We propose and realize a definition of security for password-based key exchange within the framework of universal composability (UC), thus providing security guarantees under arbitrary composition with other protocols. In addition, our definition captures some aspects of the problem that were not adequately addressed by most prior notions. For instance, our definition does not assume any underlying probability distribution on passwords, nor does it assume independence between passwords chosen by different parties. We also formulate a definition of password-based secure channels, and show how to realize such channels given any password-based key exchange protocol. The password-based key exchange protocol shown here is in the common reference string model and relies on standard number-theoretic assumptions. The components of our protocol can be instantiated to give a relatively efficient solution which is conceivably usable in practice. We also show that it is impossible to satisfy our definition in the "plain" model (e.g., without a common reference string).

We introduce the notion of {\em resource-fair} protocols. Informally, this property states that if one party learns the output of the protocol, then so can all other parties, as long as they expend roughly the same amount of resources. As opposed to similar previously proposed definitions, our definition follows the standard simulation paradigm and enjoys strong composability properties. In particular, our definition is similar to the security definition in the universal composability (UC) framework, but works in a model that allows any party to request additional resources from the environment to deal with dishonest parties that may prematurely abort. In this model we specify the ideally fair functionality as allowing parties to ``invest resources'' in return for outputs, but in such an event offering all other parties a fair deal. (The formulation of fair dealings is kept independent of any particular functionality, by defining it using a ``wrapper.'') Thus, by relaxing the notion of fairness, we avoid a well-known impossibility result for fair multi-party computation with corrupted majority; in particular, our definition admits constructions that tolerate arbitrary number of corruptions. We also show that, as in the UC framework, protocols in our framework may be arbitrarily and concurrently composed. Turning to constructions, we define a ``commit-prove-fair-open'' functionality and design an efficient resource-fair protocol that securely realizes it, using a new variant of a cryptographic primitive known as ``time-lines.'' With (the fairly wrapped version of) this functionality we show that some of the existing secure multi-party computation protocols can be easily transformed into resource-fair protocols while preserving their security.

We study the problem of constructing secure multi-party computation (MPC) protocols that are {\em completely fair} --- meaning that either all the parties learn the output of the function, or nobody does --- even when a majority of the parties are corrupted. We first propose a framework for fair multi-party computation, within which we formulate a definition of secure and fair protocols. The definition follows the standard simulation paradigm, but is modified to allow the protocol to depend on the runing time of the adversary. In this way, we avoid a well-known impossibility result for fair MPC with corrupted majority; in particular, our definition admits constructions that tolerate up to $(n-1)$ corruptions, where $n$ is the total number of parties. Next, we define a ``commit-prove-fair-open'' functionality and construct an efficient protocol that realizes it, using a new variant of a cryptographic primitive known as ``time-lines.'' With this functionality, we show that some of the existing secure MPC protocols can be easily transformed into fair protocols while preserving their security. Putting these results together, we construct efficient, secure MPC protocols that are completely fair even in the presence of corrupted majorities. Furthermore, these protocols remain secure when arbitrarily composed with any protocols, which means, in particular, that they are concurrently-composable and non-malleable. Finally, as an example of our results, we show a very efficient protocol that fairly and securely solves the socialist millionaires' problem.

Committed Oblivious Transfer (COT) is a useful cryptographic primitive that combines the functionalities of bit commitment and oblivious transfer. In this paper, we introduce an extended version of COT (ECOT) which additionally allows proofs of relations among committed bits, and we construct an efficient protocol that securely realizes an ECOT functionality in the universal-composability (UC) framework in the common reference string (CRS) model. Our construction is more efficient than previous (non-UC) constructions of COT, involving only a constant number of exponentiations and communication rounds. Using the ECOT functionality as a building block, we construct efficient UC protocols for general two-party and multi-party functionalities (in the CRS model), each gate requiring a constant number of ECOT's.

Recently there has been an interest in zero-knowledge protocols with stronger properties, such as concurrency, unbounded simulation soundness, non-malleability, and universal composability. In this paper, we show a novel technique to convert a large class of existing honest-verifier zero-knowledge protocols into ones with these stronger properties in the common reference string model. More precisely, our technique utilizes a signature scheme existentially unforgeable against adaptive chosen-message attacks, and transforms any $\Sigma$-protocol (which is honest-verifier zero-knowledge) into an unbounded simulation sound concurrent zero-knowledge protocol. We also introduce $\Omega$-protocols, a variant of $\Sigma$-protocols for which our technique further achieves the properties of non-malleability and/or universal composability. In addition to its conceptual simplicity, a main advantage of this new technique over previous ones is that it avoids the Cook-Levin theorem, which tends to be rather inefficient. Indeed, our technique allows for very efficient instantiation based on the security of some efficient signature schemes and standard number-theoretic assumptions. For instance, one instantiation of our technique yields a universally composable zero-knowledge protocol under the Strong RSA assumption, incurring an overhead of a small constant number of exponentiations, plus the generation of two signatures.

We study the recently introduced notion of a simulation-sound trapdoor commitment (SSTC) scheme. In this paper, we present a new, simpler definition for an SSTC scheme that admits more efficient constructions and can be used in a larger set of applications. Specifically, we show how to construct SSTC schemes from any one-way functions, and how to construct very efficient SSTC schemes based on specific number-theoretic assumptions. We also show how to construct simulation-sound, non-malleable, and universally-composable zero-knowledge protocols using SSTC schemes, yielding, for instance, the most efficient universally-composable zero-knowledge protocols known. Finally, we explore the relation between SSTC schemes and non-malleable commitment schemes by presenting a sequence of implication and separation results, which in particular imply that SSTC schemes are non-malleable.

In the most strict formal definition of security for password-authenticated key exchange, an adversary can test at most one password per impersonation attempt. We propose a slightly relaxed definition which restricts an adversary to testing at most a constant number of passwords per impersonation attempt. This definition seems useful, since there is currently a popular password-authenticated key exchange protocol called SRP that seems resistant to off-line dictionary attack, yet does allow an adversary to test two passwords per impersonation attempt. In this paper we prove (in the random oracle model) that a certain instantiation of the SPEKE protocol that uses hashed passwords instead of non-hashed passwords is a secure password-authenticated key exchange protocol (using our relaxed definition) based on a new assumption, the Decision Inverted-Additive Diffie-Hellman assumption. Since this is a new security assumption, we investigate its security and relation to other assumptions; specifically we prove a lower bound for breaking this new assumption in the generic model, and we show that the computational version of this new assumption is equivalent to the Computational Diffie-Hellman assumption.

When designing password-authenticated key exchange protocols (as opposed to key exchange protocols authenticated using cryptographically secure keys), one must not allow any information to be leaked that would allow verification of the password (a weak shared key), since an attacker who obtains this information may be able to run an off-line dictionary attack to determine the correct password. Of course, it may be extremely difficult to hide all password information, especially if the attacker may pose as one of the parties in the key exchange. Nevertheless, we present a new protocol called PAK which is the first Diffie-Hellman-based password-authenticated key exchange protocol to provide a formal proof of security (in the random oracle model) against both passive and active adversaries. In addition to the PAK protocol that provides mutual explicit authentication, we also show a more efficient protocol called PPK that is provably secure in the implicit-authentication model. We then extend PAK to a protocol called PAK-X, in which one side (the client) stores a plaintext version of the password, while the other side (the server) only stores a verifier for the password. We formally prove security of PAK-X, even when the server is compromised. Our formal model for password-authenticated key exchange is new, and may be of independent interest.

We initiate the investigation of the class of relations that admit extremely efficient perfect zero knowledge proofs of knowledge: constant number of rounds, communication linear in the length of the statement and the witness, and negligible knowledge error. In its most general incarnation, our result says that for relations that have a particular three-move honest-verifier zero-knowledge (HVZK) proof of knowledge, and which admit a particular three-move HVZK proof of knowledge for an associated commitment relation, perfect zero knowledge (against a general verifier) can be achieved essentially for free, even when proving statements on several instances combined under under monotone function composition. In addition, perfect zero-knowledge is achieved with an optimal 4-moves. Instantiations of our main protocol lead to efficient perfect ZK proofs of knowledge of discrete logarithms and RSA-roots, or more generally, $q$-one-way group homomorphisms. None of our results rely on intractability assumptions.

We consider a "mobile adversary" which may corrupt all participants throughout the lifetime of the system in a non-monotonic fashion (i.e. recoveries are possible) but the adversary is unable to corrupt too many participants during any short time period. Schemes resiliant to such adverasry are called proactive. We present a proactive RSA system in which a threshold of servers applies the RSA signature (or decryption) function in a distributed manner. Employing new combinatorial and elementary number theoretic techniques, our protocol enables the dynamic updating of the servers (which hold the RSA key distributively); it is secure even when a linear number of the servers are corrupted during any time period; it efficiently "self-maintains" the security of the function and its messages (ciphertexts or signatures); and it enables continuous availability, namely, correct function application using the shared key is possible at any time.