In the model of Perfectly Secure Message Transmission (PSMT), a sender Alice is connected to a receiver Bob via $n$ parallel two-way channels, and Alice holds an $ell$ symbol secret that she wishes to communicate to Bob. There is an unbounded adversary Eve that controls $t$ of the channels, where $n=2t+1$. Eve is able to corrupt any symbol sent through the channels she controls, and furthermore may attempt to infer Alice’s secret by observing the symbols sent through the channels she controls. The transmission is required to be (a) reliable, i.e., Bob must always be able to recover Alice’s secret, regardless of Eve’s corruptions; and (b) private, i.e., Eve may not learn anything about the Alice’s secret. We focus on the two-round model, where Bob is permitted to first transmit to Alice, and then Alice responds to Bob.
In this work we provide tight upper and lower bounds for the PSMT model when the length of the communicated secret $ell$ is asymptotically large. Specifically, we first construct a protocol that allows Alice to communicate an $ell$ symbol secret to Bob by transmitting at most $2(1+o(1))nell$ symbols. We complement this with a lower bound showing that $2nell$ symbols are necessary for Alice to privately and reliably communicate her secret. Thus, we completely determine the optimal transmission rate in this regime, even up to the leading constant.