure strategies of the attacker are functions . As for the probabilistic strategies, those of the defender are defined as in simultaneous games: namely they are distributions . On the other hand, the attacker’s probabilistic strategies can be defined in two different ways: In the behavioral case, an attacker’s probabilistic strategy is a function . Namely, the attacker chooses a distribution on its actions after he sees the move of the defender. In the mixed case, an attacker’s probabilistic strategy is a probability distribution . Namely, the attacker chooses a priori a distribution on pure strategies. The defender’s and the attacker’s expected payoff functions for mixed strategies are defined, respectively, as
Behavioral case  
Mixed case 
The case of imperfect information is typically formalized by assuming an indistinguishability (equivalence) relation over the actions chosen by the leader, representing a scenario in which the follower cannot distinguish between the actions belonging to the same equivalence class. The pure strategies of the followers, therefore, are functions from the set of the equivalence classes on the actions of the leader to his own actions. Formally, a defenderfirst sequential game with imperfect information is a tuple where , , and are defined as in simultaneous games, and is a partition of . The expected payoff functions are defined as before, except that now the argument of and is the equivalence class of . Note that in the case in which all defender’s actions are indistinguishable from each other at the eyes of the attacker (totally imperfect information), we have and the expected payoff functions coincide with those of the simultaneous games. In contrast, in the games in which all defender’s actions are distinguishable from the view of the attacker (perfect information), we have .
In the standard game theory, under the assumption of perfect recall (i.e., the players never forget what they have learned), behavioral and mixed strategies are equivalent, in the sense that for any behavioral strategy there is a mixed strategy that yields the same payoff, and vice versa. This is true for both cases of perfect and imperfect information, see Osborne and Rubinstein (1994), Chapter 11.4. In our leakage games, however, this equivalence does not hold anymore, as it will be shown in Sections 5 and 6.
2.1.4 Zerosum games and Minimax Theorem
A game is zerosum if for any and any , the defender’s loss is equivalent to the attacker’s gain, i.e., . For brevity, in zerosum games we denote by the attacker’s payoff function , and by the attacker’s expected payoff .^{2}^{2}2Conventionally in game theory the payoff is set to be that of the first player, but we prefer to look at the payoff from the point of view of the attacker to be in line with the definition of payoff as vulnerability. Consequently, the goal of the defender is to minimize , and the goal of the attacker is to maximize it.
In simultaneous zerosum games the Nash equilibrium corresponds to the solution of the minimax problem (or equivalently, the maximin problem), namely, the strategy profile such that . The von Neumann’s minimax theorem, in fact, ensures that such solution (which always exists) is stable.
[von Neumann’s minimax theorem] Let and be compact convex sets, and be a continuous function such that is a convex function in and a concave function in . Then
A related property is that, under the conditions of Theorem 2.1.4, there exists a saddle point s.t., for all and : .
The solution of the minimax problem can be obtained by using convex optimization techniques. In case is affine in and in , we can also use linear optimization.
In case and contain two elements each, there is a closed form for the solution. Let and respectively. Let be the payoff of the defender on . Then the Nash equilibrium is given by:
(1) 
if these values are in . Note that, since there are only two elements, the strategy is completely specified by its value in , and analogously for .
2.2 Quantitative information flow
Finally, we briefly review the standard framework of quantitative information flow, which is concerned with measuring the amount of information leakage in a (computational) system.
2.2.1 Secrets and vulnerability
A secret is some piece of sensitive information the defender wants to protect, such as a user’s password, social security number, or current location. The attacker usually only has some partial knowledge about the value of a secret, represented as a probability distribution on secrets called a prior. We denote by the set of possible secrets, and we typically use to denote a prior belonging to the set of probability distributions over .
The vulnerability of a secret is a measure of the payoff that it represents for the attacker. In this paper we consider a very general notion of vulnerability, following Alvim et al. (2016), and we define a vulnerability to be any continuous and convex function of type . It has been shown in Alvim et al. (2016) that these functions coincide with the set of vulnerabilities, and are, in a precise sense, the most general information measures w.r.t. a set of fundamental informationtheoretic axioms. ^{3}^{3}3More precisely, if posterior vulnerability is defined as the expectation of the vulnerability of posterior distributions, the measure respects the fundamental informationtheoretic properties of dataprocessing inequality (i.e., that postprocessing can never increase information, but only destroy it), and of nonnegativity of leakage (i.e., that by observing the output of a channel an actor cannot, in average, lose information) if, and only if, vulnerability is convex. This notion, hence, subsumes all information measures that respect such fundamental principles, including the widely adopted measures of Bayes vulnerability (aka minvulnerability, aka (the converse of) Bayes risk) Smith (2009); Chatzikokolakis et al. (2008), Shannon entropy Shannon (1948), guessing entropy Massey (1994), and vulnerability Alvim et al. (2012).
2.2.2 Channels, posterior vulnerability, and leakage
Computational systems can be modeled as information theoretic channels. A channel is a function in which is a set of input values, is a set of output values, and represents the conditional probability of the channel producing output when input is provided. Every channel satisfies for all and , and for all .
A distribution and a channel with inputs and outputs
induce a joint distribution
on, producing joint random variables
with marginal probabilities and , and conditional probabilities if . For a given (s.t. ), the conditional probabilities for each form the posterior distribution .A channel in which is a set of secret values and is a set of observable values produced by a system can be used to model computations on secrets. Assuming the attacker has prior knowledge about the secret value, knows how a channel works, and can observe the channel’s outputs, the effect of the channel is to update the attacker’s knowledge from to a collection of posteriors , each occurring with probability .
Given a vulnerability , a prior , and a channel , the posterior vulnerability is the vulnerability of the secret after the attacker has observed the output of the channel . Formally: .
Consider, for instance, the example of the passwordchecker with a timing sidechannel from the introduction (Example 1, also discussed in detail in Section 7). Here the set of secrets consists of all possible passwords (say, all strings of bits), and a natural vulnerability function is Bayesvulnerability, given by . This function expresses the adversary’s probability of guessing correctly the password in one try; assuming that the passwords are chosen uniformly, i.e. is uniform, any guess would be correct with probability , giving . Now, imagine that the timing sidechannel reveals that the adversary’s input failed on the first bit. The adversary now knows the first bit of the password (say ), hence the posterior assigns probability to all passwords with first bit , and probability to all passwords with first bit . This happens for all possible posteriors, giving posterior vulnerability ( times greater than the prior ).
It is known from the literature Alvim et al. (2016) that the posterior vulnerability is a convex function of . Namely, for any channel , any family of distributions , and any set of convex coefficients , we have:
The (information) leakage of channel under prior is a comparison between the vulnerability of the secret before the system was run—called prior vulnerability—and the posterior vulnerability of the secret. Leakage reflects by how much the observation of the system’s outputs increases the attacker’s information about the secret. It can be defined either additively (), or multiplicatively (). In the passwordchecker example, the additive leakage is and the multiplicative leakage is .
3 An illustrative example
We introduce an example which will serve as running example through the paper. Although admittedly contrived, this example is simple and yet produces different leakage measures for all different combinations of visible/hidden choice and simultaneous/sequential games, thus providing a way to compare all different scenarios we are interested in.
Consider that a binary secret must be processed by a program. As usual, a defender wants to protect the secret value, whereas an attacker wants to infer it by observing the system’s output. Assume the defender can choose which among two alternative versions of the program to run. Both programs take the secret value as high input, and a binary low input whose value is chosen by the attacker. They both return the output in a low variable . ^{4}^{4}4We adopt the usual convention in QIF of referring to secret variables, inputs and outputs in programs as high, and to their observable counterparts as low. Program 0 returns the binary product of and , whereas Program 1 flips a coin with bias (i.e., a coin which returns heads with probability ) and returns if the result is heads, and the complement of otherwise. The two programs are represented in Figure 1.
The combined choices of the defender’s and of the attacker’s determine how the system behaves. Let represent the set of the defender’s choices—i.e., the index of the program to use—, and represent the set of the attacker’s choices—i.e., the value of the low input . We shall refer to the elements of and as actions. For each possible combination of actions and , we can construct a channel modeling how the resulting system behaves. Each channel is a function of type , where is the set of possible high input values for the system, and is the set of possible output values from the system. Intuitively, each channel provides the probability that the system (which was fixed by the defender) produces output given that the high input is (and that the low input was fixed by the attacker). The four possible channels are depicted in Table 0(a).


Note that channel does not leak any information about the input (i.e., it is noninterferent), whereas channels and completely reveal . Channel is an intermediate case: it leaks some information about , but not all.
We want to investigate how the defender’s and the attacker’s choices influence the leakage of the system. For that we can just consider the (simpler) notion of posterior vulnerability, since in order to make the comparison fair we need to assume that the prior is always the same in the various scenarios, and this implies that the leakage is in a onetoone correspondence with the posterior vulnerability (this happens for both additive and multiplicative leakage).
For this example, assume we are interested in Bayes vulnerability Chatzikokolakis et al. (2008); Smith (2009), defined as for every . Assume for simplicity that the prior is the uniform prior . In this case we know from Braun et al. (2009) that the posterior Bayes vulnerability of a channel is the sum of the greatest elements of each column, divided by the total number of inputs. Table 0(b) provides the Bayes vulnerability of each channel considered above.
Naturally, the attacker aims at maximizing the vulnerability of the system, while the defender tries to minimize it. The resulting vulnerability will depend on various factors, in particular on whether the two players make their choice simultaneously (i.e., without knowing the choice of the opponent) or sequentially. Clearly, if the choice of a player who moves first is known by an opponent who moves second, the opponent will be in advantage. In the above example, for instance, if the defender knows the choice of the attacker, the most convenient choice for him is to set , and the vulnerability will be at most . Vice versa, if the attacker knows the choice of the defender, the most convenient choice for him is to set . The vulnerability in this case will be .
Things become more complicated when players make choices simultaneously. None of the pure choices of and are the best for the corresponding player, because the vulnerability of the system depends also on the (unknown) choice of the other player. Yet there is a strategy leading to the best possible situation for both players (the Nash equilibrium), but it is mixed (i.e., probabilistic), in that the players randomize their choices according to some precise distribution.
Another factor that affects vulnerability is whether or not the defender’s choice is known to the attacker at the moment in which he observes the output of the channel. Obviously, this corresponds to whether or not the attacker knows what channel he is observing. Both cases are plausible: naturally the defender has all the interest in keeping his choice (and, hence, the channel used) secret, since then the attack will be less effective (i.e., leakage will be smaller). On the other hand, the attacker may be able to identify the channel used anyway, for instance because the two programs have different running times. We will call these two cases
hidden and visible choice, respectively.It is possible to model players’ strategies, as well as hidden and visible choices, as operations on channels. This means that we can look at the whole system as if it were a single channel, which will turn out to be useful for some proofs of our technical results. Next section is dedicated to the definition of these operators. We will calculate the exact values for our example in Section 5.
4 Choice operators for protocol composition
In this section we define the operators of visible and hidden choice for protocol composition. These operators are formally defined on the channel matrices of the protocols, and, since channels are a particular kind of matrix, we use these matrix operations to define the operations of visible and hidden choice among channels, and to prove important properties of these channel operations.
4.1 Matrices, and their basic operators
Given two sets and , a matrix is a total function of type . Two matrices and are said to be compatible if . If it is also the case that , we say that the matrices have the same type. The scalar multiplication between a scalar and a matrix is defined as usual, and so is the summation of a family of matrices all of a same type.
Given a family of compatible matrices s.t. each has type , their concatenation is the matrix having all columns of every matrix in the family, in such a way that every column is tagged with the matrix it came from. Formally, , if , and the resulting matrix has type . ^{5}^{5}5 denotes the disjoint union of the sets , , , . When the family has only two elements we may use the binary version of the concatenation operator. The following depicts the concatenation of two matrices and in tabular form.
4.2 Channels, and their hidden and visible choice operators
A channel is a stochastic matrix, i.e., all elements are nonnegative, and all rows sum up to 1. Here we will define two operators specific for channels. In the following, for any real value , we denote by the value .
4.2.1 Hidden choice
The first operator models a hidden probabilistic choice among channels. Consider a family of channels of a same type. Let be a probability distribution on the elements of the index set . Consider an input is fed to one of the channels in , where the channel is randomly picked according to . More precisely, an index is sampled with probability , then the input is fed to channel , and the output produced by the channel is then made visible, but not the index of the channel that was used. Note that we consider hidden choice only among channels of a same type: if the sets of outputs were not identical, the produced output might implicitly reveal which channel was used.
Formally, given a family of channels s.t. each has same type , the hidden choice operator is defined as .
Proposition 4.1 (Type of hidden choice).
Given a family of channels of type , and a distribution on , the hidden choice is a channel of type .
See Appendix A for the proof.
In the particular case in which the family has only two elements and , the distribution on indexes is completely determined by a real value s.t. and . In this case we may use the binary version of the hidden choice operator: . The example below depicts the hidden choice between channels and , with probability .
4.2.2 Visible choice
The second operator models a visible probabilistic choice among channels. Consider a family of compatible channels. Let be a probability distribution on the elements of the index set . Consider an input is fed to one of the channels in , where the channel is randomly picked according to . More precisely, an index is sampled with probability , then the input is fed to channel , and the output produced by the channel is then made visible, along with the index of the channel that was used. Note that visible choice makes sense only between compatible channels, but it is not required that the output set of each channel be the same.
Formally, given of compatible channels s.t. each has type , and a distribution on , the visible choice operator is defined as .
Proposition 4.2 (Type of visible choice).
Given a family of compatible channels s.t. each has type , and a distribution on , the result of the visible choice is a channel of type .
See Appendix A for the proof.
In the particular case the family has only two elements and , the distribution on indexes is completely determined by a real value s.t. and . In this case we may use the binary version of the visible choice operator: . The following depicts the visible choice betwee channels and , with probability .
4.3 Properties of hidden and visible choice operators
We now prove algebraic properties of channel operators. These properties will be useful when we model a (more complex) protocol as the composition of smaller channels via hidden or visible choice.
Whereas the properties of hidden choice hold generally with equality, those of visible choice are subtler. For instance, visible choice is not idempotent, since in general . (In fact if has type , has type .) However, idempotency and other properties involving visible choice hold if we replace the notion of equality with the more relaxed notion of “equivalence” between channels. Intuitively, two channels are equivalent if they have the same input space and yield the same value of vulnerability for every prior and every vulnerability function.
[Equivalence of channels] Two compatible channels and with domain are equivalent, denoted by , if for every prior and every posterior vulnerability we have .
Two equivalent channels are indistinguishable from the point of view of information leakage, and in most cases we can just identify them. Indeed, nowadays there is a tendency to use abstract channels McIver et al. (2014); Alvim et al. (2016), which capture exactly the important behavior with respect to any form of leakage. In this paper, however, we cannot use abstract channels because the hidden choice operator needs a concrete representation in order to be defined unambiguously.
The first properties we prove regard idempotency of operators, which can be used do simplify the representation of some protocols.
Proposition 4.3 (Idempotency).
Given a family of channels s.t. for all , and a distribution on , then: (a) ; and (b) .
See Appendix A for the proof.
The following properties regard the reorganization of operators, and they will be essential in some technical results in which we invert the order in which hidden and visible choice are applied in a protocol.
Proposition 4.4 (“Reorganization of operators”).
Given a family of channels indexed by sets and , a distribution on , and a distribution on :

, if all ’s have the same type;

, if all ’s are compatible; and

, if, for each , all ’s have same type .
See Appendix A for the proof.
Finally, analogous properties of the binary operators are shown in Appendix B.
4.4 Properties of vulnerability w.r.t. channel operators
We now derive some relevant properties of vulnerability w.r.t. our channel operators, which will be later used to obtain the Nash equilibria in information leakage games with different choice operations.
The first result states that posterior vulnerability is convex w.r.t. hidden choice (this result was already presented in Alvim et al. (2017)), and linear w.r.t. to visible choice.
[Convexity/linearity of posterior vulnerability w.r.t. choices] Let be a family of channels, and be a distribution on . Then, for every distribution on , and every vulnerability :

posterior vulnerability is convex w.r.t. to hidden choice: if all ’s have the same type.

posterior vulnerability is linear w.r.t. to visible choice: if all ’s are compatible.
Proof.

Let us call the type of each channel in the family . Then:
(by definition of hidden choice) (by definition of posterior ) (by convexity of ) where .

Let us call the type of each channel in the family . Then:
(by definition of visible choice) (by definition of posterior ) (see (*) below) where , and step (*) holds because in the vulnerability of a concatenation of matrices every column will contribute to the vulnerability in proportion to its weight in the concatenation, and hence it is possible to break the vulnerability of a concatenated matrix as the weighted sum of the vulnerabilities of its submatrices.
∎
The next result is concerned with posterior vulnerability under the composition of channels using both operators.
[Convexlinear payoff function] Let be a family of channels, all with domain and with the same type, and let , and be any vulnerability. Define as follows: . Then U is convex on and linear on .
5 Information leakage games
In this section we present our framework for reasoning about information leakage, extending the notion of information leakage games proposed in Alvim et al. (2017) from only simultaneous games with hidden choice to both simultaneous and sequential games, with either hidden or visible choice.
In an information leakage game the defender tries to minimize the leakage of information from the system, while the attacker tries to maximize it. In this basic scenario, their goals are just opposite (zerosum). Both of them can influence the execution and the observable behavior of the system via a specific set of actions. We assume players to be rational (i.e., they are able to figure out what is the best strategy to maximize their expected payoff), and that the set of actions and the payoff function are common knowledge.
Players choose their own strategy, which in general may be probabilistic (i.e., behavioral or mixed) and choose their action by a random draw according to that strategy. After both players have performed their actions, the system runs and produces some output value which is visible to the attacker and may leak some information about the secret. The amount of leakage constitutes the attacker’s gain, and the defender’s loss.
To quantify the leakage we model the system as an informationtheoretic channel (cf. Section 2.2). We recall that leakage is defined as the difference (additive leakage) or the ratio (multiplicative leakage) between posterior and prior vulnerability. Since we are only interested in comparing the leakage of different channels for a given prior, we will define the payoff just as the posterior vulnerability, as the value of prior vulnerability will be the same for every channel.
5.1 Defining information leakage games
An (information) leakage game consists of:

two nonempty sets , of defender’s and attacker’s actions, respectively,

a function that associates to each pair of actions a channel ,

a prior on secrets, and

a vulnerability measure , used to define the payoff function for pure strategies as . We have only one payoff function because the game is zerosum.
Like in traditional game theory, the order of actions and the extent by which a player knows the move performed by the opponent play a critical role in deciding strategies and determining the payoff. In security, however, knowledge of the opponent’s move affects the game in yet another way: the effectiveness of the attack, i.e., the amount of leakage, depends crucially on whether or not the attacker knows what channel is being used. It is therefore convenient to distinguish two phases in the leakage game:
 Phase 1  Determination of players’ strategies, and the subsequent choice of their actions:

Each player determines the most convenient strategy (which in general is probabilistic) for himself, and draws his action accordingly. One of the players may commit first to his action, and his choice may or may not be revealed to the follower. In general, knowledge of the leader’s action may help the follower choose a more advantageous strategy.
 Phase 2  Observation of the resulting channel’s output, and payoff computation:

The attacker observes the output of the selected channel and performs his attack on the secret. In case he knows the defender’s action, he is able to determine the exact channel being used (since, of course, the attacker knows his own action), and his payoff will be the posterior vulnerability . However, if the attacker does not know exactly which channel has been used, then his payoff will be smaller.
Note that the issues raised in Phase 2 are typical of leakage games; they do not have a correspondence (to the best of our knowledge) in traditional game theory. Indeed, in traditional game theory the resulting payoff is a deterministic function of all players’ actions. On the other hand, the extra level of randomization provided by the channel is central to security, as it reflects the principle of preventing the attacker from inferring the secret by obfuscating the link between secret and observables.
Following the above discussion, we consider various possible scenarios for games, along two lines of classification. The first classification concerns Phase 1 of the game, in which strategies are selected and actions are drawn, and consists in three possible orders for the two players’ actions.
 Simultaneous:

The players choose (draw) their actions in parallel, each without knowing the choice of the other.
 Sequential, defenderfirst:

The defender draws an action, and commits to it, before the attacker does.
 Sequential, attackerfirst:

The attacker draws an action, and commits to it, before the defender does.
Note that these sequential games may present imperfect information (i.e., the follower may not know the leader’s action), and that we have to further specify whether we use behavioral or mixed strategies.
The second classification concerns Phase 2 of the game, in which some leakage occurs in consequence of the attacker’s observation of the channel’s output, and consists in two kinds of knowledge the attacker may have at this point about the channel that was used.
 Visible choice:

The attacker knows the defender’s action when he observes the output of the channel, and therefore he knows which channel is being used. Visible choice is modeled by the operator .
 Hidden choice:

The attacker does not know the defender’s action when he observes the output of the channel, and therefore in general he does not exactly know which channel is used (although in some special cases he may infer it from the output). Hidden choice is modeled by the operator .
Note that the distinction between sequential and simultaneous games is orthogonal to that between visible and hidden choice. Sequential and simultaneous games model whether or not, respectively, the follower’s choice can be affected by knowledge of the leader’s action. This dichotomy captures how knowledge about the other player’s actions can help a player choose his own action, and it concerns how Phase 1 of the game occurs. On the other hand, visible and hidden choice capture whether or not, respectively, the attacker is able to fully determine the channel representing the system, once the defender and attacker’s actions have already been fixed. This dichotomy reflects the different amounts of information leaked by the system as viewed by the attacker, and it concerns how Phase 2 of the game occurs. For instance, in a simultaneous game neither player can choose his action based on the choice of the other. However, depending on whether or not the defender’s choice is visible, the attacker will or will not, respectively, be able to completely recover the channel used, which will affect the amount of leakage.
If we consider also the subdivision of sequential games into perfect and imperfect information, there are possible different combinations. Some, however, make little sense. For instance, the defenderfirst sequential game with perfect information (by the attacker) does not combine naturally with hidden choice , since that would mean that the attacker knows the action of the defender and chooses his strategy accordingly, but forgets it at the moment of computing the channel and its vulnerability. (We assume perfect recall, i.e., the players never forget what they have learned.) Yet other combinations are not interesting, such as the attackerfirst sequential game with (totally) imperfect information (by the defender), since it coincides with the simultaneousgame case. Note that the attacker and defender are not symmetric with respect to hiding/revealing their actions and , since the knowledge of affects the game only in the usual sense of game theory (in Phase 1), while the knowledge of also affects the computation of the payoff (in Phase 2). Note that the attacker and defender are not symmetric with respect to hiding/revealing their actions and , since the knowledge of affects the game only in the usual sense of game theory, while the knowledge of also affects the computation of the payoff (cf. “Phase 2” above). Other possible combinations would come from the distinction between behavioral and mixed strategies, but, as we will see, they are always equivalent except in one scenario, so for the sake of conciseness we prefer to treat it as a case apart.
Table 2 lists the meaningful and interesting combinations. In Game V we assume imperfect information: the attacker does not know the action chosen by the defender. In all the other sequential games we assume that the follower has perfect information. In the remaining of this section, we discuss each game individually, using the example of Section 3 as a running example.
Order of action  

simultaneous  defender first  attacker first  
Defender’schoice  visible  Game I  Game II  Game III 
hidden  Game IV  Game V  Game VI 
5.1.1 Game I (simultaneous with visible choice)
This simultaneous game can be represented by a tuple . As in all games with visible choice , the expected payoff U of a mixed strategy profile is defined to be the expected value of , as in traditional game theory:
where we recall that .
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