Probabilistic Anonymity via Coalgebraic Simulations
Ichiro Hasuo1⋆ and Yoshinobu Kawabe2
1

Radboud University Nijmegen, the Netherlands
http://www.cs.ru.nl/˜ichiro
2
NTT Communication Science Laboratories, NTT Corporation, Japan
http://www.brl.ntt.co.jp/people/kawabe

Abstract. There is a growing concern on anonymity and privacy on the Internet,
resulting in lots of work on formalization and verification of anonymity. Especially, importance of probabilistic aspect of anonymity is claimed recently bymany authors. Among them are Bhargava and Palamidessi who present the definition of probabilistic anonymity for which, however, proof methods are not yet
elaborated. In this paper we introduce a simulation-based proof method for probabilistic anonymity. It is a probabilistic adaptation of the method by Kawabe etal. for non-deterministic anonymity: anonymity of a protocol is proved by findingout a forward/backward simulation between certain automata. For the jump from
non-determinism to probability we fully exploit a generic, coalgebraic theory of
traces and simulations developed by Hasuo and others. In particular, an appropriate notion of probabilistic simulations is obtained by instantiating a genericdefinition with suitable parameters.

1 Introduction
Nowadays most human activities rely on communication on the Internet, hence on communication protocols. This has made verification of communication protocols a trend incomputer science. At the same time, the variety of purposes of communication protocols has identified new verification goals—or security properties—such as anonymity,in addition to rather traditional ones like secrecy or authentication.
Anonymity properties have attracted growing concern from the public. There are
emerging threats as well: for example, the European Parliament in December 2005 approved rules forcing ISPs to retain access records. Consequently more and more research activities—especially from the formal methods community—are aiming at verification of anonymity properties (see [2]).
Formal verification of anonymity properties is at its relative youth compared to authentication or secrecy. The topic still allows for definitional work (such as [4,7,8,11,16])
pointing out many different aspects of anonymity notions. Notably many authors [4,8,20,21]
claim the significant role of probability in anonymity notions. This is the focus of thispaper.
Bhargava and Palamidessi [4] define the notion of probabilistic anonymity which ismathematically precise and which subsumes many competing notions of anonymity⋆

This work was done during the first author’s stay at NTT Communication Science Laboratories
in September–October 2006.

in probabilistic settings. However, it is not yet elaborated how we can verify if ananonymizing protocol satisfies this notion of probabilistic anonymity.
In this paper we introduce a simulation-based proof method for probabilistic anonymity
as defined by Bhargava and Palamidessi. It is a probabilistic extension of the method by
Kawabe et al. [13,12] for a non-deterministic (as opposed to probabilistic) setting. The
basic scenario is common in both non-deterministic and probabilistic cases:
1. First we model an anonymizing protocol to be verified as a certain kind of automaton X .
2. Second we construct the anonymized version an(X ) of X . The automaton an(X )satisfies the appropriate notion of anonymity because of the way it is constructed.
3. We prove that(trace semantics of X ) = (trace semantics of an(X )) .
Then, since the notion of anonymity is defined in terms of traces, anonymity ofan(X ) yields anonymity of X . The equality is proved by showing that the (appropriate notion of) inclusion order ⊑ holds in both directions.
– ⊑ holds because of the construction of an(X ).
– ⊒ is proved by finding a (forward or backward) simulation from an(X ) to X .
Here we appeal to soundness theorem of simulations—existence of a simulation yields trace inclusion.
Hence the anonymity proof of X is reduced to finding a suitable forward/backward
simulation.
There is an obvious difficulty in conducting this scenario in a probabilistic setting.
The theory of traces and simulations in a non-deterministic setting is well studied e.g.
by [14]; however appropriate definitions of probabilistic traces and simulations are far
from trivial.
For the jump from non-determinism to probability we exploit a generic, coalgebraic
theory of traces and simulations developed by Hasuo, Jacobs and Sokolova [9,10]. In thegeneric theory, fundamental notions such as systems (or automata), trace semantics andforward/backward simulations are identified as certain kinds of coalgebraic constructs.
On this level of abstraction the general soundness theorem—existence of a (coalgebraic)
simulation yields (coalgebraic) trace inclusion—is proved by categorical arguments.
The theory is generic in that, by fixing two parameters appearing therein, it instantiates to a concrete theory for various kinds of systems. In particular, according
to the choice of one parameter, systems can be non-deterministic or probabilistic.3 In
this work a complex definition of probabilistic simulations is obtained as an instanceof the general, coalgebraic definition. Moreover, this definition is an appropriate one:soundness theorem comes for free from the general soundness theorem.
The paper is organized as follows. In Section 2 we illustrate the probabilistic aspect of anonymity properties using the well-known example of Dining Cryptographers.
3

Unfortunately the combination of both non-determinism and probability—which is e.g. inprobabilistic automata [19]—is not covered in this paper. In fact this combination is a notorious one [6,23]: many mathematical tools that are useful in a purely non-deterministic orprobabilistic setting cease to work in the presence of both.

We model anonymizing protocols as a special kind of automata called (probabilistic)
anonymity automata. This notion is introduced in Section 3; the definition of probabilistic anonymity following [4] is also there. Finally in Section 4 we describe oursimulation-based proof method for anonymity and prove its correctness. In Section 5
we conclude.
Notations In the sequel the disjoint union of sets X and Y is denoted by X + Y .
The set of lists over an alphabet X with length ≥ 1 is denoted by X ∗ X in a regularexpression-like manner: obviously we have X ∗ = X ∗ X + {hi}. This appears as adomain of trace semantics for anonymity automata.

2 Motivating example: dining cryptographers (DC)In this section—following [4]—we shall illustrate the probabilistic aspect of anonymity,
using the well-known dining cryptographers (DC) protocol [5].
2.1

The DC protocol

There are three cryptographers (or users) dining together. The payment will be madeeither by one of the cryptographers, or NSA (U.S. National Security Agency) whichorganizes the dinner. Who is paying is determined by NSA; if one of the cryptographers
is paying, she has been told so beforehand.
The goal of the DC protocol is as follows. The three cryptographers announce
whether one of them is paying or not; but if it is the case, the information on which
cryptographer is paying should be disguised from the viewpoint of an observer (called
the adversary in the sequel) and also from that of the cryptographers who are not paying. This is where anonymity is involved.
The protocol proceeds in the following way. Three cryptographers Crypti for i =0, 1, 2 sit in a circle, each with a coin Coini . The coins are held in such a way that they
can be seen by the owner and one of the other two: in the following figure → denotes
the “able-to-see-her-coin” relation.
Crypt0
Crypt1

Crypt2

Then the coins are flipped; each cryptographer, comparing the two coins she can see,announces to the public whether they agree (showing the same side) or disagree. The
trick is that the one who is paying—if there is—lies on the announcement. For example,
given that Crypt0 is paying, then the configuration of coins(h, t, h)that ishth,results in the announcement(a, d, a)that isada .

This announcement is the only thing the adversary can observe; occurrence of an odd
number of d’s reveals the presence of a liar, hence the presence of a payer among thecryptographers.
Can the adversary say which cryptographer is paying? No. In fact, given an announcement with an odd number of d’s and any payer Crypti , we can construct a coin
configuration which yields the given announcement. For example, the announcement(a, d, a) above can be yielded by any of the following configurations.
Crypt0 pays, and coins are (h, t, h) or (t, h, t)
Crypt1 pays, and coins are (h, h, h) or (t, t, t)
Crypt2 pays, and coins are (h, h, t) or (t, t, h)2.2

Probabilistic anonymity in DC

Up to now the arguments have been non-deterministic: now we shall explain how probabilistic aspects in DC emerge. Assume that the coins are biased: each of three Coini ’s
gives head with the probability 9/10. Provided that Crypt0 is paying, the announcement(a, d, a) occurs with the probability (9 · 1 · 9 + 1 · 9 · 1)/103 , because it results from(h, t, h) or (t, h, t). Similar calculations lead to the following table of probabilities.
(d, a, a) (a, d, a) (a, a, d) (d, d, d)
Crypt0 pays 0.73
0.09
0.09
0.09
0.73
0.09
0.09
Crypt1 pays 0.09
Crypt2 pays 0.09
0.09
0.73
0.09
Are the cryptographers still “anonymous”? We would not say so. For example, if theadversary observes an announcement (d, a, a), it is reasonable for her to suspect Crypt0
more than the other two.
Nevertheless, if the coins are not biased, we cannot find any symptom of broken
anonymity. Therefore we want to obtain the following two things.
The first is an appropriate notion of “probabilistic anonymity” which holds withfair coins but is violated with biased coins—this is done in [4]. The intuition is quite
similar to the one behind the notion of conditional anonymity [8]. The adversary has apriori knowledge on “who is likely to be blamed”; however, after observing a run of ananonymizing protocol, the adversary should not gain any additional information—eachuser looks as suspicious as it did before the actual execution.
The second is an effective proof method to verify this notion of anonymity: this iswhat we aim at in the current work.

3 Probabilistic anonymity
3.1

Anonymity automata: models of anonymizing protocols

In this work anonymizing protocols are formalized as a specific kind of probabilistic
systems which we shall call (probabilistic) anonymity automata. The notion is similar to probabilistic automata [19]: however, in anonymity automata branching is purely

probabilistic without any non-determinism. This modification, together with other minor ones, is made so that the coalgebraic framework in [9] applies.
The features of an anonymity automaton are as follows.
– By making a transition it can eithera• execute an action and successfully terminate (x → X), ora• execute an action and move to another state (x → y).
Internal, silent actions are not explicitly present.
– An action a can be either• an observable action o which can be seen by the adversary, or• an actor action blame(i) which denotes that a user i has performed the action
whose performer we want to disguise (such as payment in DC).
– Each state comes with a probability subdistribution over the set of possible transitions. By “sub”distribution it is meant that the sum of all the probabilities is ≤ 1rather than = 1: the missing probability is understood as the probability for deadlock.
Here is a formal definition.
Definition 3.1 (Anonymity automata) An anonymity automaton is a 5-tuple (X, U, O, c, s)
where:– X is a non-empty set called the state space.
– U is a non-empty set of users.4– O is a non-empty set of observable actions.
– c : X → D A × {X} + A × X is a function which assigns to each state x ∈ X
a probability subdistribution c(x) over possible transitions. The set A is the set ofactions and defined by
A = O + { blame(i) | i ∈ U} .
The operation D gives the set of subdistributions: for a set Y ,X
DY = d : Y → [0, 1] |d(y) ≤ 1 .

(1)

y∈Y

This operation D canonically extends to a monad5 which we shall call the subdistribution monad.
For example, the value c(x)(a, X)6 in [0, 1] is the probability with which a state x
aexecutes a and then successfully terminate (i.e. x → X).
– s is a probability subdistribution over the state space X. This specifies which state
would be a starting (or initial) one.
4

A user is called an anonymous user in [4].
Monads are a categorical notion. Interested `readers are´ referred to [3] for the details.
6
To be precise this should be written as c(x) κ1 (a, X) , where κ1 : A × {X} → A × {X} +A × X is the inclusion map.

5

Example 3.2 (Anonymity automaton XDC for DC) To model the DC protocol, wetake
U = {0, 1, 2} , O = {a, d} × {a, d} × {a, d} = (x, y, z) | x, y, z ∈ {a, d} .
We need to fix the a priori probability distribution on who will make a payment, in viewof the conditional notion of probabilistic anonymity. Let us denote by pi the probability
with which a user i pays.
The DC protocol (with its a priori probability distribution given by pi ’s) is naturally
described as follows. Probability for each transition is presented in square brackets;otherwise the transition occurs with probability 1.
↓
blame(0) [p0 ]blame(1) [p1 ] .
..
1t0 [ 2 ]h0 [ 12 ]h1 [ 12 ]t1 [ 12 ]h2 [ 12 ]t1 [ 12 ]t1 [ 21 ]t2 [ 12 ]h2 [ 21 ]X(a, a, a)Xt2 [ 12 ](a, d, d)X(d, d, a)X(d, a, d)X(d, a, d)Xh1 [ 12 ]h2 [ 12 ]t2 [ 21 ](d, d, a)X(a, d, d)Xt2 [ 21 ]h2 [ 21 ](a, a, a)Xt1 [ 21 ]h2 [ 12 ](d, a, a)Xt0 [ 12 ]h1 [ 12 ]t2 [ 21 ](d, d, d)X(a, d, a)Xt2 [ 21 ]h2 [ 12 ](a, a, d)X(a, a, d)Xh1 [ 12 ]h2 [ 12 ]t2 [ 21 ](a, d, a)X(d, d, d)(d, a, a)Xt2 [ 12 ]h2 [ 12 ]τ [1 − p0 − p1 − p2 ].. blame(2) [p2 ].
h0 [ 21 ]Here τ denotes an internal action with the intention of “NSA pays”.
However, the actions hi and ti —with their obvious meanings—must not be present
because they are not observable by the adversary. These actions are replaced by τ ’s.
Moreover, for technical simplicity we do not allow τ ’s to appear in an anonymity automaton. Hence we take the “closure” of the above automaton in an obvious way, andobtain the following.

1 −p2(a, a, a)[ 1−p0 −p
]
4

↓

1 −p2
(a, d, d)[ 1−p0 −p]4xblame(0) [p0 ]

X

X
X
X

(d, d, d)[ 41 ]X(a, a, d)[ 41 ]X(a, d, a)[ 14 ]X

(d, a, a)[ 41 ]

X

1 −p2
]
(d, d, a)[ 1−p0 −p4y2(d, d, d)[ 41 ]X(a, a, d)[ 14 ]X(a, d, a)[ 41 ]X(d, a, a)[ 41 ]X

y1(d, d, d)[ 41 ](a, a, d)[ 41 ](a, d, a)[ 14 ](d, a, a)[ 14 ]X

blame(2) [p2 ]blame(1) [p1 ]

y0

1 −p2
(d, a, d)[ 1−p0 −p
]
4

X

X

X

The start state distribution s is: x 7→ 1. This anonymity automaton we shall refer to as
XDC .
3.2

Anonymity automata reconciled as coalgebras

The generic, coalgebraic theory of traces and simulations in [9] applies to anonymity
automata. The generic theory is developed with two parameters T and F :– a monad T on Sets specifies the branching-type, such as non-determinism or probability;– a functor F on Sets specifies the transition-type, i.e., what a system can do bymaking a transition.
Systems for which traces/simulations are defined are called (T, F )-systems in the generictheory, making the parameters explicit. The theory is coalgebraic because a (T, F )system is essentially a coalgebra in a suitable category.
Anonymity automata fit in the generic theory. They are (T, F )-systems with thefollowing choice of parameters T and F .
– T is the subdistribution monad D, modeling purely probabilistic branching.
– F X = A × {X} + A × X, modeling the transition-type of “(action and terminate)or (action and next state)”.
It is immediately seen that for this choice of F , the set A∗ A carries the following initial
algebra in Sets. We denote its structure map by α.
A × {X} + A × (A∗ A)
α ∼
=
A∗ A

κ1 (a, X)

κ2 (a, a)

hai

a·a

,

where hai denotes a list of length 1, and a · a is what would be written as (cons a a) in
L ISP. Therefore [9, Corollary 5.2] suggests that the set A∗ A is the appropriate domainof (finite) trace semantics for anonymity automata: this is actually the case later in
Definition 3.3.
3.3

Trace semantics for anonymity automata

The trace semantics for anonymity automata is used in defining probabilistic anonymity.
In a non-deterministic setting, trace semantics yields a set of lists of actions which canpossibly occur. In contrast, trace semantics of a probabilistic system is a probability
subdistribution over lists.
Definition 3.3 (Trace semantics for anonymity automata) Given an anonymity automaton X = (X, U, O, c, s), its trace
PX ∈ D(A∗ A)
is defined as follows. For a list of actions ha0 , a1 , . . . , an i with a finite length n ≥ 1,
X
an−1
an
a
a
PX (ha0 , a1 , . . . , an i) =X) ,PX (x0 →0 x1 →1 · · · → xn →x0 ,x1 ,...,xn ∈X

where the probability
a

a

an−1

a

n
X)
PX (x0 →0 x1 →1 · · · → xn →= s(x0 ) · c(x0 )(a0 , x1 ) · · · · · c(xn−1 )(an−1 , xn ) · c(xn )(an , X)

a

a

an−1

is for the event that X starts at x0 , follows the path →0 x1 →1 · · · → xn and finallyanX.
terminates with →Intuitively the value PX (a) ∈ [0, 1] for a list a ∈ A∗ A is the probability with which
the system X executes actions in a successively and then terminates. Our concern is on
actions (observable actions or actor actions) the system makes but not on the states it
exhibits.
The following alternative characterization allows us to apply the generic, coalgebraic theory of traces in [9,10].
Lemma 3.4 (Trace semantics via the generic theory) Given an anonymity automaton X , let (s, c) be a (T, F )-system identified with X as in Section 3.2.
The trace PX of X coincides with the coalgebraic trace tr(s,c) defined in the generictheory [9, Definition 5.7] for (s, c).
⊓⊔Example 3.5 (Dining cryptographers) For the anonymity automaton XDC in Example 3.2, its trace PXDC is the following probability subdistribution.
h (a, a, a) i 7→ (1 − p0 − p1 − p2 )/4
h blame(i), (d, a, a) i 7→ pi /4h (a, d, d) i 7→ (1 − p0 − p1 − p2 )/4
h blame(i), (a, d, a) i 7→ pi /4h (d, a, d) i 7→ (1 − p0 − p1 − p2 )/4
h blame(i), (a, a, d) i 7→ pi /4h (d, d, a) i 7→ (1 − p0 − p1 − p2 )/4
h blame(i), (d, d, d) i 7→ pi /4(for i = 0, 1, 2)

The other lists in A∗ A have probability 0.
In this work we assume that in each execution of an anonymizing protocol there
appears at most one actor action. This is the same assumption as [4, Assumption 1] andis true in all the examples in this paper.
Assumption 3.6 (At most one actor action) Let X = (X, U, O, c, s) be an anonymity
automaton and a ∈ A∗ A. If a contains more than one actor actions, thenPX (a) = 0 .
3.4

Definition of probabilistic anonymity

In this section we formalize the notion of probabilistic anonymity following [4]. First,
for the sake of simplicity of presentation, we shall introduce the following notations forpredicates (i.e. subsets) on A∗ A.
Definition 3.7 (Predicates [blame(i)] and [o])on A∗ A is defined as follows.

– For each i ∈ U, a predicate [blame(i)][blame(i)] = {a ∈ A∗ A | blame(i) appears in a}
By Assumption 3.6, it is the set of lists obtained by augmenting blame(i) withobservable actions: in a regular-expression-like notation,[blame(i)] = O∗ blame(i) O∗ .
Moreover, [blame(i)] ∩ [blame(j)] = ∅ if i 6= j.

– For each o ∈ O∗ , a predicate [o] on A∗ A is defined as follows.
[o] = {a ∈ A∗ A | removeActor(a) = o} ,where the function removeActor : A∗ A → O∗ —which is defined by a suitableinduction—removes actor actions appearing in a list. The set [o] ⊆ A∗ A consistsof those lists which yield o as the adversary’s observation. It is emphasized that [o]is not the set of lists which contain o as sublists: we remove only actor actions, but
not observable actions.
Note that we are overriding the notation [ ]: no confusion would arise since the arguments are of different types. Values such as PX ( [blame(i)] ) are defined in a straightforward manner:X
PX ( [blame(i)] ) =PX (a) .
a∈[blame(i)]This is the probability with which X yields an execution in which a user i is to be
blamed.
We follow [4] and adopt the following definition of anonymity.
Definition 3.8 (Probabilistic anonymity [4]) We say an anonymity automaton X isanonymous if for each i, j ∈ U and o ∈ O∗ ,PX ( [blame(i)] ) > 0 ∧ PX ( [blame(j)] ) > 0=⇒ PX ( [o] | [blame(i)] ) = PX ( [o] | [blame(j)] ) .
Here PX ( [o] | [blame(i)] ) is a conditional probability: it is given by
PX ( [o] | [blame(i)] ) =PX ( [o] ∩ [blame(i)] ).
PX ( [blame(i)] )The intuition behind this notion—sketched in Section 2.2—is similar to the one behindconditional anonymity [8]. In fact, it is shown in [4] that under reasonable assumptions
the two notions of anonymity coincide. For completeness the definition of conditional
anonymity (adapted to the current setting) is also presented.
Definition 3.9 (Conditional anonimity [8]) An anonymity automaton X satisfies conditional anonymity if for each i ∈ U and o, o′ ∈ O∗ ,PX ( [blame(i)] ∩ [o] ) > 0=⇒

PX ( [blame(i)] | [o] ) = PX ( [blame(i)] |[[blame(j)] ) .

j∈UThe notion in Definition 3.8 is (one possibility of) probabilistic extension of trace
anonymity in [18]. It is emphasized that these anonymity notions are based on trace
semantics which is at the coarsest end in the linear time-branching time spectrum [22].
Hence our adversary has less observation power than one in [1] for example where security notions are bisimulation-based. A justification for having such a weaker adversary
is found in [13].

4 Anonymity proof via probabilistic simulations
In this section we extend the proof method [13,12] for anonymity to the probabilistic
setting. In the introduction we have presented the basic scenario. Now we shall describe
its details, with all the notions therein (traces, simulations, etc.) interpreted probabilistically.
4.1

Anonymized automaton an(X )We start with the definition of an(X ), the anonymized version of an anonymity automaton X . Recall that our notion of anonymity is conditional: the adversary has a priori knowledge on who is more suspicious. In an anonymity automaton X , the a priori
probability with which a user i does wrong is given by PX ( [blame(i)] ). Its normalized,
conditional version
S[
PX ( [blame(i)] ∩ j∈U [blame(j)] )def.
Sri = PX ( [blame(i)] |[blame(j)] ) =PX ( j∈U [blame(j)] )j∈U=PPX ( [blame(i)] )j∈U PX ( [blame(j)] )(the equalities are due to Assumption 3.6) plays an important role in the following
definition of an(X ). The value ri is the conditional probabilityPwith which a user i is tobe blamed, given that there is any user to be blamed; we have i∈U ri = 1. Of course,
for the values ri to be well-defined, the anonymity automaton X needs to satisfy thefollowing reasonable assumption.
Assumption 4.1 (There is someone to blame) For an anonymity automaton X ,X
PX ( [blame(j)] ) 6= 0 .
j∈UIntuitively, an(X ) is obtained from X by distributing an actor action blame(i) toeach user j, with the probability distributed in proportion to rj .
Definition 4.2 (Anonymized anonymity automaton an(X )) Given an anonymity automaton X = (X, U, O, c, s), its anonymized automaton an(X ) is a 5-tuple (X, U, O, can , s),where can is defined as follows. For each x ∈ X,
P
can (x)(blame(i), u) = j∈U ri · c(x)(blame(j), u) for i ∈ U and u ∈ {X} + X,
can (x)(o, u)= c(x)(o, u)
for o ∈ O and u ∈ {X} + X.
On the first equation, the summand ri · c(x)(blame(j), u) results from distributing theblame(j)probability c(x)(blame(j), u) for a transition x −→ u, to a user i. This is illustrated
in the following figure: here U = {0, 1, . . . , n} and q = c(x)(blame(j), u).
In X•

blame(j) [q]•In an(X )•blame(0) [r0 · q]···•

blame(n) [rn · q] (2)The automaton an(X ) is “anonymized” in the sense of the following lemmas.
Lemma 4.3 Let X be an anonymity automaton. In its anonymized version an(X ) =(X, U, O, can , s) we have
rj · can (x)(blame(i), u) = ri · can (x)(blame(j), u)
for any i, j ∈ U, x ∈ X and u ∈ {X} + X.
Proof. Obvious from the definition of can .

⊓⊔Lemma 4.4 (an(X ) is anonymous) For an anonymity automaton X , an(X ) is anonymous in the sense of Definition 3.8.
Proof. Let o = ho1 , o2 , . . . , on i ∈ O∗ and i, j ∈ U. Moreover, assume
Pan(X ) ( [blame(i)] ) 6= 0

and

Pan(X ) ( [blame(j)] ) 6= 0 ,hence ri 6= 0 and rj 6= 0. Then
Pan(X ) ( [o] ∩ [blame(i)] )= Pan(X ) ( hblame(i), o1 , o2 , . . . , on i )+ Pan(X ) ( ho1 , blame(i), o2 , . . . , on i )+ · · · + Pan(X ) ( ho1 , o2 , . . . , on , blame(i)i )X=
s(x0 ) · can (x0 )(blame(i), x1 ) · can (x1 )(o1 , x2 ) · · · · · can (xn )(on , X)
x0 ,x1 ,...,xn ∈X+Xs(x0 ) · can (x0 )(o1 , x1 ) · can (x1 )(blame(i), x2 ) · · · · · can (xn )(on , X)X

s(x0 ) · can (x0 )(o1 , x1 ) · can (x1 )(o2 , x2 ) · · · · · can (xn )(blame(i), X) .

x0 ,x1 ,...,xn ∈X

+ ···
+

x0 ,x1 ,...,xn ∈X

We have the same equation for j instead of i. Hence by Lemma 4.3 we have
rj · Pan(X ) ( [o] ∩ [blame(i)] ) = ri · Pan(X ) ( [o] ∩ [blame(j)] ) .

(3)

This is used to show the equality of two conditional probabilities.
Pan(X ) ( [o] | [blame(i)] ) =Pan(X ) ( [o] ∩ [blame(i)] )Pan(X ) ( [blame(i)] )=ri Pan(X ) ( [o] ∩ [blame(j)] )·rj
Pan(X ) ( [blame(i)] )By (3)=

Pan(X ) ( [o] ∩ [blame(j)] )Pan(X ) ( [blame(j)] )By definition of ri , rj

= Pan(X ) ( [o] | [blame(j)] ) .

⊓
⊔

4.2

Forward/backward simulations for anonymity automata

We proceed to introduce appropriate notions of forward and backward simulations. The(tedious) definition and soundness theorem—existence of a forward/backward simulation implies trace inclusion—come for free from the generic theory in [9]. This forms a
crucial part of our simulation-based proof method.
Definition 4.5 (Forward/backward simulations for anonymity automata) Let X =(X, U, O, c, s) and Y = (Y, U, O, d, t) be anonymity automata which have the same
sets of users and observable actions.
A forward simulation from X to Y—through which Y forward-simulates X —is afunctionf : Y −→ DX
which satisfies the following inequalities in [0, 1].
Ps(x)≤ y∈Y t(y) · f (y)(x) for any x ∈ X,P
f (y)(x) · c(x)(e, X) ≤ d(y)(e, X)
for any y ∈ Y and e ∈ A,
P
Px∈X
′
′
′
′
y ′ ∈Y d(y)(e, y ) · f (y )(x )x∈X f (y)(x) · c(x)(e, x ) ≤for any y ∈ Y , e ∈ A and x′ ∈ X.
A backward simulation from X to Y—through which Y backward-simulates X —isa function
b : X −→ DY
which satisfies the following inequalities in [0, 1].
P≤ t(y)
for any y ∈ Y ,x∈X s(x) · b(x)(y)Pc(x)(e, X)≤ y∈Y b(x)(y) · d(y)(e, X) for any x ∈ X and e ∈ A,
P
P
′
′
′
′
x′ ∈X c(x)(e, x ) · b(x )(y ) ≤y∈Y b(x)(y) · d(y)(e, y )for any x ∈ X, e ∈ A and y ′ ∈ Y .
The definition definitely looks puzzling. Why does a forward simulation have the type
Y → DX? Why is a backward simulation not of the same type? How come the complex
inequalities? How do we know that the inequalities are in the correct direction?
In fact, this definition is an instantiation of the general, coalgebraic notions of forward/backward simulations [9, Definitions 4.1, 4.2]. More specifically, the two parameters T and F in the generic definition are instantiated as in Section 3.2.
Theorem 4.6 (Soundness of forward/backward simulations) Assume there is a forward (or backward) simulation from one anonymity automaton X to another Y. Then
we have trace inclusion
PX ⊑ PY ,where the order ⊑ is defined to be the pointwise order: for each a ∈ A∗ A,
PX (a) ≤ PY (a) .
Proof. We know (Lemma 3.4) that the notions of traces and simulations for anonymity
automata are instantiations of the general, coalgebraic notions in [9,10]. Therefore wecan appeal to the general soundness theorem [9, Theorem 6.1].
⊓
⊔

4.3

Probabilistic anonymity via simulations

We shall use the materials in Sections 4.1 and 4.2 to prove the validity of our simulationbased proof method (Theorem 4.11).
The following lemma—which essentially says PX ⊑ Pan(X ) —relies on the wayan(X ) is constructed. The proof is a bit more complicated than in the non-deterministic
setting [13,12].
Lemma 4.7 Let X be an anonymity automaton. Assume there exists a forward or backward simulation from an(X ) to X —through which X simulates an(X ). Then their trace
semantics are equal:
PX = Pan(X ) .
Proof. By the soundness theorem (Theorem 4.6) we have
PX ⊒ Pan(X ) ,(4)

where ⊒ refers to the pointwise order between functions A∗ A ⇉ [0, 1]. We shall showthat this inequality is in fact an equality.
First we introduce an operation obs which acts on anonymity automata. Intuitively,
obs(Y) is obtained from Y by replacing all the different actor actions blame(i) with single blame(sb)—sb is for “somebody”. This conceals actor actions in Y; hence obs(Y)only carries information on the observable actions of Y.
In X•

blame(0) [q0 ]···•In obs(X )blame(n) [qn ](5)blame(sb) [q0 + · · · + qn ]••Formally,
Definition 4.8 (Anonymity automaton obs(Y)) Given an anonymity automaton Y =(Y, U, O, d, t), we define an anonymity automaton obs(Y) as the 5-tuple (Y, {sb}, O, dobs , t)where:– sb is a fresh entity,– dobs is a function
dobs : Y −→ D Aobs × {X} + Aobs × Y
where Aobs = O + {blame(sb)}, defined by:P
dobs (y)(blame(sb), u) = i∈U d(y)(blame(i), u)
dobs (y)(o, u)= d(y)(o, u)for y ∈ Y and u ∈ {X} + Y ,for y ∈ Y , o ∈ O and u ∈ {X} + Y .

The following fact is obvious.
Sublemma 4.9 For an anonymity automaton X , obs(X ) and obs(an(X )) are identical.
⊓⊔

The following sublemma is crucial in the proof of Lemma 4.7. Two automata Y andobs(Y), although their trace semantics distributes over different sets, have the same
sum of probabilities taken over all executions.
Sublemma 4.10 For an anonymity automaton Y,
X
X
Pobs(Y) (a′ ) .
PY (a) =a∈A∗ Aa′ ∈(Aobs )∗ Aobs

Recall that A = O + {blame(i) | i ∈ U} and Aobs = O + {blame(sb)}.
Proof. From the definition of trace semantics (Definition 3.3), the sublemma is provedby easy calculation.
⊓⊔
We turn back to the proof of Lemma 4.7. We argue by contradiction—assume thatthe inequality in (4) is strict. ThatPis, there exists a0 P
∈ A∗ A such that PX (a0 )Pan(X ) (a0 ). Then, by (4) we have a∈A∗ A PX (a)a∈A∗ A Pan(X ) (a). However,PP′
By Sublemma 4.10
a′ ∈(Aobs )∗ Aobs Pobs(X ) (a )a∈A∗ A PX (a) =P′
By Sublemma 4.9= a′ ∈(Aobs )∗ Aobs Pobs(an(X )) (a )P
By Sublemma 4.10
= a∈A∗ A Pan(X ) (a) .
⊓⊔This contradiction concludes the proof of Lemma 4.7.
Now we are ready to state the main result.

Theorem 4.11 (Main theorem: probabilistic anonymity via simulations) If there exists a forward or backward simulation from an(X ) to X , then X is anonymous.
Proof. By Lemma 4.7 we have PX = Pan(X ) . Moreover, by Lemma 4.4, an(X ) isanonymous. This proves anonymity of X : recall that probabilistic anonymity is a property defined in terms of traces (Definition 3.8).
⊓⊔
Example 4.12 (Dining cryptographers) We demonstrate our proof method via simulations by applying it to the DC protocol.
Let X = {x, y0 , y1 , y2 } be the state space of XDC . Its anonymized version an(XDC )has the same state space: for notational convenience the state space of an(XDC ) isdenoted by X ′ = {x′ , y0 ′ , y1 ′ , y2 ′ }. It is verified by easy calculation that the following
function f : X → D(X ′ ) is a forward simulation from an(XDC ) to XDC .
f (x) = [x′ 7→ 1]

y0 ′ 7→ p0 +pp10 +p2
p
f (y0 ) = f (y1 ) = f (y2 ) =  y1 ′ 7→ p0 +p11 +p2 ′
y2 7→ p0 +pp21 +p2


By Theorem 4.11 this proves (probabilistic) anonymity of XDC , hence of the DC protocol.

5 Conclusion and future work
We have extended the simulation-based proof method [13,12] for non-deterministic
anonymity to apply to the notion of probabilistic anonymity defined in [4]. For themove we have exploited a generic theory of traces and simulations [9,10] in which
the difference between non-determinism and probability is just a different choice of aparameter.
The DC example in this paper fails to demonstrate the usefulness of our proof
method: for this small example direct calculation of trace distribution is not hard. A
real benefit would arise in theorem-proving anonymity of an unboundedly large system (which we cannot model-check). In fact, the non-deterministic version of our proof
method is used to theorem-prove anonymity of a voting protocol with arbitrary many
voters [12]. A probabilistic case study of such kind is currently missing.
In [4] the probabilistic π-calculus is utilized as a specification language for automata. We have not yet elaborated which subset of the calculus is suitable for describing our notion of anonymity automata.
There is a well-established body of work on verification of probabilistic informationhiding properties such as non-interference [24,17]. Our proof method could be reconciled in this context by, for example, finding a translation of anonymity into a noninterference property.
The significance of having both non-deterministic and probabilistic branching inconsidering anonymity is claimed in [15]. However the current method cannot handle
this combination due to the lack of suitable coalgebraic framework. Elaboration in thisdirection would also help better understanding of the nature of the (notorious) combination of non-determinism and probability.
Acknowledgments Thanks are due to Ken Mano, Peter van Rossum, Hideki Sakurada,
Ana Sokolova, Yasuaki Tsukada and the anonymous referees for helpful discussions
and comments. The first author is grateful to his supervisor Bart Jacobs for encouragement.

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