Abstract. This paper describes a simple candidate one-way hash func-tion which satisfies a quasi-commutative property that allows it to be used aa an accumulator. This property allows protocols to be developed in which the need for a trusted central authority can be eliminated. Space-efficient distributed protocols are given for document time stamping and for membership testing, and many other applications are possible. 1

We present zero-knowledge proofs and arguments for arithmetic circuits over finite prime fields, namely given a circuit, show in zero-knowledge that inputs can be selected leading to a given output. For a field GF (q), where q is an n-bit prime, a circuit of size O(n), and error probability 2 , our protocols require communication of O(n ) bits. This is the same worst-cast complexity as the trivial (non zero-knowledge) interactive proof where the prover just reveals the input values. If the circuit involves n multiplications, the best previously known methods would in general require communication of \Omega\Gamma n log n) bits. Variations of the

Signature schemes are fundamental cryptographic primitives, useful as a stand-alone application, and as a building block in the design of secure protocols and other cryptographic objects. In this thesis, we study both the uses that signature schemes find in protocols, and the design of signature schemes suitable for a broad range of applications. An important

) Martin Hirt 1 , Ueli Maurer 1 , and Bartosz Przydatek 2?? 1 ETH Zurich, Switzerland, fhirt,maurerg@inf.ethz.ch 2 Carnegie Mellon University, USA, bartosz@cs.cmu.edu Asiacrypt 2000 Abstract. Since the introduction of secure multi-party computation, all proposed protocols that provide security against cheating players suer from very high communication complexities. The most ecient unconditionally secure protocols among n players, tolerating cheating by up to t < n=3 of them, require communicating O(n 6 ) eld elements for each multiplication of two elements, even if only one player cheats. In this paper, we propose a perfectly secure multi-party protocol which requires communicating O(n 3 ) eld elements per multiplication. In this protocol, the number of invocations of the broadcast primitive is independent of the size of the circuit to be computed. The proposed techniques are generic and apply to other protocols for robust distributed computations. Furthe...

Abstract. We present a very efficient multi-party computation protocol unconditionally secure against an active adversary. The security is maximal, i.e., active corruption of up to t < n/3 of the n players is tolerated. The communication complexity for securely evaluating a circuit with m multiplication gates over a finite field is O(mn 2) field elements, including the communication required for simulating broadcast, but excluding some overhead costs (independent of m) for sharing the inputs and reconstructing the outputs. This corresponds to the complexity of the best known protocols for the passive model, where the corrupted players are guaranteed not to deviate from the protocol. The complexity of our protocol may well be optimal. The constant overhead factor for robustness is small and the protocol is practical. 1

Abstract. Sander, Young and Yung recently exhibited a protocol for computing on encrypted inputs, for functions computable in NC 1. In their variant of secure function evaluation, Bob (the “CryptoComputer”) accepts homomorphically-encrypted inputs (x) from client Alice, and then returns a string from which Alice can extract f(x, y) (where y is Bob’s input, or e.g. the function f itself). Alice must not learn more about y than what f(x, y) reveals by itself. We extend their result to encompass NLOGSPACE (nondeterministic log-space functions). In the domain of multiparty computations, constant-round protocols have been known for years [BB89,FKN95]. This paper introduces novel parallelization techniques that, coupled with the [SYY99] methods, reduce the constant to 1 with preprocessing. This resolves the conjecture that NLOGSPACE subcomputations (including log-slices of circuit computation) can be evaluated with latency 1 (as opposed to just O(1)). 1

Abstract. The notion of non-interactive zero-knowledge (NIZK) is of fundamental importance in cryptography. Despite the vast attention the concept of NIZK has attracted since its introduction, one question has remained very resistant: Is it possible to construct NIZK schemes for any NPlanguage with statistical or even perfect ZK? Groth, Ostrovsky and Sahai recently answered this question in the affirmative. However, in order to achieve adaptive soundness, i.e., soundness against dishonest provers who may choose the target statement depending on the common reference string (CRS), their schemes require some restriction to be put upon the statements to be proven, e.g. an a-priori bound on its size. In this work, we first present a very simple and efficient adaptively-sound perfect NIZK argument system for any NP-language. Besides being the first adaptively-sound statistical NIZK argument for all NP that does not pose any restriction on the statements to be proven, it enjoys a number of additional desirable properties: it allows to re-use the CRS, it can handle arithmetic circuits, and the CRS can be set-up very efficiently without the need for an honest party. We then show an application of our techniques in constructing efficient NIZK schemes for proving arithmetic relations among committed secrets, whereas previous methods required expensive generic NP-reductions. The security of the proposed schemes is based on a strong non-standard assumption, an extended version of the so-called Knowledge-of-Exponent Assumption (KEA) over bilinear groups. We give some justification for using such an assumption by showing that the commonly-used approach for proving NIZK arguments sound does not allow for adaptively-sound statistical NIZK arguments (unless NP ⊂ P/poly). Furthermore, we show that the assumption used in our construction holds with respect to generic adversaries that do not exploit the specific representation of the group elements. We also discuss how to avoid the non-standard assumption in a pre-processing model.

This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5, and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E. Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the notion of provable security and its usage for the design of secure protocols. Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science. Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography and network security course at UCSD.

Abstract. Secure multi-party computation (MPC) is an active research area, and a wide range of literature can be found nowadays suggesting improvements and generalizations of existing protocols in various directions. However, all current techniques for secure MPC apply to functions that are represented by (boolean or arithmetic) circuits over finite fields. We are motivated by two limitations of these techniques: – Generality. Existing protocols do not apply to computation over more general algebraic structures (except via a brute-force simulation of computation in these structures). – Efficiency. The best known constant-round protocols do not efficiently scale even to the case of large finite fields. Our contribution goes in these two directions. First, we propose a basis for unconditionally secure MPC over an arbitrary finite ring, an algebraic object with a much less nice structure than a field, and obtain efficient MPC protocols requiring only a black-box access to the ring operations and to random ring elements. Second, we extend these results to the constant-round setting, and suggest efficiency improvements that are relevant also for the important special case of fields. We demonstrate the usefulness of the above results by presenting a novel application of MPC over (non-field) rings to the round-efficient secure computation of the maximum function. 1

Abstract. Essentially all work studying the round complexity of secure computation assumes broadcast as an atomic primitive. Protocols constructed under this assumption tend to have very poor round complexity when compiled for a point-to-point network due to the high overhead of emulating each invocation of broadcast. This problem is compounded when broadcast is used in more than one round of the original protocol due to the complexity of handling sequential composition (when using round-efficient emulation of broadcast). We argue that if the goal is to optimize round complexity in point-topoint networks, then it is preferable to design protocols — assuming a broadcast channel — minimizing the number of rounds in which broadcast is used rather than minimizing the total number of rounds. With this in mind, we present protocols for secure computation in a number of settings that use only a single round of broadcast. In all cases, we achieve optimal security threshold for adaptive adversaries, and obtain protocols whose round complexity (in a point-to-point network) improves on prior work. 1