Abstract
Oblivious Linear Evaluation (OLE) is the arithmetic analogue of the well-know oblivious transfer primitive. It allows a sender, holding an affine function \(f(x)=a+bx\) over a finite field or ring, to let a receiver learn f(w) for a w of the receiver’s choice. In terms of security, the sender remains oblivious of the receiver’s input w, whereas the receiver learns nothing beyond f(w) about f. In recent years, OLE has emerged as an essential building block to construct efficient, reusable and maliciously-secure two-party computation.
In this work, we present efficient two-round protocols for OLE over large fields based on the Learning with Errors (LWE) assumption, providing a full arithmetic generalization of the oblivious transfer protocol of Peikert, Vaikuntanathan and Waters (CRYPTO 2008). At the technical core of our work is a novel extraction technique which allows to determine if a non-trivial multiple of some vector is close to a q-ary lattice.
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Notes
- 1.
It is easy to see that, if we consider the affine function \(f:\{0,1\}\rightarrow \{0,1\}\) such that \(f(b)=m_0+b(m_1-m_0)\), OLE trivially implements OT.
- 2.
While two-party reusable non-interactive secure computation (NISC) is impossible in the OT-hybrid model [11], reusable NISC for general Boolean circuits is known to be possible in the (reusable) OLE-hybrid model assuming one-way functions [11]. The result stated above is meaningful only if we have access to a reusable two-round OLE protocol. The only efficient realizations of this primitive are based on the Decisional Composite Residuosity (DCR) and the Quadratic Residuosity assumptions [11].
- 3.
As an example consider the work of [11], where the Paillier cryptosystem is extended into an OLE protocol with malicious security and the construction is highly non-trivial.
- 4.
Despite numerous efforts, HPS in the lattice setting fall short in efficiency when comparing to their group-based counterpart.
- 5.
The protocol presented in Sect. 5 is presented in a slightly, but equivalent, form.
- 6.
The approach is similar in spirit as previous works, e.g. [25].
- 7.
The construction actually works for any OLE scheme that is secure against semi-honest senders and malicious receivers. In the technical sections we present the generic construction.
- 8.
The matrix \(\mathbf {B}\) is called a basis of \(\varLambda (\mathbf {B})\).
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Acknowledgment
Pedro Branco thanks the support from DP-PMI and FCT (Portugal) through the grant PD/BD/135181/2017. Part of the work was done while the author was at CISPA.
Pedro Branco and Paulo Mateus are partially supported by national funds through Fundação para a Ciência e a Tecnologia (FCT) with reference UIDB/50008/2020 (Instituto de Telecomunicações via actions QuRUNNER, QUESTS) and Projects QuantumMining POCI-01-0145-FEDER-031826, PREDICT PTDC/CCI-CIF/29877/2017 and QuantumPrime PTDC/EEI-TEL/8017/2020.
Nico Döttling was supported by the Helmholtz Association within the project “Trustworthy Federated Data Analytics” (TFDA) (funding number ZT-I- OO1 4).
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Branco, P., Döttling, N., Mateus, P. (2022). Two-Round Oblivious Linear Evaluation from Learning with Errors. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds) Public-Key Cryptography – PKC 2022. PKC 2022. Lecture Notes in Computer Science(), vol 13177. Springer, Cham. https://doi.org/10.1007/978-3-030-97121-2_14
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