Doubly Efficient Interactive Proofs over Infinite and Non-Commutative Rings

We introduce the first proof system for layered arithmetic circuits over an arbitrary ring $R$ that is (possibly) non-commutative and (possibly) infinite, while only requiring black-box access to its arithmetic and a subset $A \subseteq R$. Our construction only requires limited commutativity and regularity properties from $A$, similar to recent work on efficient information theoretic multi-party computation over non-commutative rings by Escudero and Soria-Vazquez (CRYPTO 2021), but furthermore covering infinite rings. We achieve our results through a generalization of GKR-style interactive proofs (Goldwasser, Kalai and Rothblum, Journal of the ACM, 2015). When $A$ is a subset of the center of $R$, generalizations of the sum-check protocol and other building blocks are not too problematic. The case when the elements of $A$ only commute with each other, on the other hand, introduces a series of challenges. In order to overcome those, we need to introduce a new definition of polynomial ring over a non-commutative ring, the notion of left (and right) multi-linear extensions, modify the layer consistency equation and adapt the sum-check protocol. Despite these changes, our results are compatible with recent developments such as linear time provers. Moreover, for certain rings our construction achieves provers that run in sublinear time in the circuit size. We obtain such result both for known cases, such as matrix and polynomial rings, as well as new ones, such as for some rings resulting from Clifford algebras. Besides efficiency improvements in computation and/or round complexity for several instantiations, the core conclusion of our results is that state of the art doubly efficient interactive proofs do not require much algebraic structure. This enables exact rather than approximate computation over infinite rings as well as agile proof systems, where the black-box choice of the underlying ring can be easily switched through the software life cycle

New Primitives for Actively-Secure MPC over Rings with Applications to Private Machine Learning

At CRYPTO 2018 Cramer et al. presented SPDZ2k, a new secret-sharing based protocol for actively secure multi-party computation against a dishonest majority, that works over rings instead of fields. Their protocol uses slightly more communication than competitive schemes working over fields. However, their approach allows for arithmetic to be carried out using native 32 or 64-bit CPU operations rather than modulo a large prime. The authors thus conjectured that the increased communication would be more than made up for by the increased efficiency of implementations. In this work we answer their conjecture in the affirmative. We do so by implementing their scheme, and designing and implementing new efficient protocols for equality test, comparison, and truncation over rings. We further show that these operations find application in the machine learning domain, and indeed significantly outperform their field-based competitors. In particular, we implement and benchmark oblivious algorithms for decision tree and support vector machine (SVM) evaluation

Using secret sharing for searching in encrypted data

When outsourcing data to an untrusted database server, the data should be encrypted. When using thin clients or low-bandwidth networks it is best to perform most of the work at the server. We present a method, inspired by secure multi-party computation, to search efficiently in encrypted data. XML elements are translated to polynomials. A polynomial is split into two parts: a random polynomial for the client and the difference between the original polynomial and the client polynomial for the server. Since the client polynomials are generated by a random sequence generator only the seed has to be stored on the client. In a combined effort of both the server and the client a query can be evaluated without traversing the whole tree and without the server learning anything about the data or the query

SWIFT: Super-fast and Robust Privacy-Preserving Machine Learning

Performing machine learning (ML) computation on private data while maintaining data privacy, aka Privacy-preserving Machine Learning~(PPML), is an emergent field of research. Recently, PPML has seen a visible shift towards the adoption of the Secure Outsourced Computation~(SOC) paradigm due to the heavy computation that it entails. In the SOC paradigm, computation is outsourced to a set of powerful and specially equipped servers that provide service on a pay-per-use basis. In this work, we propose SWIFT, a robust PPML framework for a range of ML algorithms in SOC setting, that guarantees output delivery to the users irrespective of any adversarial behaviour. Robustness, a highly desirable feature, evokes user participation without the fear of denial of service. At the heart of our framework lies a highly-efficient, maliciously-secure, three-party computation (3PC) over rings that provides guaranteed output delivery (GOD) in the honest-majority setting. To the best of our knowledge, SWIFT is the first robust and efficient PPML framework in the 3PC setting. SWIFT is as fast as (and is strictly better in some cases than) the best-known 3PC framework BLAZE (Patra et al. NDSS'20), which only achieves fairness. We extend our 3PC framework for four parties (4PC). In this regime, SWIFT is as fast as the best known fair 4PC framework Trident (Chaudhari et al. NDSS'20) and twice faster than the best-known robust 4PC framework FLASH (Byali et al. PETS'20). We demonstrate our framework's practical relevance by benchmarking popular ML algorithms such as Logistic Regression and deep Neural Networks such as VGG16 and LeNet, both over a 64-bit ring in a WAN setting. For deep NN, our results testify to our claims that we provide improved security guarantee while incurring no additional overhead for 3PC and obtaining 2x improvement for 4PC.Comment: This article is the full and extended version of an article to appear in USENIX Security 202

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder

Pika: Secure Computation using Function Secret Sharing over Rings

Machine learning algorithms crucially depend on non-linear mathematical functions such as division (for normalization), exponentiation (for softmax and sigmoid), tanh (as an activation function), logarithm (for cross-entropy loss), and square root (for back-propagation of normalization layers). However, when machine learning is performed over secure computation, these protocols incur a large communication overhead and high round complexity. In this work, we propose new multi-party computation (MPC) protocols for such functions. Our protocols achieve constant round complexity (3 for semi-honest, 4 for malicious), an order of magnitude lower communication (54-121x lower than prior art), and high concrete efficiency (2-1163x faster runtime). We rely on recent advances in function secret sharing (FSS) to construct these protocols. Our contributions can be summarized as follows: (1) A constant round protocol to securely evaluate non-linear functions such as division, exponentiation, logarithm, and tanh (in comparison to prior art which uses round complexity proportional to the rounds of iterative methods/required precision) with high accuracy. This construction largely follows prior work in look-up style secure computation. (2) Our main contribution is the extension of the above protocol to be secure in the presence of malicious adversaries in the honest majority setting. We provide a malicious sketching protocol for FSS schemes that works over rings and in order to prove its security, we extend (and prove) a corresponding form of Schwartz-Zippel lemma over rings. This is the first such extension of the lemma and it can be of independent interest in other domains of secure computation. (3) We implement our protocol and showcase order of magnitude improvements in runtime and communication. Given the low round complexity and substantially lower communication, our protocols achieve even better performance over network constrained environments such as WAN. Finally, we showcase how such functions can lead to scalability in machine learning. Note that techniques presented are applicable beyond the application of machine learning as the protocols effectively present an efficient 1-out-of-N oblivious transfer or an efficient private information retrieval protocol

Computing on Anonymous Quantum Network

This paper considers distributed computing on an anonymous quantum network, a network in which no party has a unique identifier and quantum communication and computation are available. It is proved that the leader election problem can exactly (i.e., without error in bounded time) be solved with at most the same complexity up to a constant factor as that of exactly computing symmetric functions (without intermediate measurements for a distributed and superposed input), if the number of parties is given to every party. A corollary of this result is a more efficient quantum leader election algorithm than existing ones: the new quantum algorithm runs in O(n) rounds with bit complexity O(mn^2), on an anonymous quantum network with n parties and m communication links. Another corollary is the first quantum algorithm that exactly computes any computable Boolean function with round complexity O(n) and with smaller bit complexity than that of existing classical algorithms in the worst case over all (computable) Boolean functions and network topologies. More generally, any n-qubit state can be shared with that complexity on an anonymous quantum network with n parties.Comment: 25 page