Key-agreement exists if and only if the "interactive vs non interactive Kolmogorov problem" is not in ioBPP: a short proof
Bruno Bauwens, Bruno Loff
TL;DR
The paper investigates the existence of key-agreement protocols through the lens of $\mathsf{ioBPP}$-decidability of the promise problem $(\mathcal{Y}^t_c,\mathcal{N}_e)$, and provides a short, self-contained proof of the hard direction of the equivalence: if $(\mathcal{Y}^t_c,\mathcal{N}_e) \in \mathsf{ioBPP}$ for suitable parameters, then secure key agreement cannot exist, and conversely the existence of a key-agreement protocol implies the promise problem lies outside $\mathsf{ioBPP}$. The construction uses a Goldreich–Levin leakage reduction and hashes to simulate a DH-like adversary, clarifying the link between key exchange and interactive Kolmogorov complexity with implications for program-size complexity and randomness extraction.
Abstract
Ball, Liu, Mazor and Pass proved that the existence of key-agreement protocols is equivalent to the hardness of a certain problem about interactive Kolmogorov complexity. We generalize the statement and give a short proof of the difficult implication.