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In a public-key threshold encryption scheme, the sender produces a single ciphertext, and any \(t+1\) out of \(n\) intended recipients can combine their partial decryptions to obtain the plaintext. Ad hoc threshold encryption (ATE) schemes require no correlated setup, enabling each party to simply generate its own key pair. In this paper, we initiate a systematic study of the possibilities and limitations of ad-hoc threshold encryption, and introduce a key application to scalable multiparty computation (MPC).

We consider the problem of designing scalable, robust protocols for computing statistics about sensitive data. Specifically, we look at how best to design differentially private protocols in a distributed setting, where each user holds a private datum. The literature has mostly considered two models: the “central” model, in which a trusted server collects users’ data in the clear, which allows greater accuracy; and the “local” model, in which users individually randomize their data, and need not trust the server, but accuracy is limited. Attempts to achieve the accuracy of the central model without a trusted server have so far focused on variants of cryptographic multiparty computation (MPC), which limits scalability.

In this chapter we study the out-of-time-ordered correlators (OTOCs) for the gauge field in our model. While originally introduced in the context of quasiclassical approaches to quantum systems (Larkin and Ovchinnikov in Sov Phys JETP 28:1200–1205, 1969, [1]), OTOCs have recently received renewed interest due to their connections with the emergence of quantum chaotic behaviour [2, 3, 4].