Which computational complexity assumptions are inherent to cryptography? We present a broad framework to pose and investigate this question. We first aim to understand the “cryptographic complexity ” of various tasks, independent of any computational assumptions. In our framework the cryptographic tasks are modeled as multi-party computation functionalities. We consider a universally composable secure protocol for one task given access to another task as the most natural complexity reduction between the two tasks. Some of these cryptographic complexity reductions are unconditional, others are unconditionally non-existent, but the vast majority appear to depend on computational assumptions; it is this relationship with computational assumptions that we study. In our detailed investigation of large classes of 2-party functionalities, we find that every reduction we are able to classify turns out to be unconditionally true or false, or else equivalent to the existence of one-way functions (OWF) or of semi-honest (equivalently, standalone-secure) oblivious transfer protocols (sh-OT). This leads us to conjecture that there are only a small finite number of distinct (i.e., black-box separable) computational assump-tions that are inherent among the infinite number of different cryptographic reductions in our framework. On moving to 3-party functionalities, there does exist a natural computational assumption, distinct from the OWF and sh-OT assumptions (namely, the existence of a key agreement protocol) that manifests itself as a cryptographic
Which computational complexity assumptions are inherent to cryptography? We present a broad framework to pose and investigate this question. We first aim to understand the “cryptographic complexity ” of various tasks, independent of any computational assumptions. In our framework the cryptographic tasks are modeled as multi-party computation functionalities. We consider a universally composable secure protocol for one task given access to another task as the most natural complexity reduction between the two tasks. Some of these cryptographic complexity reductions are unconditional, others are unconditionally non-existent, but the vast majority appear to depend on computational assumptions; it is this relationship with computational assumptions that we study. In our detailed investigation of large classes of 2-party functionalities, we find that every reduction we are able to classify turns out to be unconditionally true or false, or else equivalent to the existence of one-way functions (OWF) or of semi-honest (equivalently, standalone-secure) oblivious transfer protocols (sh-OT). This leads us to conjecture that there are only a small finite number of distinct (i.e., black-box separable) computational assump-tions that are inherent among the infinite number of different cryptographic reductions in our framework. On moving to 3-party functionalities, there does exist a natural computational assumption, distinct from the OWF and sh-OT assumptions (namely, the existence of a key agreement protocol) that manifests itself as a cryptographic
In cryptographic protocols, honest parties would prefer that their security is assured even in presence of adversarial parties who have unbounded computational power. Information theoretic secure realization of cryptographic primitives provides such guarantees; but for most tasks such strong security guarantees cannot be provided for any reasonable notion of security. The standard technique used in cryptography is to assume the existence of some puzzle, whose hard instances are easy to generate but no efficient algorithm can solve them. Such assumptions, which define intractable problems for efficient algorithms, are called computational intractability assumptions.
In this work, we motivate a study of computational intractability assumptions which is goal driven and lends support to the fundamental nature of some of the traditional assumptions beyond being historical accidents. Secure multi-party computation deals with the study of constructing secure protocols for general cryptographic tasks which conform to various notions of security. Inspired by complexity theory, we use the notion of reduction to further our understanding of computational intractability assumptions. Our framework explores the hardness of natural assumptions of the form: “Task F can be securely computed given an ideally secure facility for computing G ”.
Formally, we characterize the minimal computational intractability assumption which is sufficient to securely realize a functionality using trusted copies of some other functionality. A functionality F reduces to G , similar to the complexity-theoretic notion of reduction, if F can be securely realized when parties can access trusted copies of G . Thus, if F does not reduce to G information-theoretically, we are interested in characterizing the minimal assumption, which is sufficient, to securely realize the reduction of F to G .
In our framework, we explore the relative strengths of these minimal computational assumptions. The fundamental problem, then, is to establish relations, like implication, equivalence and separation, among the assumptions which correspond to various reductions. In this work, we further our understanding of the minimal computational assumptions corresponding to such reductions by showing the following results:
1. We identify a spectrum for the set of assumptions that correspond to reduction between a pair of tasks under different security notions. These reductions are implied by the computational intractability assumption: “semi-honest secure protocol for oblivious transfer exists” (sh-OT assumption). Additionally, the information theoretically impossible reductions considered in this thesis imply the assumption: “one-way functions exist” (OWF assumption). We conjecture that OWF assumption is necessary for these reductions. In fact, the reductions considered in this thesis are either information theoretically true, false, equivalent to OWF assumption or equivalent to sh-OT assumption [MPR10a].
We expand our study to encompass reductions involving randomized functionalities and study the consequences of providing parties with a trusted source of common unbiased coins. For a strong notion of security they turn out to be useless and unless sh-OT assumption holds these randomness sources are not useful [MOPR11, MP11].
2. We [MPS10] seek to lend additional support to the widely believed premise that “non-trivial cryptography entails the existence of one-way functions”. We show that (constant round) weak coin-tossing protocols, imply OWF assumption. For the general problem, i.e. secure weak coin-tossing protocols with polynomial round complexity, we show a slightly weaker implication NP n BPP.
3. Finally, we indicate the possibility of a wide range of intractability assumptions in our spectrum, intermediate to OWF assumption and sh-OT assumption. We show that a class of natural assumptions corresponding to reductions that are relativistically separated from OWF assumption. Similar techniques have been used to show their separation from the assumption that “public-key encryption exists” (PKE assumption) [MMP11]; but these assumptions are not known to imply sh-OT assumption. We conjecture that, in fact, sh-OT assumption is relativistically separated from these assumptions.
We use security in the Universal Composition framework as a means to study the “cryptographic complexity ” of 2-party secure computation tasks (functionalities). We say that a functionality F reduces to another functionality G if there is a UC-secure protocol for F using ideal access to G. This reduction is a natural and fine-grained way to compare the relative complexities of cryptographic tasks. There are two natural “extremes ” of complexity under the reduction: the trivial functionalities, which can be reduced to any other functionality; and the complete functionalities, to which any other functionality can be reduced. In this work we show that under a natural computational assumption (the existence of a protocol for oblivious transfer secure against semi-honest adversaries), there is a zero-one law for the cryptographic complexity of 2-party deterministic functionalities. Namely, every such functionality is either trivial or complete. No other qualitative distinctions exist among functionalities, under this computational assumption. While nearly all previous work classifying multi-party computation functionalities has been restricted to the case of secure function evaluation, our results are the first to consider completeness of arbitrary reactive functionalities, which receive input and give output repeatedly throughout several rounds of interaction. One important technical contribution in this work is to initiate the comprehensive study of the cryptographic properties of reactive functionalities. We model these functionalities as finite automata and develop an automata-theoretic methodology for classifying and studying their cryptographic properties. Consequently, we completely characterize the reactive behaviors that lead to cryptographic non-triviality.
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.
Abstract. Understanding the minimal assumptions required for carrying out cryptographic tasks is one of the fundamental goals of theoretical cryptography. A rich body of work has been dedicated to understanding the complexity of cryptographic tasks in the context of (semi-honest) secure two-party computation. Much of this work has focused on the characterization of trivial and complete functionalities (resp., functionalities that can be securely implemented unconditionally, and functionalities that can be used to securely compute all functionalities). All previous works define reductions via an ideal implementation of the functionality; i.e., f reduces to g if one can implement f using an ideal box (or oracle) that computes the function g and returns the output to both parties. Such a reduction models the computation of f as an atomic operation. However, in the real-world, protocols proceed in rounds, and the output is not learned by the parties simultaneously. In this paper we show that this distinction is significant. Specifically, we show that there exist symmetric functionalities (where both parties receive the same outcome), that are neither trivial nor complete under “ideal-box reductions”, and yet the existence of a constant-round protocol for securely computing such a functionality implies infinitely-often oblivious transfer (meaning that it is secure for infinitely-many n’s). In light of the above, we propose an alternative definitional infrastructure for studying the triviality and completeness of functionalities. 1
It is well-known that most cryptographic tasks do not have universally composable (UC) secure protocols, if no trusted setup is available in the framework. On the other hand, if a task like fair coin-tossing is available as a trusted setup, then all cryptographic tasks have UC-secure protocols. What other trusted setups allow UC-secure protocols for all tasks? More generally, given a particular setup, what tasks have UC-secure protocols?We show that, surprisingly, every trusted setup is either useless (equivalent to having no trusted setup) or all-powerful (allows UC-secure protocols for all tasks). There are no “intermediate” trusted setups in the UC framework. We prove this zero-one law under a natural intractability assumption, and consider the class of deterministic, finite, 2-party functionalities as candidate trusted setups.One important technical contribution in this work is to initiate the comprehensive study of the cryptographic properties of reactive functionalities. We model these functionalities as finite automata and develop an automata-theoretic methodology for classifying and studying their cryptographic properties. Consequently, we completely characterize the reactive behaviors that lead to cryptographic non-triviality. Another contribution of independent interest is to optimize the hardness assumption used by Canetti et al. (STOC 2002) in showing that the common random string functionality is complete (a result independently obtained by Damgård et al. (TCC 2010)).
Two main goals of modern cryptography are to identify the minimal assumptions necessary to construct secure cryptographic primitives as well as to construct secure protocols in strong and realistic adversarial models. In this thesis, we address both of these fundamental questions. In the first part of this thesis, we present results on the black-box complexity of two basic cryptographic primitives: non-malleable encryption and optimally-fair coin tossing. Black-box reductions are reductions in which both the underlying primitive as well as the adversary are accessed only in an input-output (or black-box) manner. Most known cryptographic reductions are black-box. Moreover, black-box reductions are typically more efficient than non-black-box reductions. Thus, the black-box complexity of cryptographic primitives is a meaningful and important area of study which allows us to gain insight into the primitive. We study the black box complexity of non-malleable encryption and optimally-fair coin tossing, showing a positive result for the former and a negative one for the latter. Non-malleable encryption is a strong security notion for public-key encryption, guaranteeing that it is impossible to "maul" a ciphertext of a message m into a ciphertext of a related message. This security guarantee is essential for many applications such as auctions. We show how to transform, in a black-box manner, any public-key encryption scheme satisfying a weak form of security, semantic security, to a scheme satisfying non-malleability. Coin tossing is perhaps the most basic cryptographic primitive, allowing two distrustful parties to flip a coin whose outcome is 0 or 1 with probability 1/2. A fair coin tossing protocol is one in which the outputted bit is unbiased, even in the case where one of the parties may abort early. However, in the setting where parties may abort early, there is always a strategy for one of the parties to impose bias of Omega(1/r) in an r-round protocol. Thus, achieving bias of O(1/r) in r rounds is optimal, and it was recently shown that optimally-fair coin tossing can be achieved via a black-box reduction to oblivious transfer. We show that it cannot be achieved via a black-box reduction to one-way function, unless the number of rounds is at least Omega(n/log n), where n is the input/output length of the one-way function. In the second part of this thesis, we present protocols for multiparty computation (MPC) in the Universal Composability (UC) model that are secure against malicious, adaptive adversaries. In the standard model, security is only guaranteed in a stand-alone setting; however, nothing is guaranteed when multiple protocols are arbitrarily composed. In contrast, the UC model, introduced by (Canetti, 2000), considers the execution of an unbounded number of concurrent protocols, in an arbitrary, and adversarially controlled network environment. Another drawback of the standard model is that the adversary must decide which parties to corrupt before the execution of the protocol commences. A more realistic model allows the adversary to adaptively choose which parties to corrupt based on its evolving view during the protocol. In our work we consider the the adaptive UC model, which combines these two security requirements by allowing both arbitrary composition of protocols and adaptive corruption of parties. In our first result, we introduce an improved, efficient construction of non-committing encryption (NCE) with optimal round complexity, from a weaker primitive we introduce called trapdoor-simulatable public key encryption (PKE). NCE is a basic primitive necessary to construct protocols secure under adaptive corruptions and in particular, is used to construct oblivious transfer (OT) protocols secure against semi-honest, adaptive adversaries. Additionally, we show how to realize trapdoor-simulatable PKE from hardness of factoring Blum integers, thus achieving the first construction of NCE from hardness of factoring. In our second result, we present a compiler for transforming an OT protocol secure against a semi-honest, adaptive adversary into one that is secure against a malicious, adaptive adversary. Our compiler achieves security in the UC model, assuming access to an ideal commitment functionality, and improves over previous work achieving the same security guarantee in two ways: it uses black-box access to the underlying protocol and achieves a constant multiplicative overhead in the round complexity. Combining our two results with the work of (Ishai et al., 2008), we obtain the first black-box construction of UC and adaptively secure MPC from trapdoor-simulatable PKE and the ideal commitment functionality.