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In this paper, Ueli Maurer uses a very cool model to generalize a number of results about broadcast, secret sharing and secure multiparty computation. Rather than talking about an adversary being either active (malicious) or passive (semi-honest) and having a threshold on the number of parties it controls, instead there are sets of sets of nodes $\Delta$ and $\Sigma$ such that the adversary may have active control of any set $A \in \Delta$ and passive control of any set $B$ such that $A \cup B \in \Sigma$.

Maurer states (Theorem 1) that:

"The simulation of a broadcast channel is secure against a $(\Sigma,\Delta)$-adversary is possible if and only if $P \notin \Delta \sqcup \Delta \sqcup \Delta$."

Here $P$ is the set of all nodes and $X\sqcup Y = \{x \in X, y\in Y : x \cup y\}$.

A corollary of this is that for a fixed threshold, we require $t < |P|/3$, where $t$ is the maximum number of nodes that may be actively controlled by the adversary.

Is Maurer just referring to the information theoretic case? Because I seem to remember reading somewhere that assuming cryptographic signatures you can have broadcast with up to $t < |P|/2$ malicious parties.

If so what is the tight limit for broadcast using Maurer's notation in the case of a polynomially-bound adversary, secure signatures and some public key infrastructure?

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This is a somewhat standard method for generalizing results based on threshold adversaries. Let $n$ be the number of parties and let $\mathcal{C} \subseteq 2^{[n]}$ denote the family of subsets that the adversary can corrupt. For example, in the case of honest majority, $\mathcal{C} = \{ C \subseteq [n] : |C| < n/2 \}$.

  • $\mathcal{C}$ has the Q2 property if no two corruptible sets $A, B \in \mathcal{C}$ satisfy $A \cup B = [n]$. In Maurer's notation, $[n] \not\in \mathcal{C} \sqcup \mathcal{C}$.

  • $\mathcal{C}$ has the Q3 property if no three corruptible sets $A, B,C \in \mathcal{C}$ satisfy $A \cup B \cup C = [n]$. In Maurer's notation, $[n] \not\in \mathcal{C} \sqcup \mathcal{C} \sqcup \mathcal{C}$.

Generally speaking, classical results that require honest majority will generalize to any Q2 corruption pattern. Results that require honest two-thirds majority generalize to Q3.

I think the first appearance of this generalization is in Complete Characterization of Adversaries Tolerable in Secure Multi-Party Computation by Hirt & Maurer. It's generalized even farther in MPC vs. SFE: Perfect Security in a Unified Corruption Model by Beerliova-Trubiniova et al. They consider an adversary that can corrupt any $S \in \mathcal{P}$ passively, or corrupt any $S \in \mathcal{A}$ actively, or corrupt any $S \in \mathcal{F}$ in a fail-stop way. These results are harder to state in a stackexchange answer.

Regarding the question about broadcast, the actual tight bound depends very heavily on the model (poly-time vs information-theoretic, [im]perfect security, PKI setup). Also agreement, consensus, and broadcast are all slightly different. But in general, information-theoretic broadcast with no setup requires honest 2/3rds majority (Q3 corruptions).

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