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Information-Theoretically Secure Oblivious Polynomial Evaluation in the Commodity-Based Model

Rafael Tonicelli, Anderson C. A. Nascimento, Rafael Dowsley, Jörn Müller‐Quade, Hideki Imai, Goichiro Hanaoka, Akira Otsuka

Published Oct 28, 2013• 4 citations
OpenAlex

Abstract

Abstract. Oblivious polynomial evaluation (OPE) consists of a two-party protocol where a sender inputs a polynomial p(x), and a receiver inputs a single value x0. At the end of the protocol, the sender learns nothing and the receiver learns p(x0). This paper deals with the problem of oblivious polynomial evaluation under an information-theoretic perspective, which is based on recent definitions of Unconditional Security developed by Crépeau et al. [11]. In this paper, we propose an information-theoretic model for oblivious polynomial evaluation relying on pre-distributed data, and prove very general lower bounds on the size of the pre-distributed data, as well as the size of the communications in any protocol. It is demonstrated that these bounds are tight by obtaining a round-optimal OPE protocol, which meets the lower bounds simultaneously. We present a natural generalization to OPE called oblivious linear functional evaluation.

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•Jan 1, 2012•2 citations
Abstract. Oblivious polynomial evaluation (OPE) consists of a two-party protocol where a sender inputs a polynomial p(x), and a receiver inputs a single value x0. At the end of the protocol, the sender learns nothing and the receiver learns p(x0). This paper deals with the problem of oblivious polynomial evaluation under an information-theoretic perspective, which is based on recent definitions of Unconditional Security developed by Crépeau et al. [11]. In this paper, we propose an information-theoretic model for oblivious polynomial evaluation relying on pre-distributed data, and prove very general lower bounds on the size of the pre-distributed data, as well as the size of the communications in any protocol. It is demonstrated that these bounds are tight by obtaining a round-optimal OPE protocol, which meets the lower bounds simultaneously. We present a natural generalization to OPE called oblivious linear functional evaluation.
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.
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.
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.
We introduce a new approach to actively secure two-party computation based on so-called oblivious linear function evaluation (OLE), a natural generalisation of oblivious transfer (OT) and a special case of the notion of oblivious polynomial evaluation introduced by Naor and Pinkas at STOC 1999. OLE works over a finite field F. In an OLE the sender inputs two field elements a ƒ F and b ƒ F, and the receiver inputs a field element x ∈ F and learns only ƒx) = ax + b. Our protocol can evaluate an arithmetic circuit over a finite field F given black-box access to OLE for F. The protocol is unconditionally secure and consumes only a constant number of OLEs per multiplication gate. An OLE over a field F of size O(2κ) be implemented with communication O(κ). This gives a protocol with communication complexity O(C κ) for large enough fields, where C is an arithmetic circuit computing the desired function.
We introduce a new approach to actively secure two-party computation based on so-called oblivious linear function evaluation (OLE), a natural generalisation of oblivious transfer (OT) and a special case of the notion of oblivious polynomial evaluation introduced by Naor and Pinkas at STOC 1999. OLE works over a finite field F. In an OLE the sender inputs two field elements a ƒ F and b ƒ F, and the receiver inputs a field element x ∈ F and learns only ƒx) = ax + b. Our protocol can evaluate an arithmetic circuit over a finite field F given black-box access to OLE for F. The protocol is unconditionally secure and consumes only a constant number of OLEs per multiplication gate. An OLE over a field F of size O(2κ) be implemented with communication O(κ). This gives a protocol with communication complexity O(C κ) for large enough fields, where C is an arithmetic circuit computing the desired function. This asymptotically matches the best previous protocols, but our protocol at the same time obtains significantly smaller constants hidden by the big-O notation, yielding a highly practical protocol. Conceptually our techniques lift the techniques for basing practical actively secure 2PC of Boolean circuits on OT introduced under the name TinyOT by Nielsen, Nordholt, Orlandi and Burra at Crypto 2012 to the arithmetic setting. In doing so we develop several novel techniques for generating various flavours of OLE and combining these. We believe that the efficiency of our protocols, both in asymptotic and practical terms, establishes OLE and its variants as an important foundation for efficient actively secure 2PC.