The SPDZ multiparty computation protocol \citeC:DPSZ12 allows n parties to securely compute arithmetic circuits over a finite field, while tolerating up to n-1 active corruptions. A line of work building upon SPDZ has made considerable improvement to the protocol's performance, typically focusing on concrete efficiency. However, the communication complexity of each of these protocols is Ømega(n^2|C|).

We present gOTzilla, a protocol for interactive zero-knowledge proofs for very large disjunctive statements of the following format: given publicly known circuit C, and set of values Y = {y1 , . . . , yn }, prove knowledge of a witness x such that C(x) = y1 ∨ C(x) = y2 ∨ · · · ∨ C(x) = yn . These type of statements are extremely important for the proof of assets (PoA) problem in cryptocurrencies where a prover wants to prove the knowledge of a secret key sk that associates with the hash of a public key H(pk) posted on the ledger. We note that the size of n in popular cryptocurrencies, such as Bitcoin, is estimated to 80 million. For the construction of gOTzilla, we start by observing that if we restructure the proof statement to an equivalent of proving knowledge of (x, y) such that (C(x) = y) ∧ (y = y1 ∨ · · · ∨ y = yn )), then we can reduce the disjunction of equalities to 1-out-of-N oblivious transfer (OT). Our overall protocol is based on the MPC in the head (MPCitH) paradigm. We additionally provide a concrete, efficient extension of our protocol for the case where C combines algebraic and non-algebraic statements (which is the case in the PoA application). We achieve an asymptotic communication cost of O(log n) plus the proof size of the underlying MPCitH protocol. While related work has similar asymptotic complexity, our approach results in concrete performance improvements. We implement our protocol and provide benchmarks. Concretely, for a set of size 1 million entries, the total run-time of our protocol is 14.89 seconds using 48 threads, with 6.18 MB total communication, which is about 4x faster compared to the state of the art when considering a disjunctive statement with algebraic and non-algebraic elements.

We design several new protocols for private set intersection (PSI) with active security: one for the two party setting, and two protocols for the multi-party setting. In recent years, the state-of-the-art protocols for two party PSI have all been built from OT-extension. This has led to extremely efficient protocols that provide correct output to one party; seemingly inherent to the approach, however, is that there is no efficient way to relay the result to the other party with a provable correctness guarantee. Furthermore, there is no natural way to extend this line of works to more parties. We consider a new instantiation of an older approach. Using the MPC-in-the-head paradigm of Ishai et al. [IPS08], we construct a polynomial with roots that encode the intersection, without revealing the inputs. Our reliance on this paradigm allows us to base our protocol on passively secure Oblivious Linear Evaluation (OLE) (requiring 4 such amortized calls per input element). Unlike state-ofthe-art prior work, our protocols provide correct output to all parties. We have implemented our protocols, providing the first benchmarks for PSI that provides correct output to all parties. Additionally, we present a variant of our multi-party protocol that provides output only to a central server.

We construct new protocols for two parties to securely compute on the items in their intersection. Our protocols make use of an untrusted third party that has no input. The use of this party allows us to construct highly efficient protocols that are secure against a single malicious corruption.