=Paper= {{Paper |id=Vol-469/paper-5 |storemode=property |title=A Method to Evaluate Uncertain and Conflicting Trust and Authenticity Statements |pdfUrl=https://ceur-ws.org/Vol-469/paper5.pdf |volume=Vol-469 }} ==A Method to Evaluate Uncertain and Conflicting Trust and Authenticity Statements== https://ceur-ws.org/Vol-469/paper5.pdf

A Method to Evaluate Uncertain and
Conflicting Trust and Authenticity Statements

Andreas Gutscher

Institute of Communication Networks and Computer Engineering,
Universität Stuttgart, 70569 Stuttgart, Germany
andreas.gutscher@ikr.uni-stuttgart.de ?

Abstract. Countless Internet applications and platforms make it easy
to communicate, to collaborate and to exchange information and opin-
ions with a huge number of individuals. However, it is getting more and
more difficult to distinguish honest and reliable individuals from mali-
cious users distributing false information or malware. Digital signatures,
webs of trust and reputation systems can help to securely identify com-
munication partners and to estimate the trustworthiness of others, but
there is a lack of trust and authenticity evaluation methods that do not
show counterintuitive effects in the case of conflicting opinions.
This article proposes a new integrated method to evaluate uncertain
and conflicting trust and authenticity statements. It introduces a set of
operators and inference rules for combining and reasoning with these
statements, it defines an approach to compute the resulting confidence
values of derived statements and it compares different computation al-
gorithms. The computation is based on a probability theoretical model
in order to exclude inconsistencies and counter-intuitive effects.

1 Introduction

An increasing number of different Internet applications, platforms and social
networks makes it easy to communicate with a huge number of individuals, to
exchange and share information, news, photos, files and product recommenda-
tions and to socialize with people sharing similar interests. However, for users
participating in a large number of social networks, discussion forums, etc. it is
getting more and more difficult to find out who their new “friends” actually are
and whether they can trust them.
With a reputation system users can share their knowledge and opinions about
other users. The reputation system collects and evaluates the opinions of all
users about the trustworthiness of others. On request it computes the resulting
confidence value for the requested entity according to a trust model.
?
D. Chadwick, I. You and H. Chang (Eds.): Proceedings of the 1st International
Workshop on Managing Insider Security Threats (MIST2009), Purdue University,
West Lafayette, USA, June 16, 2009. *Copyright is held by the author(s)*
Evaluation of Trust and Authenticity Statements 63

The authenticity of all opinion statements should be protected, e. g., with
digital signatures, to prevent manipulations and to make the evaluation verifi-
able to the users. Digital signatures are only useful if the users can identify the
signature key holder. If no global trusted public key infrastructure is available,
users can share their knowledge about the ownership of keys in a so-called web
of trust (e. g., the PGP/GnuPG web of trust [1]) by exchanging digitally signed
identity certificates. However, these authenticity statements are only useful if the
users can verify that the issuer is trustworthy to verify the ownership of public
keys. Trust and authenticity evaluation are thus highly interdependent.
Various different computational trust models, reputation systems and ap-
plications using trust have been proposed [1–10]. To obtain an intuitive and
consistent trust model one must define clearly what a confidence value repre-
sents and find a sound mathematical basis for the computation with confidence
values. Confidence values have strong similarities to probability values. The most
sophisticated trust models are therefore based on probability theory. Maurer [4]
proposed a probabilistic model in which confidence values for trust and authen-
ticity relations correspond to probability values. However, it does not model
negative opinions (distrust) and entities cannot have more than one key. Cre-
dential Networks and related models proposed by Haenni [6], Jonczy [11] and
Kohlas [10] also model confidence values as probabilities. Besides degrees of sup-
port and uncertainty the confidence values can express also degrees of refutation
(e. g., distrust). However, confidence values may contain either degrees of belief
and ignorance1 , disbelief and ignorance, or belief and disbelief, but they can-
not contain degrees of belief, disbelief and ignorance at the same time. Jøsang’s
Subjective Logic [5] can express opinions with degrees of belief, ignorance and
disbelief (at the same time), but the approach to compute the resulting confi-
dence values is (although based on probability theory, too) quite different. In
the model of Maurer and in Credential Networks the initial confidence values
are uncertain and the inference rules are deterministic, whereas in Subjective
Logic the uncertainty is modeled in the operators of the inference rules, i. e., the
confidence value of a conclusion is computed from the confidence values of the
preconditions. Unfortunately, this leads to the problem that the resulting confi-
dence value generally depends on the order in which the operators are applied.
It seems that Subjective Logic can not be used to evaluate arbitrary networks of
trust and authenticity statements without using questionable workarounds [12].
If users can express both positive (supporting) and negative (refuting) opin-
ions, then the combination of contradictory opinions can lead to conflicts. In
Credential Networks and Subjective Logic the probability mass associated with
conflicting combinations is eliminated and the remaining probability mass is re-
normalized. Zadeh [13] has shown that conflict elimination and re-normalization
approaches (like Dempster’s rule of combination [14]) can produce counter-
intuitive effects.
This article proposes a new integrated approach to evaluate uncertain and
conflicting trust and authenticity statements without eliminating conflict. This
1
the terms uncertainty and ignorance are used interchangeable
64 Gutscher

avoids the counter-intuitive effects of re-normalizations. The trust model is based
on a combination of the inference rules from [15] and the calculus and operators
from [16], extended by a new operator for reasoning with authenticity statements.
Sect. 2 describes the representation of trust and authenticity statements, Sect. 3
the representation of confidence values and the corresponding operators. The new
inference rules are formulated in Sect. 4. Sect. 5 proposes different algorithms to
compute the resulting confidence value, Sect. 6 compares the computation time
of these algorithms and Sect. 7 concludes the article.

2 Model of Trust and Authenticity Statements
An opinion refers to a trust or authenticity statement Hj with an associated
confidence value tj . A first-hand opinion is an opinion that is based only on
the experience and knowledge of a single entity (the trustor or issuer ) and that
is independent of other opinions. A second-hand opinion is an opinion that is
derived from other opinions and that is thus not independent.
We define trust as “a unidirectional relation between a trustor and a trustee
expressing the strong belief of the trustor that the trustee will behave as expected
with respect to a particular capability within a particular context” [15]. Therefore
we represent the standard form of a trust statement as follows:
Trust(trustor , trustee, r, hmin ..hmax ) (1)
The trustor can be an entity or a key, the trustee an entity, a description or a key
(see Tab. 1). An entity (EA , EB , . . . ) can be a person, an organization, a network
node, etc. referred to by a local identifier. To exchange opinions with others
users have to use unique descriptions or public keys to refer to other entities.
A description (DA , DB , . . . ) consists of a list of names, identifiers or attributes
that uniquely identifies the described entity. Entities may have several different
descriptions. A public key (KA , KB , . . . ) is the public part of an asymmetric
key pair. The holder uses the key pair to sign trust or authenticity statements
(certificates). An entity can use several different key pairs at the same time.

Table 1. Trust and authenticity statements (relations and certificates)

Trust statements Authenticity
Standard form Internal form statements
Trust(EA , EB , r, hmin ..hmax ) Trust(EA , EB , r, h, l) Auth(EA , KB , EB )
Relation Trust(EA , KB , r, hmin ..hmax ) Trust(EA , KB , r, h, l) Auth(EA , DB , EB )
Trust(EA , DB , r, hmin ..hmax ) Trust(EA , DB , r, h, l) Auth(EA , KB , DB )
Trust(KA , KB , r, hmin ..hmax ) Trust(KA , KB , r, h, l) Auth(KA , KB , DB )
Certificate
Trust(KA , DB , r, hmin ..hmax ) Trust(KA , DB , r, h, l)

The capability r refers to an application specific capability (r1 , r2 , . . . ) or
to the capability rPKI , which represents the capability to honestly and carefully
verify that a description uniquely refers to the holder of a particular key pair.
Evaluation of Trust and Authenticity Statements 65

We distinguish different types of trust identified by a different number of
recommendation hops (h): Functional trust expresses the belief that the trustee
has the capability r and is described by h = 0. Recommendation trust for h = 1
hop expresses the belief that the trustee can recommend someone with capability
r, recommendation trust for h = 2 hops that the trustee can recommend someone
who can recommend someone with capability r, etc. Each standard form trust
statement can specify the desired range of recommendation hops hmin ..hmax .
For the evaluation of trust statements we need in addition trust statements in
the slightly different internal form. These trust statements refer not to a range,
but to a single recommendation hop value h ≥ 0 and they have an additional
parameter, the chain length l ≥ 1:

Trust(trustor , trustee, r, h, l) (2)

Trust is not transitive in general, but trust statements can be combined in certain
cases to trust chains according to the transitive trust inference rule (7) described
in Sect. 4.1. The chain length l of the derived trust statement refers to the number
of first-hand trust statements in the trust chain.
Authenticity statements express the strong belief of the issuer that a de-
scription belongs to an entity, that a public key belongs to an entity or that a
description belongs to the holder of a public key:

Auth(issuer , actor 1 , actor 2 ) (3)

The issuer is an entity or a public key, actor 1 and actor 2 are entities, descriptions
or public keys. All four possible combinations are listed in Tab. 12 .

3 Confidence Values

This section introduces discrete and continuous confidence values as well as oper-
ators for reasoning with discrete confidence values. Users express their opinions
with continuous confidence values while the discrete confidence values are used
internally only for reasoning with opinions.

3.1 Representation of Discrete and Continuous Confidence Values

Users can have different and possibly conflicting opinions about trust and au-
thenticity statements. Therefore, we can not definitively decide whether a state-
ment H is “true” or “false”. We can only evaluate known indications that support
or refute H. It is possible that neither supporting nor refuting or that both sup-
porting and refuting indications for H are found. Therefore we describe knowl-
edge of supporting and refuting indications independently. For each statement
H we introduce the propositions H + and H − to describe that the reputation
2
certificates can not contain local identifiers for entities (EA , EB , . . .) because they
would be meaningless to other entities
66 Gutscher

system is aware of indications that imply that H must be true and that H must
be false, respectively. We also introduce the four discrete confidence values belief
(+), ignorance (∅), disbelief (−) and conflict (±) to represent the four possi-
ble combinations of these propositions (see Tab. 2). They can be seen as “truth
values” of a paraconsistent logic [16].

Table 2. Discrete confidence values

Propositions Discrete confidence value Semantics
{H + } t0 = + (belief ) “the indications imply that H must be true”
{} t0 = ∅ (ignorance) “there are no relevant indications about H”
{H − } t0 = − (disbelief ) “the indications imply that H must be false”
{H , H } t0 = ± (conflict)
+ −
“the indications imply that H must be true
and that H must be false at the same time”

As statements can in fact not be both true and false at the same time we
can conclude that first-hand opinions can not have the confidence value conflict.
However, if we combine statements of different (disagreeing) entities, it is possi-
ble to find both H + and H − , i. e., the confidence value of derived (second-hand )
opinions can be conflict. Conflict must not be confused with partial support and
partial refutation (ambivalent opinions). An entity that has for example expe-
rienced some positive and some negative interactions can express this opinion
with continuous confidence values.
Continuous confidence values t = (b, i, d, c) with b, i, d, c ∈ [0, 1] and b + i +
d + c = 1 express degrees of belief, ignorance, disbelief and conflict. The value
b represents the issuer’s subjective estimation of the probability that there are
indications supporting (but no refuting) H. Similarly, d represents the subjec-
tive estimation of the probability that there are indications refuting (but no
supporting) H. c represents the subjective estimation of the probability that
there are both supporting and refuting indications for H at the same time, and
i represents the subjective estimation of the probability that there are neither
supporting nor refuting indications for H. For the same reason as before, c must
be zero in all first-hand opinions, whereas second-hand opinions can contain
conflict. Nevertheless, ambivalent first-hand opinions can be expressed by con-
tinuous confidence values with both b > 0 and d > 0. A user that has made
many positive and few negative experiences can choose, for example, a first-
hand confidence value with b = 0.7 and d = 0.1 (i. e., t = (0.7, 0.2, 0.1, 0). Thus,
in first-hand statements b can be seen as the lower bound and 1 − d as the upper
bound for the estimated subjective probability that H must be true.
The degrees of ignorance and conflict in resulting confidence values have dif-
ferent meanings, and applications should handle high degrees of ignorance and
conflict differently: A high degree of ignorance indicates that the reputation sys-
tem has little information about the requested statement and suggests searching
more relevant statements, if possible. A high degree of conflict, however, shows
that the requested statement H is controversial. This suggests that the requester
Evaluation of Trust and Authenticity Statements 67

should verify whether the trust and authenticity assignments he made and that
cause the conflict are correct.
Continuous confidence value can be condensed to a single value w, if desired:

w = b + wi i + wd d + wc c (4)

The parameters wi , wd and wc represent weights for the degrees of ignorance,
disbelief and conflict, e. g., wi = 0.5, wd = −1 and wc = 0. They can be chosen
according to the preferences of the application and allow for rather optimistic
or rather pessimistic behavior in the cases of uncertainty and conflict.

3.2 Operators to Combine Discrete Confidence Values

This section describes the recommendation and authentication operators. The
operators define whether Hz+ and Hz− can be derived from a set of premises (Hx+ ,
Hx− , Hy+ , Hy− ). The short notation with statements is provided for convenience
and will be used to formulate the inference rules in Sect. 4.

Recommendation Operator The recommendation operator (⊗) is used to
concatenate two trust statements or a trust with an authenticity statement. It
is reasonable for a user to adopt the opinions of trustworthy entities. However,
it is not reasonable (it is in fact even dangerous) to assume that untrustworthy
(malicious or incompetent) entities always tell the opposite of the truth. Instead,
opinions of untrustworthy entities should be ignored. Therefore, we do not draw
any conclusions from Hx− . The operator is thus defined as follows:

Hx ⊗ Hy Hx+ Hy+ Hx+ Hy−
⇔ , (5)
Hz Hz+ Hz−

This reads as follows: Hz follows from a combination of Hx and Hy with the
recommendation operator. If there are supporting indications for Hx and for Hy ,
then infer Hz+ . If there are supporting indications for Hx and refuting indications
for Hy , then infer Hz− . Fig. 1 (left) shows the corresponding “truth table”.

t0x
t0z = t0x ⊗ t0y
+∅−± +∅−±
+ +∅∅+ ++∅−±
∅ ∅∅∅∅ ∅∅∅∅∅
t0y
− −∅∅− −−∅∅−
± ±∅∅± ±±∅−±

Fig. 1. Recommendation and authentication operator truth tables
68 Gutscher

Authentication Operator The authentication operator ( ) is used to reason
with two authenticity relations between entities, descriptions and public keys:

Hx Hy Hx+ Hy+ Hx+ Hy− Hx− Hy+
⇔ , , (6)
Hz Hz+ Hz− Hz−
The operator definition can be understood as follows: Assume Hx and Hy repre-
sent statements like “A and B belong together” and “B and C belong together”,
respectively. If we have supporting indications for both statements, then this
supports that A and C belong together (Hz ). If we have indications that A and
B belong together but that B does not belong to C, then we conclude that A
does not belong to C either. If neither A belongs to B nor does B belong to
C, then we can draw no conclusion about A and C. Fig. 1 (right) shows the
corresponding truth table.

4 Inference Rules
The inference rules specify which conclusions the reputation system can draw
from a set of given trust and authenticity propositions.

4.1 Transitive Trust Inference Rule
This inference rule describes the transitivity property of trust statements. It
defines in which cases two trust statements for the same capability r can be
combined with the recommendation operator in order to derive a new trust
statement from the trustor of the first statement (A) to the trustee of the second
statement (C). The trustor A can be an entity (EA ) or a public key (KA ). The
second statement can be a trust statement or a trust certificate, i. e., B can be
an entity (EB ) or a public key (KB ). The final trustee C can be an entity (EC ),
a public key (KC ) or a description (DC ).

Trust(A, B, r, h + l2 , l1 ) ⊗ Trust(B, C, r, h, l2 )
(7)
Trust(A, C, r, h, l1 + l2 )

This inference rule differs from other proposed transitive trust inference rules
in that it allows the combination of trust statements only if the number of
recommendation hops matches: The number of recommendation hops of the
first statement must equal the sum of the recommendation hops plus the chain
length of the second statement. The chain length of the resulting statement is
the sum of the chain lengths of the input statements. This ensures that the
recommendation hop value of the trust statements decreases by one throughout
the chain of first-hand trust relations (e. g., h = 2, h = 1, h = 0).
The example in Fig. 2 illustrates the inference rule. The transitive trust
inference rule allows to combine H1+ = Trust+ (EA , EB , r, 2, 1) with H2+ =
Trust+ (EB , EC , r, 1, 1) to H4+ = Trust+ (EA , EC , r, 1, 2) and then H4+ with H3− =
Trust− (EC , ED , r, 0, 1) to H5− = Trust− (EA , ED , r, 0, 3).
Evaluation of Trust and Authenticity Statements 69

H1+ H2+ H3−
h = 2, l = 1 h = 1, l = 1 h = 0, l = 1
EA EB EC ED
H4+
h = 1, l = 2
EA EC
H5−
h = 0, l = 3
EA ED

Fig. 2. Example for application of the transitive trust inference rule

4.2 Trust in Entities, Keys and Descriptions
A number of simple rules allow to infer from trust assigned to an entity to trust
assigned to the holder of a key and to trust assigned to an entity identified by a
description, and vice versa. If an entity is trustworthy, then the holder of a key
that belongs to this entity is trustworthy, too, and vice versa:

Auth(EA , KC , EC ) ⊗ Trust(EA , EC , r, h, l)
(8)
Trust(EA , KC , r, h, l)
Auth(EA , KC , EC ) ⊗ Trust(EA , KC , r, h, l)
(9)
Trust(EA , EC , r, h, l)
If an entity is trustworthy, then the entity identified by a description that
belongs to this entity is trustworthy, too, and vice versa:

Auth(EA , DC , EC ) ⊗ Trust(EA , EC , r, h, l)
(10)
Trust(EA , DC , r, h, l)
Auth(EA , DC , EC ) ⊗ Trust(EA , DC , r, h, l)
(11)
Trust(EA , EC , r, h, l)
If the holder of a key is trustworthy, then the entity identified by a description
that belongs to this key holder is trustworthy, too, and vice versa. This applies
to trust relations and trust certificates:

Auth(EA , KC , DC ) ⊗ Trust(EA , KC , r, h, l)
(12)
Trust(EA , DC , r, h, l)
Auth(EA , KC , DC ) ⊗ Trust(EA , DC , r, h, l)
(13)
Trust(EA , KC , r, h, l)
Auth(KA , KC , DC ) ⊗ Trust(KA , KC , r, h, l)
(14)
Trust(KA , DC , r, h, l)
Auth(KA , KC , DC ) ⊗ Trust(KA , DC , r, h, l)
(15)
Trust(KA , KC , r, h, l)
70 Gutscher

4.3 Local Authenticity Inference Rule

If an entity EA has partial knowledge about whether an entity EB is the holder
of a key KB , whether a description DB refers to the entity EB or whether the
description DB refers to the holder of the key KB , then it can draw further con-
clusions about the confidence values of the authenticity statements between EB ,
KB and DB . If the confidence values of two corresponding authenticity relations
are known, then the confidence value of the third authenticity relation can be
derived with the authentication operator:

Auth(EA , KC , DC ) Auth(EA , KC , EC )
(16)
Auth(EA , DC , EC )
Auth(EA , KC , DC ) Auth(EA , DC , EC )
(17)
Auth(EA , KC , EC )
Auth(EA , KC , EC ) Auth(EA , DC , EC )
(18)
Auth(EA , KC , DC )

4.4 Authenticity Inference with Authenticity Confirmation

If a trustor (EA or KA ) trusts a trustee (EB or KB ) to issue only correct au-
thenticity relations or identity certificates (property rPKI ), then the trustor can
conclude that authenticity relations or identity certificates of the trustee are cor-
rect:

Trust(EA , EB , rPKI , 0, l) ⊗ Auth(EB , KC , DC )
(19)
Auth(EA , KC , DC )
Trust(EA , KB , rPKI , 0, l) ⊗ Auth(KB , KC , DC )
(20)
Auth(EA , KC , DC )
Trust(KA , KB , rPKI , 0, l) ⊗ Auth(KB , KC , DC )
(21)
Auth(KA , KC , DC )

4.5 Uniqueness Conditions

Two further conclusions can be drawn from the condition that each public key
has only one holder and that each description refers to only one entity. If A
knows that EB is the holder of KB , then it can infer that all other entities are
not the holder of KB . Similarly, if A knows that EB has the description DB ,
then it can infer that all other entities do not have the description DB (A can
be an entity or a key).

Auth+ (A, KB , EB ) Auth+ (A, DB , EB )
, ∀Ej 6= EB (22)
Auth− (A, KB , Ej ) Auth− (A, DB , Ej )
Evaluation of Trust and Authenticity Statements 71

5 Confidence Value Computation

The reputation system collects all issued first-hand trust and authenticity opin-
ions Hj with associated continuous confidence value tj (with cj = 0). Users can
then send requests in the form of a standard form trust statement or an authen-
ticity statement to the reputation system. The reputation system then processes
all collected opinions. It applies the inference rules to derive trust and authen-
ticity statements and it computes the resulting continuous confidence value t0 of
the requested statement H0 from the confidence values of the relevant first-hand
statements. As the components of the continuous first-hand confidence values
(b, i and d) represent probabilities, we define the resulting confidence value by a
random experiment and propose different algorithms for the computation of the
resulting confidence value.

5.1 Probabilistic Model for the Confidence Value Computation

The components of the computed resulting confidence value t0 = (b0 , i0 , d0 , c0 )
for H0 are computed from the combination of all available first-hand opinions
with the inference rules under the assumption that the confidence values of the
opinions of the requestor are correct. In short, b0 is the computed lower bound for
the probability that the combination of the available first-hand opinions leads to
the conclusion that H0 must be true (but not that H0 must be false). Similarly,
d0 is the computed lower bound for the probability that the combination of the
available first-hand opinions leads to the conclusion that H0 must be false (but
not that H0 must be true). The degree of conflict c0 is the computed probability
that the combination of the first-hand opinions leads to the contradicting con-
clusion that H0 must be both true and false at the same time. The degree of
ignorance is the remaining probability i0 = 1 − b0 − d0 − c0 .
The following description of a random experiment provides a more detailed
definition for t0 : We assume that the reputation system has collected J first-hand
opinions, i. e., the statements Hj (j = 1, 2, . . . J) with associated continuous
confidence values tj = (bj , ij , dj , 0). For each first-hand statement Hj choose
a discrete confidence value t0j from {+, ∅, −} according to the weights bj , ij
and dj , i. e., choose t0j = + with probability bj , t0j = ∅ with probability ij
and t0j = − with probability dj . Statements with the discrete confidence value
ignorance don’t contribute knowledge and can be discarded3 . Each remaining
first-hand statement Hj with associated discrete confidence value t0j corresponds
to a set of first-hand propositions according to Tab. 2.
The inference rules always operate on trust propositions in the internal rep-
resentation. We therefore have to replace each standard-form trust statement
Trust(A, B, r, hmin ..hmax ) by a list of single-hop trust statements in internal
form with chain length l = 1: Trust(A, B, r, hmin , l), Trust(A, B, r, hmin + 1, l),
. . . , Trust(A, B, r, hmax , l). The internal trust statements inherit their assigned
3
this optimization does not change the resulting confidence value, the resulting con-
tinuous confidence value t0 nevertheless contains the correct degree of ignorance
72 Gutscher

discrete confidence value from the standard-form trust statement. Next, we ap-
ply all inference rules (see Sect. 4) to derive all (positive and negative) deducible
propositions from the set of all known first-hand propositions and all already de-
rived propositions. To get back to trust statements in standard form we conclude
H0+ = Trust+ (A, B, r, hmin ..hmax ) if we have been able to derive a proposition
+
H0,h = Trust+ (A, B, r, h, l) with hmin ≤ h ≤ hmax . Similarly, we conclude H0− if
−
we have been able to derive a proposition H0,h .
To obtain the resulting continuous confidence value of a requested trust or
authenticity statement we compute the probability that the random experiment
leads to a set of first-hand propositions from which we can derive positive and
negative propositions for the requested statement H0 . The components of the
resulting confidence value t0 = (b0 , i0 , d0 , c0 ) are defined as follows: b0 is the
probability that H0+ (but not H0− ) can be derived and d0 is the probability that
H0− (but not H0+ ) can be derived. The probability that neither H0+ nor H0− can
be derived is i0 , and c0 is the probability that both H0+ and H0− can be derived.
In contrast to other trust models (e. g., [5, 6, 10, 11]) we propose not to elim-
inate the degree of conflict, not only to avoid counter-intuitive effects of re-
normalizations but also because it provides valuable information to the request-
ing user or application (see Sect. 3.1).

5.2 Approximation and Exact Computation Algorithms

This section presents different possibilities to implement the computation of an
approximation or of the exact value of the resulting continuous confidence value
t0 according to Sect. 5.1. All exact algorithms return the same resulting confi-
dence value t0 , but differ in computation time. The result of the approximation
gets arbitrarily close to the exact result if the number of iterations is sufficiently
large.
To keep the computation time small we recommend for all algorithms to
precompute all possible paths: We first set up a “superposition” of possible first-
hand propositions. For each statement Hj with continuous confidence value tj =
(bj , ij , dj , 0) we select Hj+ if bj > 0 and we select (possibly in addition) Hj− if
dj > 0. Then we translate all trust propositions into the internal form, apply all
inference rules and record the dependencies, i. e., we trace which sets of first-hand
propositions (premises) allow to derive which conclusions. Each set of first-hand
propositions that allows to (directly or indirectly) derive the positive requested
proposition H0+ is called a positive path for H0 , each set that allows to derive
the negative proposition H0− a negative path for H0 . Next, we select the set of
positive paths and the set of negative paths for H0 and minimize these paths,
i. e., we remove all paths that contain at least one other path in the set. We
finally obtain the set of minimal positive paths A+ = {a+ + +
1 , a2 , . . . ak+ } and the
set of minimal negative paths A− = {a− − −
1 , a2 , . . . ak− }.

Approximation with Monte-Carlo Simulation An obvious approach to
determine an approximation for the resulting confidence value is to run the
Evaluation of Trust and Authenticity Statements 73

described random experiment N times and to count in how many experiments
the selected set of first-hand propositions contains at least one positive (but
no negative) path (nb ), no paths (ni ), at least one negative (but no positive)
path (nd ) or both positive and negative paths (nc ). The approximation for the
confidence value is t̄0 = N1 (nb , ni , nd , nc ). The choice of N allows to adjust the
trade-off between precision and computation time.

Possible Worlds Algorithm An simple algorithm to compute the exact value
is to go through the list of all possible combinations of first-hand propositions
(so-called possible worlds), to compute the probability of each of those possible
worlds and to check for each world whether the set of first-hand propositions of
this world contains the minimal paths. The sum of all probabilities of all worlds
that contain at least one positive and at least one negative path is c0 , b0 is
the sum of probabilities of all worlds that contain at least one positive, but no
negative path, and d0 the sum of probabilities of all worlds that contain at least
one negative, but no positive path. The degree of ignorance is i0 = 1−b0 −d0 −c0 .

H1,1 H2,0 Probability H0
+ +
H1 H2 H1,1 H2,0 b1 b2 H0+
+
H1,1 b1 i2
−
+
H1,1 H2,0 b1 d 2 H0−
EA EB EC +
H2,0 i1 b2
i1 i2
−
j Hj bj ij dj H2,0 i1 d2
− +
1 Trust(EA , EB , r1 , 1) 0.5 0.15 0.35 H1,1 H2,0 d 1 b2
−
2 Trust(EB , EC , r1 , 0) 0.7 0.1 0.2 H1,1 d1 i2
− −
H1,1 H2,0 d1 d2

Fig. 3. Scenario and possible worlds table of Example 1

Example 1: Simple Trust Chain with Possible Worlds Algorithm The
example scenario in Fig. 3 (left) consists of two trust statements: H1 is a rec-
ommendation trust statement for one recommendation hop (h = 1), and H2 is a
functional trust statement (h = 0). EA wants to compute the resulting functional
trustworthiness of EC (H0 = Trust(EA , EC , r1 , 0)).
First, the trust statements in standard form have to be replaced by corre-
sponding trust statements in internal form: H1 by H1,1 = Trust(EA , EB , r1 , 1, 1)
and H2 by H2,0 = Trust(EB , EC , r1 , 0, 1). Both refer to the same property r1 , it
is therefore possible to combine H1,1 and H2,0 with the transitive trust inference
rule (7) to the new functional trust statement H0,0 = Trust(EA , EC , r1 , 0, 2):
+ + + + + + + − −
H1,1 , H2,0 ` H0,0 (H1,1 and H2,0 allow to drive H0,0 ) and H1,1 , H2,0 ` H0,0 .
+ + +
Thus, there is only one positive path a1 = {H1,1 , H2,0 } and one negative path
a− + −
1 = {H1,1 , H2,0 } for H0,0 and thus for H0 .
74 Gutscher

Fig. 3 (right) shows all possible combinations of the first-hand propositions,
the probability that this world occurs and the propositions that can be derived
in this world. There are no worlds in which both H0+ and H0− can be derived,
thus c0 = 0. H0+ can be derived only in the first world, therefore b0 = b1 b2 .
Similarly, H0− can be derived only in the third world, therefore d0 = b1 d2 . The
degree of ignorance is the remaining probability mass i0 = 1 − b0 − d0 − c0 . With
the values in Fig. 3 we obtain t0 = (0.35, 0.55, 0.1, 0).

Grouped Possible Worlds Algorithm The possible worlds algorithm can
be improved by subdividing the set of relevant first-hand statements into as few
groups g1 , . . . gu as possible. Two statements, Hj and Hm , belong to the same
group if and only if the following condition holds for each positive, negative and
conflicting path: If the path contains a proposition for Hj (Hj+ or Hj− ), then it
+ −
must also contain a proposition for Hm (Hm or Hm ).
In the preparation step we construct for each group a list of all relevant
combinations of propositions of the statements in the group. This list contains
all combinations that contain exactly one proposition (i. e., either Hj+ or Hj− )4 for
each statement and that is identical to the corresponding section of at least one
(positive or negative) path. An additional element of this list consists of an empty
set. It represents all remaining possible combinations of propositions, i. e., all
combinations that contain neither Hj+ nor Hj− for at least one statement Hj of
the group. We can subsume these combinations because the have the same effect
on the derivability of propositions of H0 . For each element of the list we compute
the probability that this combination will occur (within the group) from the
continuous confidence values of the statements. The probability associated with
the empty set is the sum of the probabilities of the contained combinations of
propositions (i. e., the remaining probability). Thus, the sum of all probabilities
is one.
Next, in the main step, we go through all possible worlds. Each world consists
of one possible combination of these prepared proposition-combinations of the
groups, i. e., for each groups we select one proposition-combination from the
prepared list of the group. We multiply the precomputed probabilities of the
chosen proposition-combinations to obtain the resulting probability of the world.
Finally, we compute b0 , i0 , d0 and c0 just as in the possible worlds algorithm.

Example 2: Parallel Trust Chain with Grouped Possible Worlds Algo-
rithm The scenario in Fig. 4 consists of two parallel trust chains from EA to ED .
EA requests the confidence value for the resulting functional trustworthiness of
ED (H0 = Trust(EA , ED , r1 , 0)). The trust statements in standard form are re-
placed by statements in internal form: H1 by H1,1 = Trust(EA , EB , r1 , 1, 1),
H2 by H2,0 = Trust(EB , ED , r1 , 0, 1), H3 by H3,1 = Trust(EA , EC , r1 , 1, 1)
and H4 by H4,0 = Trust(EC , ED , r1 , 0, 1). We can combine H1,1 with H2,0
and H3,1 with H4,0 with the transitive trust inference rule (7). We obtain
4
no combination can contain both Hj+ and Hj− because cj = 0
Evaluation of Trust and Authenticity Statements 75

H1 H2
j Hj bj ij dj
EB 1 Trust(EA , EB , r1 , 1) 0.8 0.15 0.05
2 Trust(EB , ED , r1 , 0) 0.7 0.1 0.2
EA ED 3 Trust(EA , EC , r1 , 1) 0.9 0.1 0
H3 H4 4 Trust(EC , ED , r1 , 0) 0.8 0.1 0.1

Fig. 4. Scenario of Example 2

+ + + +
two positive paths A+ = {{H1,1 , H2,0 }, {H3,1 , H4,0 }} and two negative paths
+ − + −
A− = {{H1,1 , H2,0 }, {H3,1 , H4,0 }}. Propositions for H1,1 and H2,0 appear al-
ways together in paths, the same holds for H3,1 and H4,0 . Therefore we can
divide the statements into two groups g1 = {H1,1 , H2,0 } and g1 = {H3,1 , H4,0 }.
In the preparation step we set up a list for each group that contains all
relevant possible combinations of the propositions and their probabilities (see
Tab. 3). For each group we find three relevant combinations: one combination
supports a positive path and one a negative path. The third entry with the
empty set represents the remaining combinations.

Table 3. Preparation step for the groups in Example 2

Propositions g1 Probability Propositions g2 Probability
+ + + +
{H1,1 , H2,0 } b1 b2 {H3,1 , H4,0 } b 3 b4
+ − + −
{H1,1 , H2,0 } b1 d 2 {H3,1 , H4,0 } b3 d4
{} 1 − b1 b2 − b1 d 2 {} 1 − b3 b4 − b3 d 4

Table 4. Confidence value computation in the parallel trust chain example

Table 5. Confidence value computation in Example 2

g1 g2 Probability H0
+ + + +
{H1,1 , H2,0 } {H3,1 , H4,0 } b1 b2 b3 b4 H0+
−
+
{H1,1 +
, H2,0 +
} {H3,1 , H4,0 } b1 b2 b3 d 4 H0+ , H0−
+ +
{H1,1 , H2,0 } {} b1 b2 (1 − b3 b4 − b3 d4 ) H0+
−
+
{H1,1 , H2,0 +
} {H3,1 +
, H4,0 } b1 d 2 b3 b4 H0+ , H0−
− −
+
{H1,1 , H2,0 +
} {H3,1 , H4,0 } b1 d 2 b3 d 4 H0−
−
+
{H1,1 , H2,0 } {} b1 d2 (1 − b3 b4 − b3 d4 ) H0−
+ +
{} {H3,1 , H4,0 } (1 − b1 b2 − b1 d2 )b3 b4 H0+
−
{} +
{H3,1 , H4,0 } (1 − b1 b2 − b1 d2 )b3 d4 H0−
{} {} (1 − b1 b2 − b1 d2 )(1 − b3 b4 − b3 d4 )
76 Gutscher

To compute the resulting confidence value t0 for H0 we set up Tab. 5 with
all 3 · 3 = 9 possible combinations of the entries in the lists (possible worlds), the
probabilities of each world and the derivable propositions for H0 . Then we add
the probabilities and obtain b0 = b1 b2 b3 b4 + b1 b2 (1 − b3 b4 − b3 d4 ) + (1 − b1 b2 −
b1 d2 )b3 b4 , i0 = (1 − b1 b2 − b1 d2 )(1 − b3 b4 − b3 d4 ), d0 = b1 d2 b3 d4 + b1 d2 (1 − b3 b4 −
b3 d4 ) + (1 − b1 b2 − b1 d2 )b3 d4 and c0 = b1 b2 b3 d4 + b1 d2 b3 b4 . With the values in
Fig. 4 we obtain t0 = (0.7112, 0.0532, 0.07, 0.1656).

Computation with Inclusion-exclusion Formula This algorithm computes
the exact resulting confidence value directly from the minimal positive and neg-
ative paths for H0 . In addition, we need the set of minimal conflicting paths.
Therefore we set up all possible combinations consisting of one positive and one
negative path (a± + − + −
x = ay ∪ az with y = 1, . . . k , z = 1, . . . k ), minimize the set
± ± ± ± ± + −
and obtain A = {a1 , a2 , . . . ak± } (with k ≤ k k ). A useful optimization is
to eliminate all paths that contain both Hj+ and Hj− (because cj = 0).
First, we compute the degree of conflict c0 from the confidence values of the
first-hand statements in the set of minimal paths with the inclusion-exclusion-
formula (I(A)): c0 is the probability that a possible world chosen according to
Sect. 5.1 will contain at least one conflicting path. Thus, we add the probabilities
of all minimal paths, subtract the probabilities of all unions of two minimal paths,
add the probabilities of all unions of three minimal paths, etc.:
±
k
X X
c0 =I(A ) = ±
(−1)n+1 P (a± ±
j1 ∪ · · · ∪ ajn )
n=1 1≤j1 <··· 0,
i > 0 and d > 0 is possible, no restriction to directed series-parallel graphs).
However, we do not eliminate the degree of conflict, because this can cause
counter-intuitive effects: Consider Example 2 (Sect. 5.2). If we choose t1 = t2 =
t3 = t4 = (1, 0, 0, 0) (full trust), then the resulting confidence value is t0 =
(1, 0, 0, 0), too (in all models). If t4 changes to t4 = (0.01, 0, 0.99, 0) (almost
complete distrust), then the resulting confidence value in our model changes
to t0 = (0.01, 0, 0, 0.99), which shows EA that the trustworthiness of ED is
now highly disputed. However, in Subjective Logic and Credential Networks the
resulting confidence value does not change. This gives EA the wrong impression
that there are no trustworthy entities who distrust ED .

6 Computation Time

Computation time is a very (although not the most) important issue for repu-
tation systems. The computation time to find the minimal paths appears to be
uncritical because it is possible to check the inference rules efficiently and be-
cause the paths can be computed incrementally and in advance. Furthermore, the
paths usually remain unchanged when the confidence values of existing opinions
are updated.
The number of possible worlds to consider in the possible worlds algorithm
increases exponentially with the number of relevant first-hand statements. It is
therefore applicable if the number of relevant statements is small. It is important
to emphasize that the computation time depends only on the number of relevant
statements or paths, not on the total number. It is sufficient to consider only
statements that are issued by the requester or by authentic entities or keys that
Evaluation of Trust and Authenticity Statements 79

have been found to be trustworthy. Moreover, we can ignore all statements that
are not part of a valid path, i. e., that do not contribute to answer the trust or
authenticity request. Furthermore, most trust chains will not be longer than two
or three statements. Therefore, the number of relevant statements or paths will
usually be reasonably small. Although a trust and authenticity network similar
to the PGP/GnuPG web of trust [1] can contain more than 100 000 trust and
authenticity statements, the number of statements that are directly or indirectly
(via valid paths) related to the requester will probably be below 100, and the
number of statements that are part of valid paths from the requester to the
requested statement is likely to be not higher than 10 or 20 in typical scenarios.
The number of possible worlds in the grouped possible worlds algorithm in-
creases exponentially with the number of groups. Thus, the computation time can
be reduced significantly if the statements can be grouped. Even large scenarios
can be evaluated efficiently as long as the relevant statements can be subdivided
into a small number of groups. In the inclusion-exclusion algorithm the number
of summands increases exponentially with the number of relevant paths. This
algorithm is therefore well suited for all scenarios with a small number of paths,
even if the number of statements is large.
We illustrate the influence of the scenario (i. e., the number of relevant state-
ments, paths and groups) on the computation time of the algorithms on two
examples5 . The scenarios are constructed in order to emphasize the large influ-
ence on the computation time and are not meant to be representative examples.
For simplicity the scenarios consist only of trust statements between entities and
refer to the same capability r. All confidence values contain degrees of belief, ig-
norance and disbelief (b > 0, i > 0, d > 0).

trustor trustee

Fig. 6. Scenario of the simple chain example

The scenario with e entities in Fig. 6 consists of a simple chain and e −
1 trust statements with h = 0..e recommendation hops. The possible worlds
algorithm has to evaluate 3e−1 worlds. The scenario contains one positive and one
negative path, therefore the inclusion-exclusion algorithm has to compute only
two summands. The grouped possible world algorithm creates one group with
three possible proposition-combinations: the positive path leads to belief, the
negative path to disbelief, all other combinations lead to ignorance. It thus has
to evaluate only three worlds. The diagram in Fig. 7 shows that the computation
time of the possible world algorithm increases exponentially with the number of
trust statements, whereas the computation time of the other algorithms increases
linearly and thus remains insignificant even for long trust chains.
5
implementation in Java 1.6; measurements on Intel Pentium M with 1.73 GHz
80 Gutscher

1000
Pos. worlds
Gr. pos. worlds
Incl.-excl.
100

computation time [ms]
10

0.1

0.01
5 10 15 20 25 30
number of entities

Fig. 7. Computation time in the simple chain example

trustor trustee

Fig. 8. Scenario of the full mesh example

The scenario in Fig. 8 is a full mesh. All entities trust each other for h = 0..1
hops. The number of relevant statements is 2e − 3, the number of positive and
negative paths is e − 1 and the number of conflicting paths is (e − 1)(e − 2).
Thus, the computation time of the possible worlds algorithm increases slower
than of the inclusion-exclusion algorithm because the number of conflicting paths
increases faster than the number of relevant statements (Fig. 9). The grouped
possible worlds algorithm subdivides the statements into e − 1 groups, which
reduces the number of possible worlds from 32e−3 to 3e−1 worlds. Therefore the
computation time remains acceptable for a larger number of entities than with
the other algorithms.
The computation time heavily depends on the scenario. It is therefore difficult
to give a general recommendation for one of the algorithms. It is possible that
one algorithm outperforms an other by orders of magnitude in one scenario, but
is much slower in an other scenario. The presented results and measurements in
other scenarios suggest that the grouped possible worlds algorithm is in most
scenarios the fastest (or at least close to the fastest) algorithm. However, further
investigations are necessary.
Evaluation of Trust and Authenticity Statements 81

1000

100

computation time [ms]
10

0.1
Pos. worlds
Gr. pos. worlds
Incl.-excl.
0.01
2 4 6 8 10 12
number of entities

Fig. 9. Computation time in the full mesh example

An estimation for the computation time of the algorithms can be computed
from the number of relevant statements, paths and groups. If the expected com-
putation of all exact algorithms exceeds an acceptable limit, then a Monte-Carlo
simulation can be used to compute an approximation. The number of iterations
can be chosen according to the required accuracy and the acceptable computa-
tion time.

7 Summary and Conclusions
We presented a detailed model to represent trust and authenticity statements as
well as confidence values and we proposed an integrated approach to reason with
these statements and to compute resulting confidence values. The model distin-
guishes clearly between entities, descriptions and keys, allows multiple keys and
descriptions per entity, distinguishes between functional and recommendation
trust and allows to specify ranges of recommendation hops in trust statements.
Confidence values allow to express degrees of belief, ignorance and disbelief. The
system is able to reason with conflicting opinions because the presented infer-
ence rules are based on a paraconsistent logic. The computation of the resulting
confidence values is based on a probability theoretical model in order to pro-
duce consistent results. In conflict-free scenarios our model is consistent with
the Model of Maurer, Subjective Logic and Credential Networks, but overcomes
several restrictions of these models. In conflicting scenarios we do not elimi-
nate the degree of conflict in order to avoid counter-intuitive effects caused by
re-normalizations.
We proposed different algorithms to implement the confidence value compu-
tation. Although the computation time increases exponentially with the number
82 Gutscher

of relevant statements, groups or paths it can be expected that an acceptable
computation time can be reached in the majority of realistic scenarios. In the
other cases, we propose to compute an approximation with Monte-Carlo simu-
lations.

Acknowledgements This work was funded by the German Research Founda-
tion (DFG) through the Collaborative Research Center (SFB) 627.

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