Tell me who you are friends with and I will tell you who you are: Unique neighborhoods in random graphs

Abstract

The identity of a physical entity in a network is traditionally defined by some secret knowledge, personal possession, or (biometric) property. What if no such unique property exists or can be defined “safely”, for example, in the interest of anonymity? In this work, we study the problem of defining an identity for an entity u under the constraint that there is no subjective property that u carries by itself, or in other words, any property that u has, may also be likewise present in another entity v. In the absence of an intrinsic property of u to define its identity, we thus consider the possibility of an “extrinsically defined identity” via the connections that u maintains to other entities. Letting u∈V be part of a graph G=(V,E), we study the question of whether the links that u has to its neighbors lend themselves to uniquely distinguish u from all other nodes v in the graph.A practical instance of this setting appears in symmetric cryptography, if we assume an edge between two nodes u,v if and only if u and v share a common secret. A single shared secret known by u is, in symmetric cryptography, also known to some other entity v. However, is the set of all secrets that u has likewise known to another node v in a network? If not, we can implement end-to-end authentication without shared secrets and exclusively using symmetric cryptography. This concept is here reviewed as peer-authentication. It works in a graph class that we call Unique-Neighborhood Network (UNN), which has been introduced in prior literature for the purpose of emulating public-key digital signatures with symmetric cryptography only. Our findings suggest that some networks naturally grow into UNNs, but even if not, we can efficiently identify substructures that allow to pin down a node's identity based on its “friend-nodes” in a graph.