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InfoCommit: Information-Theoretic Polynomial Commitment and Verification

Saeid Sahraei, Salman Avestimehr

Published Jan 1, 2020• 2 citations
OpenAlex

Abstract

We introduce InfoCommit, a protocol for polynomial commitment and verification. InfoCommit consists of two phases. An initial commitment phase and an evaluation phase. During the commitment phase, the verifier and the prover engage in a private two-party computation algorithm so that the verifier extracts a private verification key. In the evaluation phase, the verifier is interested in learning the evaluations of the polynomial at several input points. InfoCommit has four main features. Firstly, the verifier is able to detect, with high probability, if the prover has responded with evaluations of the same polynomial that he has initially committed to. Secondly, InfoCommit provides rigorous privacy guarantees for the prover: upon observing the initial commitment and the response provided by the prover to $m$ evaluation requests, the verifier only learns $O(m^2)$ symbols about the coefficients of the polynomial. Thirdly, the verifiability guarantee is unconditional and without the need for a trusted party, while bounded storage is the only assumption underlying the privacy of the algorithm. In particular, both properties hold regardless of the computation power of the two parties. Lastly, InfoCommit is doubly-efficient in the sense that in the evaluation phase, the verifier runs in $O(\sqrt{d})$ and the prover runs in $O(d)$, where $d-1$ is the degree of the polynomial.

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We introduce InfoCommit, a protocol for polynomial commitment and verification. InfoCommit consists of two phases. An initial commitment phase and an evaluation phase. During the commitment phase, the verifier and the prover engage in a private two-party computation algorithm so that the verifier extracts a private verification key. In the evaluation phase, the verifier is interested in learning the evaluations of the polynomial at several input points. InfoCommit has four main features. Firstly, the verifier is able to detect, with high probability, if the prover has responded with evaluations of the same polynomial that he has initially committed to. Secondly, InfoCommit provides rigorous privacy guarantees for the prover: upon observing the initial commitment and the response provided by the prover to mmm evaluation requests, the verifier only learns O(m2)O(m^2)O(m2) symbols about the coefficients of the polynomial. Thirdly, the verifiability guarantee is unconditional and without the need for a trusted party, while bounded storage is the only assumption underlying the privacy of the algorithm. In particular, both properties hold regardless of the computation power of the two parties. Lastly, InfoCommit is doubly-efficient in the sense that in the evaluation phase, the verifier runs in O(d)O(\sqrt{d})O(d​) and the prover runs in O(d)O(d)O(d), where d−1d-1d−1 is the degree of the polynomial.
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
A major open problem in information-theoretic cryptography is to obtain a super-polynomial lower bound for the communication complexity of basic cryptographic tasks. This question is wide open even for very powerful non-interactive primitives such as private information retrieval (or locally-decodable codes), general secret sharing schemes, conditional disclosure of secrets, and fully-decomposable randomized encoding (or garbling schemes). In fact, for all these primitives we do not even have super-linear lower bounds. Furthermore, it is unknown how to relate these questions to each other or to other complexity-theoretic questions.In this note, we relate all these questions to the classical topic of query/space trade-offs, lifted to the setting of interactive proof systems. Specifically, we consider the following Advisor-Verifier-Prover (AVP) game: First, a function f is given to the advisor who computes an advice a. Next, an input x is given to the verifier and to the prover who claims that f(x)=1.f(x) \quad =1.f(x)=1. The verifier should check this claim via a single round of interaction based on the private advice a and without having any additional information on f. We focus on the case where the prover is laconic and communicates only a constant number of bits, and, mostly restrict the attention to the simplest, purely information-theoretic setting, where all parties are allowed to be computationally unbounded. The goal is to minimize the total communication complexity which is dominated by the length of the advice plus the length of the verifier’s query.As our main result, we show that a super-polynomial lower bound for AVPs implies a super-polynomial lower bound for a wide range of information-theoretic cryptographic tasks. In particular, we present a communication-efficient transformation from any of the above primitives into an AVP protocol. Interestingly, each primitive induces some additional property over the resulting protocol. Thus AVP games form a new common yardstick that highlights the differences between all the above primitives.Equipped with this view, we revisit the existing (somewhat weak) lower bounds for the above primitives, and show that many of these lower bounds can be unified by proving a single counting-based lower bound on the communication of AVPs, whereas some techniques are inherently limited to specific domains. The latter is shown by proving the first polynomial separations between the complexity of secret-sharing schemes and conditional disclosure of secrets and between the complexity of randomized encodings and conditional disclosure of secrets.
Abstract—Oblivious polynomial evaluation is a protocol involving two parties, a sender whose input is a polynomial P, and a receiver whose input is a value x. At the end of the protocol, the receiver learns P (x) and nothing more about P, while the sender remains oblivious of both x and P (x). It is used as a primitive in many applications including protocols for private comparison of data, for mutually authenticated key exchange based on (possibly weak) passwords, and for anonymous coupons. In this paper, we describe a novel unconditionally secure oblivious polynomial evaluation protocol. Keywords—Multi-Party Computation, Oblivious Transfer, Oblivious Polynomial Evaluation.
In recent years, lattice-based secure multi-party computation (MPC) has seen a rise in popularity and is used more and more in large scale applications like privacy-preserving cloud computing, electronic voting, or auctions. Many of these applications come with the following high security requirements: a computation result should be publicly verifiable, with everyone being able to identify a malicious party and hold it accountable, and a malicious party should not be able to corrupt the computation, force a protocol restart, or block honest parties or an honest third-party (client) that provided private inputs from receiving a correct result. The protocol should guarantee verifiability and accountability even if all protocol parties are malicious. While some protocols address one or two of these often essential security features, we present the first publicly verifiable and accountable, and (up to a threshold) robust SPDZ-like MPC protocol without restart. We propose protocols for accountable and robust online, offline, and setup computations. We adapt and partly extend the lattice-based commitment scheme by Baum et al. (SCN 2018) as well as other primitives like ZKPs. For the underlying commitment scheme and the underlying BGV encryption scheme we determine ideal parameters. We give a performance evaluation of our protocols and compare them to state-of-the-art protocols both with and without our target security features: public accountability, public verifiability and robustness.
Secure multiparty computation (MPC) enables mutually distrustful parties to compute on private digital inputs. We initiate the study of spatiotemporal MPC, extending this paradigm to functionalities whose inputs additionally depend on physical facts such as the parties' locations, times, or trajectories. Such protocols must simultaneously hide spatiotemporal information and ensure its physical consistency: a malicious party should not be able to make the functionality operate on a spatiotemporal input inconsistent with its actual physical state. The main conceptual challenge is to formulate extraction of spatiotemporal information within the simulation-based security framework. We introduce arguments of spatiotemporal knowledge, following the principle "I prove, therefore I am:" rather than defining physical presence directly through a mathematical relation, we define it operationally through the ability to complete a sound spatiotemporal verification protocol. Accordingly, an extractor recovers a spatiotemporal point from a successful prover and certifies its physical validity by using the extracted prover to succeed in an auxiliary spatiotemporal verification protocol. Building on this notion, we define universally composable (UC) security for spatiotemporal MPC, capturing privacy, physical consistency, and composability. We provide constructions achieving this new MPC notion. We first construct UC-secure commit-and-prove of spatiotemporal knowledge: in the CRS model under LWE against quantum provers without pre-shared entanglement, and in the QROM against quantum provers with unbounded pre-shared entanglement. Using these protocols, we obtain UC-secure spatiotemporal MPC from semi-honest post-quantum MPC. We also extend our framework to support quantum functionalities with classical spatiotemporal input.
Oblivious polynomial evaluation is a protocol involving two parties: a sender whose input is a polynomial and a receiver whose input is a value a .Most of existing Oblivious Polynomial Evaluation protocols were built based on some cryptographically intractable problems which made them a higher computation complexity.By using some random and hiding secure information in them, this paper presents an unconditionally secure oblivious polynomial evaluation protocol.It does not rely on any cryptographic assumption and involves only simple computation.Further, our protocol does not include any third party and is more practice.