KE ⇒ Multi-party key exchange

Fully-black-box reduction · standard model · folklore · security loss: factor $n-1$ over the two-party KE advantage (one hybrid per session)

Statement

Two-party key exchange secure against eavesdroppers implies $n$-party multi-party key exchange secure against eavesdroppers: a designated party runs an independent two-party session with each of the other $n-1$ parties, samples a group key $K$ uniformly, and sends $K$ one-time-padded under each session key — folklore.

Sketch

A hybrid over the $n-1$ sessions replaces each session key by an independent uniform key; the reduction embeds its two-party challenge in one session and runs the others itself. In the last hybrid every pad is uniform and independent of $K$, so $K$ is independent of the transcript.