We show that the tensor product of a high-rate globally list recoverable code is (approximately) locally list recoverable. List recovery has been a useful building block in the design of list decodable codes, and our motivation is to use the tensor construction as such a building block. In particular, instantiating this construction with known constructions of high-rate globally list recoverable codes, and using appropriate transformations, we obtain the first capacity-achieving locally list decodable codes (over a large constant size alphabet), and the first capacity-achieving globally list decodable codes with nearly linear time list decoding algorithms. Our techniques are inspired by an approach of Gopalan, Guruswami, and Raghavendra [ SIAM J. Comput., 40 (2011), pp. 1432--1462] for list decoding tensor codes.

Recently, a number of systems have been deployed that gather sensitive statistics from user devices while giving differential privacy guarantees. One prominent example is the component in Apple's macOS and iOS devices that collects information about emoji usage and new words. However, these systems have been criticized for making unrealistic assumptions, e.g., by creating a very high "privacy budget" for answering queries, and by replenishing this budget every day, which results in a high worst-case privacy loss. However, it is not obvious whether such assumptions can be avoided if one requires a strong threat model and wishes to collect data periodically, instead of just once.

Secure multi-party computation (MPC) allows a group of mutually distrustful parties to compute a joint function on their inputs without revealing any information beyond the result of the computation. This type of computation is extremely powerful and has wide-ranging applications in academia, industry, and government. Protocols for secure computation have existed for decades, but only recently have general-purpose compilers for executing MPC on arbitrary functions been developed. These projects rapidly improved the state of the art, and began to make MPC accessible to non-expert users. However, the field is changing so rapidly that it is difficult even for experts to keep track of the varied capabilities of modern frameworks. In this work, we survey general-purpose compilers for secure multi-party computation. These tools provide high-level abstractions to describe arbitrary functions and execute secure computation protocols. We consider eleven systems: EMP-toolkit, Obliv-C, ObliVM, TinyGarble, SCALE-MAMBA (formerly SPDZ), Wysteria, Sharemind, PICCO, ABY, Frigate and CBMC-GC. We evaluate these systems on a range of criteria, including language expressibility, capabilities of the cryptographic back-end, and accessibility to developers. We advocate for improved documentation of MPC frameworks, standardization within the community, and make recommendations for future directions in compiler development. Installing and running these systems can be challenging, and for each system, we also provide a complete virtual environment (Docker container) with all the necessary dependencies to run the compiler and our example programs.

In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known. Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are `approximately' locally list-recoverable, and that the `approximately' modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.