In the model that has become known as "Perfectly Secure Message Transmission"(PSMT), a sender Alice is connected to a receiver Bob through n parallel two-way channels. A computationally unbounded adversary Eve controls t of these channels, meaning she can acquire and alter any data that is transmitted over these channels. The sender Alice wishes to communicate a secret message to Bob privately and reliably, i.e. in such a way that Eve will not get any information about the message while Bob will be able to recover it completely. In this paper, we focus on protocols that work in two transmission rounds for n= 2t+1. We break from previous work by following a conceptually simpler blueprint for achieving a PSMT protocol. We reduce the previously best-known communication complexity, i.e. the number of transmitted bits necessary to communicate a 1-bit secret, from O(n^3 log n) to O(n^2 log n). Our protocol also answers a question raised by Kurosawa and Suzuki and hitherto left open: their protocol reaches optimal transmission rate for a secret of size O(n^2 log n) bits, and the authors raised the problem of lowering this threshold. The present solution does this for a secret of O(n log n) bits.

M. Hirt (Martin) , A. Smith (Adam)
Lecture Notes in Computer Science
International Conference on Theory of Cryptography

Spini, G, & Zémor, G (Gilles). (2016). Perfectly secure message transmission in two rounds. In M Hirt & A Smith (Eds.), Theory of Cryptography (pp. 286–304).