**Almost-Everywhere Secure Computation with Edge Corruptions**

*Nishanth Chandran and Juan Garay and Rafail Ostrovsky*

**Abstract: **We consider secure multi-party computation (MPC) in a setting where
the adversary can separately corrupt not only the parties (nodes) but
also the communication channels (edges), and can furthermore choose
selectively and adaptively which edges or nodes to corrupt. Note that
if an adversary corrupts an edge, even if the two nodes that share
that edge are honest, the adversary can control the link and thus
deliver wrong messages to both players. We consider this question in
the information-theoretic setting, and require security against a
computationally unbounded adversary.

In a fully connected network the above question is simple (and we also provide an answer that is optimal up to a constant factor). What makes the problem more challenging is to consider the case of sparse networks. Partially connected networks are far more realistic than fully connected networks, which led Garay and Ostrovsky [Eurocrypt'08] to formulate the notion of (unconditional) \emph{almost everywhere (a.e.) secure computation} in the node-corruption model, i.e., a model in which not all pairs of nodes are connected by secure channels and the adversary can corrupt some of the nodes (but not the edges). In such a setting, MPC amongst all honest nodes cannot be guaranteed due to the possible poor connectivity of some honest nodes with other honest nodes, and hence some of them must be ``given up'' and left out of the computation. The number of such nodes is a function of the underlying communication graph and the adversarial set of nodes.

In this work we introduce the notion of \emph{almost-everywhere secure computation with edge corruptions}, which is exactly the same problem as described above, except that we additionally allow the adversary to completely control some of the communication channels between two correct nodes---i.e., to ``corrupt'' edges in the network. While it is easy to see that an a.e. secure computation protocol for the original node-corruption model is also an a.e. secure computation protocol tolerating edge corruptions (albeit for a reduced fraction of edge corruptions with respect to the bound for node corruptions), no polynomial-time protocol is known in the case where a {\bf constant fraction} of the edges can be corrupted (i.e., the maximum that can be tolerated) and the degree of the network is sub-linear.

We make progress on this front, by constructing graphs of degree $O(n^\epsilon)$ (for arbitrary constant $0<\epsilon<1$) on which we can run a.e. secure computation protocols tolerating a constant fraction of adversarial edges. The number of given-up nodes in our construction is $\mu n$ (for some constant $0<\mu<1$ that depends on the fraction of corrupted edges), which is also asymptotically optimal.

**Category / Keywords: **cryptographic protocols / secure computation, bounded-degree networks, edge corruptions

**Publication Info: **A version of this paper, entitled ``Edge Fault Tolerance on Sparse Networks,'' will appear in the proceedings of ICALP 2012; this is the full version of that paper with a more cryptographically oriented treatment.

**Date: **received 22 Apr 2012, last revised 23 Apr 2012

**Contact author: **nish at microsoft com

**Available format(s): **PDF | BibTeX Citation

**Note: **Added grant information and re-organized the paper slightly.

**Version: **20120423:142809 (All versions of this report)

**Short URL: **ia.cr/2012/221

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