Constant-Round Simulation-Secure Coin Tossing Extension with Guaranteed Output

Common randomness is an essential resource in many applications. However, Cleve (STOC 86) rules out the possibility of tossing a fair coin from scratch in the presence of a dishonest majority. A second-best alternative is a Coin Tossing Extension (CTE) protocol, which uses an “online” oracle that produces a few common random bits to generate many common random-looking bits. We initiate the systematic study of fully-secure CTE, which guarantees output even in the presence of malicious behavior. A fully-secure two-party statistical CTE protocol with black-box simulation was implicit in Hofheinz et al. (Eurocrypt 06), but its round complexity is nearly linear in its output length. The problem of constant-round CTE with superlogarithmic stretch remained open. We prove that statistical CTE with full black-box security and superlogarithmic stretch must have superconstant rounds. In the computational setting we prove that with \(N\ge 2\) parties and polynomial stretch: Such results were not previously known even in the two-party setting with standalone, static security. We also extend one-round CTE to sample from any efficient distribution, via strong assumptions including IO. Our one-round CTE protocols can be interpreted as explainable variants of classical randomness extractors, wherein a (short) seed and a source instance can be efficiently reverse-sampled given a random output. Such explainable extractors may be of independent interest.