Security of Graph Data: Hashing Schemes and Definitions

Use of graph-structured data models is on the rise - in graph databases, in representing biological and healthcare data as well as geographical data. In order to secure graph-structured data, and develop cryptographically secure schemes for graph databases, it is essential to formally define and develop suitable collision resistant one-way hashing schemes and show them they are efficient. The widely used Merkle hash technique is not suitable as it is, because graphs may be directed acyclic ones or cyclic ones. In this paper, we are addressing this problem. Our contributions are: (1) define the practical and formal security model of hashing schemes for graphs, (2) define the formal security model of perfectly secure hashing schemes, (3) describe constructions of hashing and perfectly secure hashing of graphs, and (4) performance results for the constructions. Our constructions use graph traversal techniques, and are highly efficient for hashing, redaction, and verification of hashes graphs. We have implemented the proposed schemes, and our performance analysis on both real and synthetic graph data sets support our claims

Cumulative object categorization in clutter

In this paper we present an approach based on scene- or part-graphs for geometrically categorizing touching and occluded objects. We use additive RGBD feature descriptors and hashing of graph configuration parameters for describing the spatial arrangement of constituent parts. The presented experiments quantify that this method outperforms our earlier part-voting and sliding window classification. We evaluated our approach on cluttered scenes, and by using a 3D dataset containing over 15000 Kinect scans of over 100 objects which were grouped into general geometric categories. Additionally, color, geometric, and combined features were compared for categorization tasks

Navigating in the Cayley graph of SL2(Fp)SL_2(F_p) and applications to hashing

Cayley hash functions are based on a simple idea of using a pair of (semi)group elements, $A$ and $B$, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the (semi)group. In this paper, we focus on hashing with $2 \times 2$ matrices over $F_p$. Since there are many known pairs of $2 \times 2$ matrices over $Z$ that generate a free monoid, this yields numerous pairs of matrices over $F_p$, for a sufficiently large prime $p$, that are candidates for collision-resistant hashing. However, this trick can "backfire", and lifting matrix entries to $Z$ may facilitate finding a collision. This "lifting attack" was successfully used by Tillich and Z\'emor in the special case where two matrices $A$ and $B$ generate (as a monoid) the whole monoid $SL_2(Z_+)$. However, in this paper we show that the situation with other, "similar", pairs of matrices from $SL_2(Z)$ is different, and the "lifting attack" can (in some cases) produce collisions in the group generated by $A$ and $B$, but not in the positive monoid. Therefore, we argue that for these pairs of matrices, there are no known attacks at this time that would affect security of the corresponding hash functions. We also give explicit lower bounds on the length of collisions for hash functions corresponding to some particular pairs of matrices from $SL_2(F_p)$.Comment: 10 page