Paper 2021/190

Decidability of Secure Non-interactive Simulation of Doubly Symmetric Binary Source

Hamidreza Amini Khorasgani, Hemanta K. Maji, and Hai H. Nguyen

Abstract

Noise, which cannot be eliminated or controlled by parties, is an incredible facilitator of cryptography. For example, highly efficient secure computation protocols based on independent samples from the doubly symmetric binary source (BSS) are known. A modular technique of extending these protocols to diverse forms of other noise without any loss of round and communication complexity is the following strategy. Parties, beginning with multiple samples from an arbitrary noise source, non-interactively, albeit securely, simulate the BSS samples. After that, they can use custom-designed efficient multi-party solutions using these BSS samples. Khorasgani, Maji, and Nguyen (EPRINT--2020) introduce the notion of secure non-interactive simulation (SNIS) as a natural cryptographic extension of concepts like non-interactive simulation and non-interactive correlation distillation in theoretical computer science and information theory. In SNIS, the parties apply local reduction functions to their samples to produce samples of another distribution. This work studies the decidability problem of whether samples from the noise $(X,Y)$ can securely and non-interactively simulate BSS samples. As is standard in analyzing non-interactive simulations, our work relies on Fourier-analytic techniques to approach this decidability problem. Our work begins by algebraizing the simulation-based security definition of SNIS. Using this algebraized definition of security, we analyze the properties of the Fourier spectrum of the reduction functions. Given $(X,Y)$ and BSS with noise parameter $\epsilon$, the objective is to distinguish between the following two cases. (A) Does there exist a SNIS from $BSS(\epsilon)$ to $(X,Y)$ with $\delta$-insecurity? (B) Do all SNIS from $BSS(\epsilon)$ to $(X,Y)$ incur $\delta'$-insecurity, where $\delta'>\delta$? We prove that there is a bounded computable time algorithm achieving this objective for the following cases. (1) $\delta=O{1/n}$ and $\delta'=$ positive constant, and (2) $\delta=$ positive constant, and $\delta'=$ another (larger) positive constant. We also prove that $\delta=0$ is achievable only when $(X,Y)$ is another BSS, where $(X,Y)$ is an arbitrary distribution over $\{-1,1\}\times\{-1,1\}$. Furthermore, given $(X,Y)$, we provide a sufficient test determining if simulating BSS samples incurs a constant-insecurity, irrespective of the number of samples of $(X,Y)$. Handling the security of the reductions in Fourier analysis presents unique challenges because the interaction of these analytical techniques with security is unexplored. Our technical approach diverges significantly from existing approaches to the decidability problem of (insecure) non-interactive reductions to develop analysis pathways that preserve security. Consequently, our work shows a new concentration of the Fourier spectrum of secure reduction functions, unlike their insecure counterparts. We show that nearly the entire weight of secure reduction functions' spectrum is concentrated on the lower-degree components. The authors believe that examining existing analytical techniques through the facet of security and developing new analysis methodologies respecting security is of independent and broader interest.