Quantum Universally Composable Oblivious Linear Evaluation

Abstract

Oblivious linear evaluation is a generalization of oblivious transfer, whereby two distrustful parties obliviously compute a linear function, f (x) = ax + b, i.e., each one provides their inputs that remain unknown to the other, in order to compute the output f (x) that becomes known to only one of them. From both a structural and a security point-of-view, oblivious linear evaluation is fundamental for arithmeticbased secure multi-party computation protocols. In the classical case it is known that oblivious linear evaluation can be generated based on oblivious transfer, and quantum counterparts of these protocols can, in principle, be constructed as straightforward extensions based on quantum oblivious transfer. Here, we present the first, to the best of our knowledge, quantum protocol for oblivious linear evaluation that, furthermore, does not rely on quantum oblivious transfer. We start by presenting a semi-honest protocol and then we extend it to the malicious setting employing a commit-and-open strategy. Our protocol uses high-dimensional quantum states to obliviously compute the linear function, f (x), on Galois Fields of prime dimension, GF (d) ∼ = Z d , or prime-power dimension, GF (d M ). These constructions utilize the existence of a complete set of mutually unbiased bases in prime-power dimension Hilbert spaces and their linear behaviour upon the Heisenberg-Weyl operators. We also generalize our protocol to achieve vector oblivious linear evaluation, where several instances of oblivious linear evaluation are generated, thus making the protocol more efficient. We prove the protocols to have static security in the framework of quantum universal composability.

Quantum Oblivious Transfer: a Short Review

“…Up until now, and to the best of our knowledge, the composable security of the protocol Π BBCS F com was only proven in the Fehr and Schaffner model [74] by Damgård et al [9] and in the quantum-UC by Unruh [81]. More recently, a generalization of Oblivious Transfer (Oblivious Linear Evaluation functionnality) developed under in the F com -hybrid model, was also proven to be secure in the quantum-UC model [99].…”

Quantum Oblivious Transfer: A Short Review

“…Additionally, this primitive can also be used for secure multi-party computation (SMC) [ 13 ], private information retrieval [ 14 ], private set intersection [ 15 ], and privacy-preserving location-based services [ 16 ]. More recently, the first direct quantum protocol for a generalization of oblivious transfer known as oblivious linear evaluation was proposed [ 17 ]. In addition, quantum versions of oblivious transfer have recently been applied to SMC systems in the field of genomics medicine [ 18 , 19 ].…”