A simple model of distributed computation that requires information exchange over a noisy channel is considered. A communication protocol is utilized that requires alternate bit exchanges between two processors. First, the case of a single public channel is considered and the number of bits that need to be exchanged between the processors to permit delta -accuracy in their goal is compared. For this computation, an error-detection-and-retransmission mechanism of error control and an error-correction-and-retransmission mixture that are consistent with the logical protocol that governs this exchange are considered. Second, the case of the availability of an additional secret channel is considered and interest in determining the minimum number of bits that need to be exchanged over a secret channel in order to maintain in -uncertainty about the computation for an eavesdropper on the public channel is shown. Various subcases under this case are considered and an upper bound on the number of secret bits when no error-control scheme is used is obtained.>
A simple model of distributed computation that requires information exchange over a noisy channel is considered. A communication protocol is utilized that requires alternate bit exchanges between two processors. First, the case of a single public channel is considered and the number of bits that need to be exchanged between the processors to permit delta -accuracy in their goal is compared. For this computation, an error-detection-and-retransmission mechanism of error control and an error-correction-and-retransmission mixture that are consistent with the logical protocol that governs this exchange are considered. Second, the case of the availability of an additional secret channel is considered and interest in determining the minimum number of bits that need to be exchanged over a secret channel in order to maintain in -uncertainty about the computation for an eavesdropper on the public channel is shown. Various subcases under this case are considered and an upper bound on the number of secret bits when no error-control scheme is used is obtained.>
A common task in parallel processing is the distributed computation of a function by a number of processors each of which possesses partial information relevant to the value of that function. In this thesis we develop protocols which allow for such computation to take place while maintaining the value of the function secret to an eavesdropper. Of interest is the communication complexity of such protocols. We begin by considering two processors and two channels, one secret and one public, and present a protocol which minimizes the number of bits exchanged over the secret channel, while maintaining ϵ-uncertainty about the value of the function for the eavesdropper. We show that all binary functions can be kept ϵ-secret using a constant number of bits independent of the size of their domain. We then generalize our results to N processors communicating over a network of arbitrary topology.
In a related problem, also of interest in parallel processing, we consider processors communicating over a mesh network with the objective of broadcasting information amongst each other. One instance of the problem involves a number of nodes all with the same message to be broadcasted. For that problem a lower-bound on the time to complete the broadcast, and an algorithm which achieves this bound are presented. In another instance, every node in the mesh has packets to be broadcast arriving independently, according to a Poisson random process. The stability region for performing such broadcasts is characterized, and broadcast algorithms which operate efficiently within that region are presented. These algorithms involve interacting queues whose analysis is known to be very difficult. Toward that end we develop an approximate model for analyzing interacting queues. This new approximation models an N-dimensional infinite Markov chain by means of two Markov chains, one being one-dimensional and infinite and the other being N-dimensional and finite. The transition probabilities of each chain are expressed in terms of statistics of the other chain. The two chains can be solved together using standard iterative methods to yield an approximation to the original N-dimensional infinite chain. The results of the approximation compare very well with simulation, especially when arrival rates are low. We show how this model can also be used to analyze the ALOHA multiple access protocol.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange, which increases the communication size in steps until the task is done. We also derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bounds of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general source sequence, such as a mixture of independent and identically distributed (IID) random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Furthermore, we propose an interactive protocol for data exchange which increases the communication size in steps until the task is done and matches the performance of our lower bound. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
In this thesis we focus on various aspects of secure message transmission protocols. Such protocols achieve the secure transmission of a message from a sender to a receiver - where the term “secure” encapsulates the notion of privacy and reliability of message transmission. These two parties are connected using an underlying network in which a static computationally unlimited active adversary able to corrupt up to t network nodes is assumed to be present. Such protocols are important to study as they are used extensively in various cryptographic protocols and are of interest to other research areas such as ad-hoc networks, military networks amongst others. Optimal bounds for the number of phases (communication from sender to receiver or vice versa), connectivity requirements (number of node disjoint network paths connecting sender and receiver - denoted by n), communication complexity (complexity of the number of field elements sent - where F is the finite field used and jFj = q) and transmission complexity (proportion of communication complexity to complexity of secrets transmitted) for secure message transmission protocols have been proven in previous work. In the one-phase model it has been shown that n 3t+1 node disjoint paths are required to achieve perfect communication. In the two phase model only n 2t + 1 node disjoint paths are necessary. This connectivity is also the required bound for almost perfectly secure one-phase protocols - protocols which achieve perfect privacy but with a negligible probability may fail to achieve reliability. In such cases the receiver accepts a different message to that transmitted by the sender or does not accept any message. The main focus of recent research in secure message transmission protocols has been to present new protocols which achieve optimal transmission complexity. This has been achieved through the transmission of multiple messages. If a protocol has a communication complexity of O(n3) field elements, to achieve optimal transmission complexity O(n2) secrets will have to be communicated. This has somewhat ignored the simplification and improvement of protocols which securely transmit a single secret. Such improvements include constructing more efficient protocols with regards to communication complexity, computational complexity and the number of field elements sent throughout the whole protocol. In the thesis we first consider one-phase almost perfectly secure message transmission and present two new protocols which improve on previous work. We present a polynomial time protocol of O(n2) communication complexity which at the time of writing this thesis, is computationally more efficient than any other protocol of similar communication complexity for the almost perfectly secure transmission of a single message. Even though our first almost perfectly secure transmission protocol is of polynomial time, it is important to study other protocols also and improve previous work presented by other researchers. This is the idea behind the second one-phase almost perfectly secure message transmission protocol we present which requires an exponential complexity of field operations but lower (O(n)) communication complexity. This protocol also improves on previous protocols of similar communication complexity, requiring in the order of O(log q) less computation to complete - where q denotes the size of the finite field used. Even though this protocol is of exponential time, for small values of n (e.g. when t = 1, t = 2 or t = 3) it may be beneficial to use this protocol for almost perfectly secure communication as opposed to using the polynomial time protocol. This is because less field elements need to be transmitted over the whole network which connects a sender and a receiver. Furthermore, an optimal almost perfectly secure transmission protocol will be one with O(n) communication complexity and with polynomial computational complexity. We hope that in the future, other researchers will be inspired by our proposed protocol, improve on our work and ideally achieve these optimal results. We also consider multi-phase protocols. By combining various cryptographic schemes, we present a new two-phase perfectly secure single message transmission protocol. At the time of writing this thesis, the protocol is the most efficient protocol when considering communication complexity. Our protocol has a communication complexity of O(n2) compared to O(n3) of previous work thus improving on the communication complexity by an order of O(n) for the perfectly secure message transmission of a single message. This protocol is then extended to a three phase protocol where a multi-recipient broadcast end channel network setting is considered. As opposed to point to point networks where a path from a sender reaches a single receiver, this network model is new in the field of message transmission protocols. In this model each path from a sender reaches multiple receivers, with all receivers receiving the same information from their common network communication channel. We show how the use of this protocol upon such a network can lead to great savings in the transmission and computation carried out by a single sender. We also discuss the importance and relevance of such a multi-recipient setting to practical applications. The first protocols in the field of perfectly secure message transmission with a human receiver are also presented. This is a topic proposed by my supervisor Professor Yvo Desmedt for which I constructed solutions. In such protocols, one of the communicating parties is considered to be a human who does not have access to a computational device. Because of this, solutions for such protocols need to be computationally efficient and computationally simple so that they can be executed by the human party. Experiments with human participants were carried out to assess how easily and accurately human parties used the proposed protocols. The experimental results are presented and these identify how well human participants used the protocols. In addition to the security of messages, we also consider how one can achieve anonymity of message transmission protocols. For such protocols, considering a single-receiver multi-sender scenario, the presence of a t-threshold bounded adversary and the transmission of multiple secrets (as many as the number of sender), once the protocols ends one should not be able to identify the sender of a received message. Considering a passive and active adversary new protocols are presented which achieve the secure and anonymous transmission of messages in the information-theoretic security model. Our proposed solutions can also be applied (with minor alterations) to the dual problem when a single-sender multi-recipient communication setting is considered. The contributions of the thesis are primarily theoretical - thus no implementation of the proposed protocols was carried out. Despite this, we reflect on practical aspects of secure message transmission protocols. We review the feasibility of implementing secure message transmission protocols in general upon various networks - focusing on the Internet which can be considered as the most important communication network at this time. We also describe in theory how concepts of secure message transmission protocols could possibly be used in practical implementations for secure communication on various existing communication networks. Open problems that remain unsolved in the research area of the proposed protocols are also discussed and we hope that these inspire research and future solutions for the design (and implementation) of better and more efficient secure message transmission protocols.
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange which increases the communication size in steps until the task is done. Next, we derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.
Multiparty computation (MPC) is a powerful and generic cryptographic framework capable of realizing essentially any cryptographic task (functionality) while preserving privacy and robustness. In the MPC model, a group of mutually distrustful parties seek to jointly compute a function f of their privately held inputs x1, ..., xn, while revealing no information about those inputs beyond what is entailed by the function output(s). Some portion of these players may be corrupted, controlled by a centralized adversary who directs their (mis)behavior and seeks to gain information on private inputs or to disrupt the protocol execution.
In this work, we examine the resource requirements of information-theoretically secure multiparty computation. In particular, we look at trade-offs involving what we call assumptions, by which we mean two aspects of the model's setup: (1) the extent to which parties are connected by secure, pairwise communication channels; and (2) the usage (as a black box) of a channel—a primitive which enforces consistency among views by allowing a single party to send a message to all other parties which is guaranteed to be consistent, even if the broadcaster is corrupted.
Part I of this work focuses on the first network-model assumption, that of the graph's connectivity by pairwise channels. We show upper and lower bounds on communication complexity in the Secure Message Transmission with Public Discussion (SMT-PD) model, as a function of the threshold of corrupted parties. The SMT-PD model, which abstracts the problem of secure communication over a partially connected network, includes two components: a public and a set of simple channels. We show significant improvements in both areas: on the channel, we reduce communication to logarithmic in terms of the message's bit-length m, where the previous best result was linear in m; on the simple channels, we reduce from O(mn) to O(mn/(n−t)), where n is the number of simple channels and t is a bound on the number of corrupted channels. Using tools from information theory, we show this amount of simple-wire communication is essentially optimal. Finally, we give an amortized solution which drastically improves communication over channel when the protocol is repeated many times sequentially.
Part II shifts focus to the second network-model assumption, the use of an atomic broadcast primitive. Here we show that three broadcast rounds suffice for general secure computation, while two broadcast rounds are necessary. In particular we show that even for the natural and important functionality of Weak Secret Sharing—a type of distributed commitment—there is no protocol which uses only a single broadcast and achieves better than an explicit constant probability of success (so the protocol's success cannot be a function of a security parameter).