Secure information hiding based on computationally intractable problems

Abstract

In this work, a method for proving copyright ownership is presented that is based on Zero Knowledge Interactive Proof (ZKIP) protocols for computationally intractable problems. The utilized problem is the 3-coloring problem, which consists in assigning one of three available colors to the vertices of a graph so that no two adjacent vertices have the same color. Using the presumed computational intractability of this problem, the construction of large signatures is proposed so that they represent adjacency matrices of random, 3-colorable graphs. Since it is easy to construct large graphs with a prescribed 3-coloring of their vertices whereas it is difficult to discover such a 3-coloring, the owner of a copyrighted digital piece of work (e.g. image, audio, video) may easily generate a random 3-colorable graph, embed it in the digital object and then use the knowledge of the 3-coloring in debates over the object’s ownership. Due to the intractability of the 3-coloring problem, only the owner is able to produce it sufficiently fast in an ownership challenge so as to convince a third party that the graph was indeed embedded in the object by herself/himself. Since graphs with maximum possible resistance to well-known coloring algorithms is required, we exploit some relatively recent experimental and theoretical findings suggesting that hard 3-coloring instances are found among graphs having a vertices to edges ratio around a specific threshold value. The proposed scheme has the additional advantage that disclosing the signature is of no consequences since it is essentially the knowledge of a characteristic of the signature, i.e. the 3-coloring of the graph it represents, that enables one to use it as proof of ownership of some digital object that contains it. Even if someone managed to locate and extract the signature, to use it would require a fast solution to a computationally intractable problem on some hard instance. Our proposal represents a shift from signatures that are simply viewed as bit sequences to signatures with properties that stem from their interpretation as instances of computationally intractable problems.