Secure Non-interactive Simulation from Arbitrary Joint Distributions

Secure non-interactive simulation (SNIS), introduced in EUROCRYPT 2022, is the information-theoretic analog of pseudo-correlation generators. SNIS allows parties, starting with samples of a source correlated private randomness (correlation), to non-interactively and securely transform them into samples from a different correlation. This work studies SNIS of binary symmetric or erasure correlations from any arbitrary source correlation. In this context, our work presents: 1. The characterization of all sources that facilitate such SNIS, 2. An upper and lower bound on their maximum achievable rate, and 3. Exemplar SNIS instances where non-linear reductions achieve optimal efficiency; however, any linear reduction is insecure. These results collectively yield the fascinating instances of computer-assisted search for secure computation protocols that identify ingenious protocols that are more efficient than all known constructions. Our work generalizes the algebraization of the simulation-based definition of SNIS as an approximate eigenvector problem. The following technical contributions are the underpinnings of the results above. 1. Characterization of Markov and adjoint Markov operators' effect on the Fourier spectrum of reduction functions. 2. A new concentration phenomenon in the Fourier spectrum of reduction functions. 3. A statistical-to-perfect lemma with broad consequences for feasibility and rate characterization of SNIS. Our technical analysis relies on Fourier analysis over large alphabets with arbitrary measure, the orthogonal Efron-Stein decomposition, and junta theorems. Our technical approach motivates the new problem of "security-preserving dimension reduction" in harmonic analysis, which may be of independent interest.