On Extremal Rates of Secure Storage over Graphs

Abstract

A secure storage code maps K source symbols, each of L w bits, to N coded symbols, each of L v bits, such that each coded symbol is stored in a node of a graph. Each edge of the graph is either associated with D of the K source symbols such that from the pair of nodes connected by the edge, we can decode the D source symbols and learn no information about the remaining K − D source symbols; or the edge is associated with no source symbols such that from the pair of nodes connected by the edge, nothing about the K source symbols is revealed. The ratio L w /L v is called the symbol rate of a secure storage code and the highest possible symbol rate is called the capacity.We characterize all graphs over which the capacity of a secure storage code is equal to 1, when D = 1. This result is generalized to D > 1, i.e., we characterize all graphs over which the capacity of a secure storage code is equal to 1/D under a mild condition that for any node, the source symbols associated with each of its connected edges do not include a common element. Further, we characterize all graphs over which the capacity of a secure storage code is equal to 2/D.

On Extremal Rates of Storage over Graphs

“…Specifically, all extremal graphs with storage code capacity 4/3 appear to go beyond the techniques of this work. Regarding generalizations, we note that our model is the most elementary, where we have focused on the highest capacity values, i.e., best rate scenarios instead of lowest capacity values, i.e., worst rate scenarios, or other physically meaningful rates; decoding constraints are placed on a pair of nodes in this work instead of an arbitrary set of nodes, i.e., we may have a hypergraph rather than a graph [2]; each edge is associated with only one source symbols instead of multiple source symbols where the decoding structure can be more diverse [1]. Finally, from an extremal rate and network perspective, we may view combinatorial objects using the metric of capacity and study further extremal (largest, densest, most (linearly) independent) graphs, set families, vector spaces etc.…”

“…Note that for a special 2-color node, at least one of V [1] , V [2] will be one symbol so that V will contain no more than three symbols (when a node V is simultaneously W 1 -special and W 2 -special, V will have only two symbols and we may zero-pad to make its length three). This completes the assignment for all 2-color nodes and the code construction is complete.…”

“…Motivated by the heterogeneity of modern distributed storage systems, a storage code problem over graphs is introduced in [1,2], where a storage code maps K independent source symbols,…”

“…The metric of pursuit is the capacity C of a storage code over a graph, i.e., the highest possible symbol rate, defined as L w /L v , where L w (L v ) is the number of bits contained in each source (coded) symbol and L w /L v represents the number of source symbol bits reliably stored in each coded symbol bit. The graph based storage code problem is not new in the sense that it can be equivalently transformed to a network coding problem [1][2][3][4] and adding further security constraints (i.e., beyond desired data decodability, leakage about other source symbols is prevented), it is intimately related to conditional disclosure of secrets [5][6][7][8] and secret sharing [9,10]. What is new is the view brought by [1] -finding extremal networks/graphs.…”

“…The graph based storage code problem is not new in the sense that it can be equivalently transformed to a network coding problem [1][2][3][4] and adding further security constraints (i.e., beyond desired data decodability, leakage about other source symbols is prevented), it is intimately related to conditional disclosure of secrets [5][6][7][8] and secret sharing [9,10]. What is new is the view brought by [1] -finding extremal networks/graphs. Instead of first fixing the network/graph and then finding its highest rate, we focus on the extremal (highest) capacity values and aim to find the networks/graphs whose capacity is equal to the extremal values (see Fig.…”