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Lecture notes in computer science

Simultaneous-Message and Succinct Secure Computation

Elette Boyle, Abhishek Jain, Sacha Servan-Schreiber, Akshayaram Srinivasan

Published Jan 1, 2025• 5 citations
Publisher
DOI
10.1007/978-3-031-91092-0_8

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We propose the notion of succinct oblivious tensor evaluation (OTE), where two parties compute an additive secret sharing of a tensor product of two vectors x λ- y, exchanging two simultaneous messages. Crucially, the size of both messages and of the CRS is independent of the dimension of x. We present a construction of OTE with optimal complexity from the standard learning with errors (LWE) problem. Then we show how this new technical tool enables a host of cryptographic primitives, all with security reducible to LWE, such as: (a) Adaptively secure laconic function evaluation for depth-D functions f:{0, 1}m→{0, 1}λ.," with communication m+λ.,"+D· poly(λ); (b) A trapdoor hash function for all functions; (c) An (optimally) succinct homomorphic secret sharing for all functions; (d) A rate-1/2 laconic oblivious transfer for batch messages, which is best possible. In particular, we obtain the first laconic function evaluation scheme that is adaptively secure from the standard LWE assumption, improving upon Quach, Wee, and Wichs (FOCS 2018). As a key technical ingredient, we introduce a new notion of adaptive lattice encodings, which may be of independent interest.
We construct protocols for secure multiparty computation with the help of a computationally powerful party, namely the “cloud”. Our protocols are simultaneously ecient in a number of metrics: • Rounds: our protocols run in 4 rounds in the semi-honest setting, and 5 rounds in the malicious setting. • Communication: the number of bits exchanged in an execution of the protocol is independent of the complexity of function f being computed, and depends only on the length of the inputs and outputs. • Computation: the computational complexity of all parties is independent of the complexity of the function f, whereas that of the cloud is linear in the size of the circuit computing f. In the semi-honest case, our protocol relies on the “ring learning with errors” (RLWE) assumption, whereas in the malicious case, security is shown under the Ring LWE assumption as well as the existence of simulation-extractable NIZK proof systems and succinct non-interactive arguments. In the malicious setting, we also relax the communication and computation requirements above, and only require that they be “small” ‐ polylogarithmic in the computation size and linear in the size of the joint size of the inputs. Our constructions leverage the key homomorphic property of the recent fully homomorphic encryption scheme of Brakerski and Vaikuntanathan (CRYPTO 2011, FOCS 2011). Namely, these schemes allow combining encryptions of messages under dierent keys to produce an encryption (of the sum of the messages) under the sum of the keys. We also design an ecient, non-interactive threshold decryption protocol for these fully homomorphic encryption schemes.