Abstract: We study the partially eponymous model of distributed computation, which simultaneously
generalizes the anonymous and the eponymous models. In this model, processors have
identities, which are neither necessarily all identical (as in the anonymous model) nor
necessarily unique (as in the eponymous model). In a decision problem formalized as a
relation, processors receive inputs and seek to reach outputs respecting the relation. We
focus on the partially eponymous ring, and we shall consider the computation of circularly
symmetric relations on it. We consider sets of rings where all rings in the set have the same
multiset of identity multiplicities.
We distinguish between solvability and computability: in solvability, processors are
required to always reach outputs respecting the relation; in computability, they must
do so whenever this is possible, and must otherwise report impossibility.
We present a topological characterization of solvability for a relation on a set of rings,
which can be expressed as an efficiently checkable, number-theoretic predicate.
We present a universal distributed algorithm for computing a relation on a set of
rings; it runs any distributed algorithm for constructing views, followed by local steps.
We derive, as our main result, a universal upper bound on the message complexity to
compute a relation on a set of rings; this bound demonstrates a graceful degradation
with the Least Minimum Base, a parameter indicating the degree of least possible
eponymity for a set of rings. Thereafter, we identify two cases where a relation can be
computed on a set of rings, with rings of size n, with an efficient number of O .n lg n/
messages.

Abstract: The concept of trust plays an important role in the operation and public acceptance of today's computing environment. Although it is a difficult concept to formalize and handle, many efforts have been made towards a clear definition of trust and the development of systematic ways for trust management. Our central viewpoint is that trust cannot be defined, anymore, as consisting of a static set of rules that define systems properties that hold eternally due to the highly dynamic nature of today's computing systems (e.g. wireless networks, ad-hoc networks, virtual communities and digital territories etc.). Our approach is an effort to define trust in terms of properties that hold with some limiting probability as the the system grows and try to establish conditions that ensure that ??good?? properties hold almost certainly. Based on this viewpoint, in this paper we provide a new framework for defining trust through formally definable properties that hold, almost certainly, in the limit in randomly growing combinatorial structures that model ??boundless?? computing systems (e.g. ad-hoc networks), drawing on results that establish the threshold behavior of predicates written in the first and second order logic. We will also see that, interestingly, some trust models have properties that do not have limiting probabilities. This fact can be used to demonstrate that as certain trust networks grow indefinitely, their trust properties are not certain to be present

Abstract: In this Phd thesis,, we try to use formallogic and threshold phenomena that asymptotically emerge with certainty in order to build new trust models and to evaluate the existing one. The departure point of our work is that dynamic, global computing systems are not amenable to a static viewpoint of the trust concept, no matter how this concept is formalized. We believe that trust should be a statistical, asymptotic concept to be studied in the limit as the system's components grow according to some growth rate. Thus, our main goal is to define trust as an emerging system property that ``appears'' or "disappears" when a set of properties hold, asymptotically with probability$ 0$ or $1$ correspondingly . Here we try to combine first and second order logic in order to analyze the trust measures of specific network models. Moreover we can use formallogic in order to determine whether generic reliability trust models provide a method for deriving trust between peers/entities as the network's components grow. Our approach can be used in a wide range of applications, such as monitoring the behavior of peers, providing a measure of trust between them, assessing the level of reliability of peers in a network. Wireless sensor networks are comprised of a vast number of ultra-small autonomous computing, communication and sensing devices, with restricted energy and computing capabilities, that co-operate to accomplish a large sensing task. Sensor networks can be very useful in practice. Such systems should at least guarantee the confidentiality and integrity of the information reported to the controlling authorities regarding the realization of environmental events. Therefore, key establishment is critical for the protection in wireless sensor networks and the prevention of adversaries from attacking the network. Finally in this dissertation we also propose three distributed group key establishment protocols suitable for such energy constrained networks. This dissertation is composed of two parts. Part I develops the theory of the first and second order logic of graphs - their definition, and the analysis of their properties that are expressible in the {\em first order language} of graphs. In part II we introduce some new distributed group key establishment protocols suitable for sensor networks. Several key establishment schemes are derived and their performance is demonstrated.

Abstract: Today we are experiencing a major reconsideration of the computing
paradigm, as witnessed by the abundance and increasing frequency
of use of terms such as {\em ambient intelligence}, {\em ubiquitous computing}, {\em disappearing computer}, {\em grid
computer}, {\em global computing} and {\em mobile ad-hoc
networks}. Systems that can be described with such terms are of a
dynamic, with no clear physical boundary, nature and it seems that
it is impossible (or, at least, difficult) to define sharply a
number of important properties holding with certainty as well as
holding throughout the whole lifetime of the system.
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One such system property, which is important for the viability of
a system, is {\em trust}. Our departure point is the assumption
that it seems very difficult to define static system properties
related to trust and expect that they hold eternally in the
rapidly changing systems falling under the new computing paradigm.
One should, rather, attempt to define trust in terms of properties
that hold with some limiting probability as the the system grows
and try to establish conditions that ensure that ``good''
properties hold {\em almost certainly}. Based on this viewpoint,
in this paper we provide a new framework for defining trust
through formally definable properties that hold, almost certainly,
in the limit in randomly growing combinatorial structures that
model ``shapeless'' computing systems (e.g. ad-hoc networks),
drawing on results that establish the threshold behavior of
predicates written in the first and second order logic.