In this work, we investigate the security of interactive computations. The main emphasis is on the mathematical methodology that is needed to formalise and analyse various security properties. Differently from many classical treatments of secure multi-party computations, we always quantify security in exact terms. Although working with concrete time bounds and success probabilities is technically more demanding, it also has several advantages. As all security guarantees are quantitative, we can always compare different protocol designs. Moreover, these security guarantees also have a clear economical interpretation and it is possible to compare cryptographic and non-cryptographic solutions. The latter is extremely important in practice, since cryptographic techniques are just one possibility to achieve practical security. Also, working with exact bounds makes reasoning errors more apparent, as security proofs are less abstract and it is easier to locate false claims. The choice of topics covered in this thesis was guided by two principles. Firstly, we wanted to give a coherent overview of the secure multi-party computation that is based on exact quantification of security guarantees. Secondly, we focused on topics that emerged from the author's own research. In that sense, the thesis generalises many methodological discoveries made by the author. As surprising as it may seem, security definitions and proofs mostly utilise principles of hypothesis testing and analysis of stochastic algorithms. Thus, we start our treatment with hypothesis testing and its generalisations. In particular, we show how to quantify various security properties, using security games as tools. Next, we review basic proof techniques and explain how to structure complex proofs so they become easily verifiable. In a nutshell, we describe how to represent a proof as a game tree, where each edge corresponds to an elementary proof step. As a result, one can first verify the overall structure of a proof by looking at the syntactic changes in the game tree and only then verify all individual proof steps corresponding to the edges. The remaining part of the thesis is dedicated to various aspects of protocol design. Firstly, we discuss how to formalise various security goals, such as input-privacy, output-consistency and complete security, and how to choose a security goal that is appropriate for a specific setting. Secondly, we also explore alternatives to exact security. More precisely, we analyse connections between exact and asymptotic security models and rigorously formalise a notion of subjective security. Thirdly, we study in which conditions protocols preserve their security guarantees and how to safely combine several protocols. Although composability results are common knowledge, we look at them from a slightly different angle. Namely, it is irrational to design universally composable protocols at any cost; instead, we should design computationally efficient protocols with minimal usage restrictions. Thus, we propose a three-stage design procedure that leads to modular security proofs and minimises usage restrictions.
In this work, we investigate the security of interactive computations. The main emphasis is on the mathematical methodology that is needed to formalise and analyse various security properties. Differently from many classical treatments of secure multi-party computations, we always quantify security in exact terms. Although working with concrete time bounds and success probabilities is technically more demanding, it also has several advantages. As all security guarantees are quantitative, we can always compare different protocol designs. Moreover, these security guarantees also have a clear economical interpretation and it is possible to compare cryptographic and non-cryptographic solutions. The latter is extremely important in practice, since cryptographic techniques are just one possibility to achieve practical security. Also, working with exact bounds makes reasoning errors more apparent, as security proofs are less abstract and it is easier to locate false claims. The choice of topics covered in this thesis was guided by two principles. Firstly, we wanted to give a coherent overview of the secure multi-party computation that is based on exact quantification of security guarantees. Secondly, we focused on topics that emerged from the author's own research. In that sense, the thesis generalises many methodological discoveries made by the author. As surprising as it may seem, security definitions and proofs mostly utilise principles of hypothesis testing and analysis of stochastic algorithms. Thus, we start our treatment with hypothesis testing and its generalisations. In particular, we show how to quantify various security properties, using security games as tools. Next, we review basic proof techniques and explain how to structure complex proofs so they become easily verifiable. In a nutshell, we describe how to represent a proof as a game tree, where each edge corresponds to an elementary proof step. As a result, one can first verify the overall structure of a proof by looking at the syntactic changes in the game tree and only then verify all individual proof steps corresponding to the edges. The remaining part of the thesis is dedicated to various aspects of protocol design. Firstly, we discuss how to formalise various security goals, such as input-privacy, output-consistency and complete security, and how to choose a security goal that is appropriate for a specific setting. Secondly, we also explore alternatives to exact security. More precisely, we analyse connections between exact and asymptotic security models and rigorously formalise a notion of subjective security. Thirdly, we study in which conditions protocols preserve their security guarantees and how to safely combine several protocols. Although composability results are common knowledge, we look at them from a slightly different angle. Namely, it is irrational to design universally composable protocols at any cost; instead, we should design computationally efficient protocols with minimal usage restrictions. Thus, we propose a three-stage design procedure that leads to modular security proofs and minimises usage restrictions.
The design of secure protocols that implement arbitrarily desired functionalities is a major part of modern cryptography. Taking the opposite perspective, the design of any cryptographic scheme may be viewed as the design of a secure protocol for implementing a suitable functionality. Still, we believe that it makes sense to differentiate between basic cryptographic primitives (which involve little interaction) like encryption and signature schemes, on the one hand, and general cryptographic protocols, on the other hand. In this chapter we consider general results concerning secure multi –party computations, where the two –party case is an important special case. In a nutshell, these results assert that one can construct protocols for securely computing any desirable multi–party functionality (see the following terminology). Indeed, what is striking about these results is their generality, and we believe that the wonder is not diminished by the (various alternative) conditions under which these results hold. Our focus on the general study of secure multi–party computation (rather than on protocols for solving specific problems) is natural in the context of the theoretical treatment of the subject matter. We wish to highlight the importance of this general study to practice. Firstly, this study clarifies fundamental issues regarding security in a multi–party environment. Secondly, it draws the lines between what is possible in principle and what is not. Thirdly, it develops general techniques for designing secure protocols. And last, sometimes it may even yield schemes (or modules) that may be incorporated in practical systems.
It seems to be a generally acknowledged fact that you should never trust a computer and that you should trust the person operating the computer even less. This in particular becomes a problem when the party that you do not trust is one which is separated from you and is one on which you depend, e.g. because that party is the holder of some data or some capability that you need in order to operate correctly. How do you perform a given task involving interaction with other parties while allowing these parties a minimal influence on you and, if privacy is an issue, allowing them to learn as little about you as possible. This is the general problem of secure multiparty computation. The usual way of formalizing the problem is to say that a number of parties who do not trust each other wish to compute some function of their local inputs, while keeping their inputs as secret as possible and guaranteeing the correctness of the output. Both goals should be obtained even if some parties stop participating or some malevolent coalition of the parties start deviating arbitrarily from the agreed protocol. The task is further complicated by the fact that besides their mutual distrust, nor do the parties trust the channels by which they communicate. A general solution to the secure multiparty computation problem is a compiler which given any feasible function describes an efficient protocol which allows the parties to compute the function securely on their local inputs over an open network. Over the past twenty years the secure multiparty computation problem has been the subject of a large body of research, both research into the models of multiparty computation and research aimed at realizing general secure multiparty computation. The main approach to realizing secure multiparty computation has been based on verifiable secret sharing as computation, where each party holds a secret share of each input and during the execution computes secret shares of all intermediate values. This approach allows the parties to keep all inputs and intermediate values secret and to pool the shares of the output values to learn exactly these values. More recently an approach based on joint homomorphic encryption was introduced, allowing for an efficient realization of general multiparty computation secure against an eavesdropper. A joint encryption scheme is an encryption scheme where all parties can encrypt, but where it takes the participation of all parties to open an encryption. The computation then starts by all parties broadcasting encryptions of their inputs and progresses through computing encryptions of the intermediary values using the homomorphic properties of the encryption scheme. The encryptions of the outputs can then be jointly decrypted. In this dissertation we extend this approach by using threshold homomorphic encryption to provide full-fledged security against an active attacker which might control some of the parties and which might take over parties during the course of the protocol execution. As opposed to a joint encryption scheme a threshold encryption scheme only requires that a given number of the parties are operating correctly for decryption to be possible. In this dissertation we show that threshold homomorphic encryption allows to solve the secure general multiparty computation problem more efficiently than previous approaches to the problem. Starting from an open point-to-point network there is a long way to general secure multiparty computation. The dissertation contains contributions at several points along the way. In particular we investigate how to realize secure channels. We also show how threshold signature schemes can be used to efficiently realize a broadcast channel and how threshold cryptography can be used to provide the parties of the network with a source of shared randomness. All three tools are well-known to be extremely powerful resources in a network, and our principle contribution is to provide efficient realizations. The dissertation also contains contributions to the definitional part of the field of multiparty computation. Most significantly we augment the recent model of universally composable security with a formal notion of what it means for a protocol to only realize a given problem under a given restricted input-output behavior of the environment. This e.g. allows us to give the first specification of a universally composable zero-knowledge proof of membership which does not at the same time require the proof system to be a proof of knowledge.
Secure multi-party computation is a conceptual framework in which distrusting parties engage in a protocol to securely perform a computational task. Depending on the precise model of security, different sets of tasks admit secure protocols. We take a complexity-theoretic approach to studying the inherent difficulty of securely realizing tasks in various standard security models. * We give the first alternate characterization of secure realizability in the framework of universally composable (UC) security. This is the first characterization in any model to consider completely arbitrary computational tasks and completely arbitrary communication channels. * The most long-standing class of computational tasks studied are those in which two parties evaluate a deterministic function. For these tasks, we give the first complete, combinatorial characterizations of secure realizability in the passive, standalone, and universally composable security models, against computationally unbounded adversaries. * Say that a task G has ``as much cryptographic complexity'' as another task F if there is a secure protocol for F that uses access to a secure implementation of G. We show that there is an infinite hierarchy of tasks with strictly increasing cryptographic complexities, with respect to computationally unbounded security. We also show that there exist tasks whose cryptographic complexities are incomparable. * In contrast, we show that under a standard cryptographic assumption, there exist only two distinct levels of cryptographic complexity with respect to polynomial-time security. Every task either has a trivial protocol using plain communication channels, or is complete (i.e., given access to a secure implementation of this task, there are secure protocols for all other tasks). This is the first result to derive a characterization of completeness for a class of arbitrary interactive tasks. In light of these characterizations, the only tasks which are securely realizable in the demanding framework of universal composition are those related to secure communication. Indeed, the framework has been used to define the security of encryption schemes, which has allowed for modular design and analysis of protocols. We consider a similar approach for homomorphic encryption schemes. A homomorphic scheme is one in which anyone can obtain an encryption of f(m_1, ..., m_n), given only the encryptions of unknown messages m_1, ..., m_n, for a specific set of functions f. * We give a construction of a homomorphic encryption scheme in which the allowed homomorphic operation is as full-featured as possible --- namely, one can derive a correctly-distributed encryption of f(m) given an encryption of unknown message m, for some functions f --- yet it is computationally infeasible to generate a ciphertext that is related to other ciphertexts in any other way. Our contributions involve developing new appropriate security definitions as well as new constructions. * We show that schemes with such powerful security guarantees can be used to build conceptually simple, efficient, UC-secure protocols for verifiable computations on encrypted data. We show protocols for two tasks related to aggregating encrypted data.
Secure communication has become an essential ingredient of our daily life. Mostly unnoticed, cryptography is protecting our interactions today when we read emails or do banking over the Internet, withdraw cash at an ATM, or chat with friends on our smartphone. Security in such communication is enabled through two components. First, two parties that wish to communicate securely engage in a key exchange protocol in order to establish a shared secret key known only to them. The established key is then used in a follow-up secure channel protocol in order to protect the actual data communicated against eavesdropping or malicious modification on the way. In modern cryptography, security is formalized through abstract mathematical security models which describe the considered class of attacks a cryptographic system is supposed to withstand. Such models enable formal reasoning that no attacker can, in reasonable time, break the security of a system assuming the security of its underlying building blocks or that certain mathematical problems are hard to solve. Given that the assumptions made are valid, security proofs in that sense hence rule out a certain class of attackers with well-defined capabilities. In order for such results to be meaningful for the actually deployed cryptographic systems, it is of utmost importance that security models capture the system's behavior and threats faced in that 'real world' as accurately as possible, yet not be overly demanding in order to still allow for efficient constructions. If a security model fails to capture a realistic attack in practice, such an attack remains viable on a cryptographic system despite a proof of security in that model, at worst voiding the system's overall practical security. In this thesis, we reconsider the established security models for key exchange and secure channel protocols. To this end, we study novel and advanced security aspects that have been introduced in recent designs of some of the most important security protocols deployed, or that escaped a formal treatment so far. We introduce enhanced security models in order to capture these advanced aspects and apply them to analyze the security of major practical key exchange and secure channel protocols, either directly or through comparatively close generic protocol designs. Key exchange protocols have so far always been understood as establishing a single secret key, and then terminating their operation. This changed in recent practical designs, specifically of Google's QUIC ("Quick UDP Internet Connections") protocol and the upcoming version 1.3 of the Transport Layer Security (TLS) protocol, the latter being the de-facto standard for security protocols. Both protocols derive multiple keys in what we formalize in this thesis as a multi-stage key exchange (MSKE) protocol, with the derived keys potentially depending on each other and differing in cryptographic strength. Our MSKE security model allows us to capture such dependencies and differences between all keys established in a single framework. In this thesis, we apply our model to assess the security of both the QUIC and the TLS 1.3 key exchange design. For QUIC, we are able to confirm the intended overall security but at the same time highlight an undesirable dependency between the two keys QUIC derives. For TLS 1.3, we begin by analyzing the main key exchange mode as well as a reduced resumption mode. Our analysis attests that TLS 1.3 achieves strong security for all keys derived without undesired dependencies, in particular confirming several of this new TLS version's design goals. We then also compare the QUIC and TLS 1.3 designs with respect to a novel 'zero round-trip time' key exchange mode establishing an initial key with minimal latency, studying how differences in these designs affect the achievable key exchange security. As this thesis' last contribution in the realm of key exchange, we formalize the notion of key confirmation which ensures one party in a key exchange execution that the other party indeed holds the same key. Despite being frequently mentioned in practical protocol specifications, key confirmation was never comprehensively treated so far. In particular, our formalization exposes an inherent, slight difference in the confirmation guarantees both communication partners can obtain and enables us to analyze the key confirmation properties of TLS 1.3. Secure channels have so far been modeled as protecting a sequence of distinct messages using a single secret key. Our first contribution in the realm of channels originates from the observation that, in practice, secure channel protocols like TLS actually do not allow an application to transmit distinct, or atomic, messages. Instead, they provide applications with a streaming interface to transmit a stream of bits without any inherent demarcation of individual messages. Necessarily, the security guarantees of such an interface differ significantly from those considered in cryptographic models so far. In particular, messages may be fragmented in transport, and the recipient may obtain the sent stream in a different fragmentation, which has in the past led to confusion and practical attacks on major application protocol implementations. In this thesis, we formalize such stream-based channels and introduce corresponding security notions of confidentiality and integrity capturing the inherently increased complexity. We then present a generic construction of a stream-based channel based on authenticated encryption with associated data (AEAD) that achieves the strongest security notions in our model and serves as validation of the similar TLS channel design. We also study the security of such applications whose messages are inherently atomic and which need to safely transport these messages over a streaming, i.e., possibly fragmenting, channel. Formalizing the desired security properties in terms of confidentiality and integrity in such a setting, we investigate and confirm the security of the widely adopted approach to encode the application's messages into the continuous data stream. Finally, we study a novel paradigm employed in the TLS 1.3 channel design, namely to update the keys used to secure a channel during that channel's lifetime in order to strengthen its security. We propose and formalize the notion of multi-key channels deploying such sequences of keys and capture their advanced security properties in a hierarchical framework of confidentiality and integrity notions. We show that our hierarchy of notions naturally connects to the established notions for single-key channels and instantiate its strongest security notions with a generic AEAD-based construction. Being comparatively close to the TLS 1.3 channel protocol, our construction furthermore enables a comparative design discussion.
The notion of secure computation is central to cryptography. Introduced in the seminal works of Yao [FOCS'82, FOCS'86] and Goldreich, Micali and Wigderson [STOC'87], secure multiparty computation allows a group of (mutually) distrustful parties to jointly compute any functionality over their individual private inputs in such a manner that the honest parties obtain the correct outputs and no group of malicious parties learn anything beyond their inputs and the prescribed outputs. General feasibility results for secure computation were given by Yao [FOCS'86] and Goldreich et al. [STOC'87] more than two decades ago. Subsequent to these works, designing secure computation protocols that can tolerate more powerful adversaries and satisfy stronger notions of security has been a very active area of research. In this dissertation, we study two such new frontiers in the area of secure computation.
In the first part of this dissertation, we initiate a study of designing leakage-resilient interactive protocols. Specifically, we consider the scenario where an adversary, in addition to corrupting a subset of parties, can leak (potentially via physical attacks) arbitrary information from the secret state of any honest party. This is in contrast to the standard notion of secure computation where it is assumed that the adversary only has black-box'' access to the honest parties. In particular, we formalize a meaningful definition of leakage-resilient zero knowledge proof systems and provide constructions that satisfy our notion. We also discuss various applications of our results.
The second part of this dissertation concerns with the general question of whether it is possible to securely run cryptographic protocols over an insecure network environment such as the Internet. It is well-known that the standard notion of secure computation is only relevant to the stand-alone setting where a single protocol is being executed in isolation; as such it does not guarantee security when multiple protocol sessions may be executed concurrently under the control of an adversary who is present across all sessions. We consider the open problem of constructing secure password-based authenticated key exchange protocols in such a setting in the plain model (i.e., without assuming any trusted infrastructure or random oracles). We give the first construction of such a protocol based on standard cryptographic assumptions. Our results are in fact much more general, and extend to other functionalities w.r.t. a (necessarily) weakened notion of concurrently secure computation.
The results presented in this dissertation stem from two papers which are joint works with Sanjam Garg and Amit Sahai, and with Vipul Goyal and Rafail Ostrovsky, respectively.
In general, it is difficult to verify security of cryptographic protocols.Indeed, flaws of designs and security proofs of some protocols were found after they had been presented.In deductive verification using formal methods, protocols and security properties are described in formal languages, and correctness of designs and security proofs are deduced by inference rules.While a number of formal frameworks and verification tools have been developed and applied to classical protocols, few formal methods have been applied to security proofs of quantum protocols.The contributions of this thesis consist of the following three results.First, we developed a software tool to verify bisimilarity of configurations (pairs of processes and quantum states) of qCCS, a quantum process calculus presented by Feng et al.Bisimilar configurations behave indistinguishably from the outside.We designed a formal framework for the verifier, which we call nondeterministic qCCS, on the basis of qCCS by Feng et al.While the transition system of qCCS is both nondeterministic and probabilistic, we presented a nondeterministic and non-probabilistic transition system for configurations, extending the definition of them.This allows the verifier to verify bisimilarity efficiently.Next, we designed the verifier to handle security parameters and quantum states symbolically.A purpose is to apply it to quantum cryptographic protocols, where the dimensions of quantum states depend on security parameters.When the verifier checks bisimulation relation of configurations, it uses user-defined equations on the symbolic representations to check the quantum states that the outsider can access are always equal.Besides, the verifier is sound with respect to qCCS, that is, when it runs with two configurations as input and returns true, a symbol representing success of the verification, the configurations are in bisimulation relation in the qCCS's definition.Second, we defined a notion of approximate bisimulation relation on configurations of nondeterministic qCCS, and extended the verifier to check the approximate bisimilarity.Approximately bisimilar configurations reveal quantum states with close trace distance to the outsider after transitions.This property is useful in security proofs of quantum key distribution protocols.We then proved the approximate bisimulation relation is closed under application of evaluation contexts of processes.Third, we formally verified Shor and Preskill's security proof of BB84 quantum key distribution protocol.They first considered another protocol (the EDP-based protocol) and proved the security of BB84 and the EDP-based protocol is equivalent.They next proved the latter is secure.For the first step of their proof, we formalized the two protocols as configurations and formally verified that they are bisimilar.For the second step, we defined a completely secure protocol (EDPideal) and formally verified that the configurations of it and the EDP-based protocol are approximately bisimilar.This is the first work where a security proof of a quantum cryptographic protocol is mechanically verified using a software tool.