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Mathematics in Computer Science

Efficient and Secure Delegation of Exponentiation in General Groups to a Single Malicious Server

Giovanni Di Crescenzo, Matluba Khodjaeva, Delaram Kahrobaei, Vladimir Shpilrain

Published Mar 28, 2020• 4 citations
Publisher
DOI
10.1007/s11786-020-00462-4

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Oblivious Pseudorandom Functions (OPRFs) allow a client to evaluate a pseudorandom function (PRF) on her secret input based on a key that is held by a server. In the process, the client only learns the PRF output but not the key, while the server neither learns the input nor the output of the client. The arguably most popular OPRF is due to Naor, Pinkas and Reingold (Eurocrypt 2009). It is based on an Oblivious Exponentiation by the server, with passive security under the Decisional Diffie-Hellman assumption. In this work, we strengthen the security guarantees of the NPR OPRF by protecting it against active attacks of the server. We have implemented our solution and report on the performance. Our main result is a new batch OPRF protocol which is secure against maliciously corrupted servers, but is essentially as efficient as the semi-honest solution. More precisely, the computation (and communication) overhead is a multiplicative factor o ( 1 ) as the batch size increases. The obvious solution using zero-knowledge proofs would have a constant factor overhead at best, which can be too expensive for certain deployments. Our protocol relies on a novel version of the DDH problem, which we call the Oblivious Exponentiation Problem (OEP), and we give evidence for its hardness in the Generic Group model. We also present a variant of our maliciously secure protocol that does not rely on the OEP but nevertheless only has overhead o ( 1 ) over the known semi-honest protocol. Moreover, we show that our techniques can also be used to efficiently protect threshold blind BLS signing and threshold ElGamal decryption against malicious attackers.
Quantum query is a basic subroutine in many quantum algorithms, and Quantum Random Access Memory (QRAM) provides a natural way to realize such coherent query access. In delegated settings, however, a standard QRAM query interface can expose sensitive information to the server. This paper introduces oblivious QRAM, a cryptographic abstraction for privacy-preserving delegated coherent query access. The protocol consists of an offline refresh phase and an online protected query phase. The database is stored in an encrypted and shuffled layout, and each query is protected by coherent address masking using either a quantum-secure pseudorandom permutation (qPRP) based method or a quantum one-time pad (qOTP) based method. In the adopted client model, the online protection adds only modest quantum overhead beyond the query register, avoiding the exponential quantum resources that would otherwise be required by an equivalent local QRAM construction. The qPRP-based variant also supports multi-query use by distributing database refresh across multiple queries to reduce classical communication. To address malicious servers, decoy checks are further incorporated to strengthen privacy protection and enable probabilistic tampering detection. Compared with fully blind quantum computing, this framework provides a lighter abstraction tailored to private delegated QRAM access, significantly reducing quantum resource requirements on both the client and server sides and achieving an exponential reduction in quantum communication.