This document presents two novel techniques for Multi-Party Computation based on secret sharing where the objects of computation are square matrices over a finite field. Matrix Triples can be consumed during the evaluation phase to obtain the product of two matrices, and they have quadratic space complexity, which makes them more efficient for this purpose than ordinary multiplication triples. Matrix Power Tuples are costly to produce during the preprocessing phase but allow the parties to take a matrix to a power with just a single round of communication during the evaluation phase. The latter technique also has a considerably simpler and cheaper variant that can be applied in cases where multiplication is commutative. We provide a bottom-up explanation of the ideas behind Matrix Power Tuples, as well as a formal specification of both techniques in the form of protocols and ideal functionalities. We use the latter to provide a security proof in the iUC model. We also provide an introduction into the most widely used techniques for Multi-Party Computation in the modern literature.
This document presents two novel techniques for Multi-Party Computation based on secret sharing where the objects of computation are square matrices over a finite field. Matrix Triples can be consumed during the evaluation phase to obtain the product of two matrices, and they have quadratic space complexity, which makes them more efficient for this purpose than ordinary multiplication triples. Matrix Power Tuples are costly to produce during the preprocessing phase but allow the parties to take a matrix to a power with just a single round of communication during the evaluation phase. The latter technique also has a considerably simpler and cheaper variant that can be applied in cases where multiplication is commutative. We provide a bottom-up explanation of the ideas behind Matrix Power Tuples, as well as a formal specification of both techniques in the form of protocols and ideal functionalities. We use the latter to provide a security proof in the iUC model. We also provide an introduction into the most widely used techniques for Multi-Party Computation in the modern literature.
Considering constant-round protocols for generating random shared values, for secure multiplication and for addition of shared values, etc are available and can be met by known techniques in all standard models of communication. Protocols are presented allowing the players to securely solve standard computational problems in linear algebra. In particular, securely, efficiently and in constant-round compute determinant of matrices product, rank of a matrix, and determine similarity between matrices. If the basic protocols (addition and multiplication, etc) are unconditionally secure, then so are our protocols. Furthermore our protocols offer more efficient solutions than previous techniques for secure linear algebra.
In this dissertation, we study the design of secure and efficient protocols for distributed linear algebra problems. All existing solutions to the problem are either inefficient in terms of round and communication complexity and/or provide security against very weak adversaries.
We design protocols for two different adversarial settings. (i) Computational: where the adversary's computational resources are bounded, but the adversary is allowed to corrupt all but one of the participants. (ii) Information theoretic: where the adversary is unconditionally powerful, but is only allowed to corrupt a third of the participants. The security in both cases is defined in the real world/ideal world simulation paradigm, where we assume that the adversary can behave arbitrarily malicious.
Our main constructions are O(s) round protocols with O(sn2+1/s) communication for testing singularity of a shared n x n matrix, for any positive integer s, in both adversarial settings. We then use and extend these constructions to design secure protocols for computing the rank of a shared matrix and solving shared linear systems with similar efficiency. Our protocols are almost optimal in terms of round and communication complexity, as they roughly match the efficiency of the best possible protocols for the case where security is not a concern.
We use a wide range of techniques from computer algebra to reduce the problem of securely deciding singularity to the problem of secure matrix multiplication. We then design new and efficient protocols for secure matrix multiplication in both adversarial environments. In the computational setting, and when there are only two parties, we apply cut-and-choose techniques to encrypted algebraic encodings of the inputs, outputs, and intermediate values in order to prevent malicious behavior. We achieve even better amortized efficiency when running multiple instances of the protocol, via use of share-packing techniques. When the number of parties is more that two, our construction relies on efficient instantiations of the inner and outer protocols in a recently introduced compiler by Ishai et al. [35]. In the information theoretic setting, our protocol takes advantage of a homomorphic commitment scheme to achieve the improved efficiency.
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
We construct the first efficient, unconditionally secure MPC protocol that only requires black-box access to a non-commutative ring R. Previous results in the same setting were efficient only either for a constant number of corruptions or when computing branching programs and formulas. Our techniques are based on a generalization of Shamir’s secret sharing to non-commutative rings, which we derive from the work on Reed Solomon codes by Quintin, Barbier and Chabot (IEEE Transactions on Information Theory, 2013). When the center of the ring contains a set A={α0,…,αn} such that ∀i=j,αi−αj∈R∗ , the resulting secret sharing scheme is strongly multiplicative and we can generalize existing constructions over finite fields without much trouble. Most of our work is devoted to the case where the elements of A do not commute with all of R, but they just commute with each other. For such rings, the secret sharing scheme cannot be linear “on both sides” and furthermore it is not multiplicative. Nevertheless, we are still able to build MPC protocols with a concretely efficient online phase and black-box access to R. As an example we consider the ring Mm×m(Z/2kZ) , for which when m>log(n+1) , we obtain protocols that require around ⌈log(n+1)⌉/2 less communication and 2⌈log(n+1)⌉ less computation than the state of the art protocol based on Circuit Amortization Friendly Encodings (Dalskov, Lee and Soria-Vazquez, ASIACRYPT 2020). In this setting with a “less commutative” A, our black-box preprocessing phase has a less practical complexity of poly(n) . We fix this by additionally providing specialized, concretely efficient preprocessing protocols for Mm×m(Z/2kZ) that exploit the structure of the matrix ring.
Abstract. As an extension of multi-party computation (MPC), we propose the concept of secure parallel multi-party computation which is to securely compute multi-functions against an adversary with multistructures. Precisely, there are m functions f1,..., fm and m adversary structures A1,..., Am, where fi is required to be securely computed against an Ai-adversary. We give a general construction to build a parallel multi-party computation protocol from any linear multi-secret sharing scheme (LMSSS), provided that the access structures of the LMSSS allow MPC at all. When computing complicated functions, our protocol has more advantage in communication complexity than the “direct sum” method which actually executes a MPC protocol for each function. The paper also provides an efficient and generic construction to obtain from any LMSSS a multiplicative LMSSS for the same multi-access structure. 1
parties to securely compute an agreed function f over some finite field in the presence of a computationally unbounded adversary, who can maliciously corrupt any t out of the n parties. Most of the known efficient MPC protocols are designed in the offline- online framework introduced in a seminal work by Beaver in CRYPTO 1991. In this framework, the parties generate shared random and private multiplication-triples during the offline phase, which are used later in the online phase for securely evaluating the multiplication gates in the circuit representing f . The efficiency of the MPC protocols in this framework then relies on efficient ways of implementing the offline phase. In this paper, we propose a new and simple framework for generating shared and private random multiplication triples with unconditional security. The existing protocols approach this problem by first producing shared pairs of private and random values, followed by securely computing the shared product of each pair of values. The latter task involves a multiplication protocol for shared values that are typically communication intensive. Our framework takes a completely different approach and shuns the use of multiplication protocol. Namely, we ask the parties to verifiably share random multiplication triples and then securely extract shared random multiplication triples unknown to the adversary, from the shared triples. Realizing our framework in the asynchronous and hybrid network setting,1 we present the first ever MPC protocols with a linear (in the number of parties) communication overhead per multiplication gate in the circuit representing f . These are significant improvements over the best known existing MPC protocols in the asynchronous and hybrid network setting with communication complexity O(n2) and O(n3), respectively. Our framework when applied to the synchronous setting results in round-efficient MPC protocols.