Parallel Privacy-Preserving Shortest Path Algorithms

Abstract

In this paper, we propose and present secure multiparty computation (SMC) protocols for single-source shortest distance (SSSD) and all-pairs shortest distance (APSD) in sparse and dense graphs. Our protocols follow the structure of classical algorithms—Bellman–Ford and Dijkstra for SSSD; Johnson, Floyd–Warshall, and transitive closure for APSD. As the computational platforms offered by SMC protocol sets have performance profiles that differ from typical processors, we had to perform extensive changes to the structure (including their control flow and memory accesses) and the details of these algorithms in order to obtain good performance. We implemented our protocols on top of the secret sharing based protocol set offered by the Sharemind SMC platform, using single-instruction-multiple-data (SIMD) operations as much as possible to reduce the round complexity. We benchmarked our protocols under several different parameters for network performance and compared our performance figures against each other and with ones reported previously.

A Parallel Privacy-Preserving Shortest Path Protocol from a Path Algebra Problem

“…The privacy-preserving Bellman-Ford protocol is more efficient then its versions 1 and 2 in [1] on SMC sharemind platform. The computation of public edges's version has the lowest round complexity among the computation parties of the sharemind because of using the public edges.…”

“…This computation can be done in the same algorithmic structure as the Floyd-Warshall algorithm with some changes based on the semiring framework ⊕. Thereby, we use the parallel version of the Floyd-Warshall algorithm presented in Algorithm 8 in [1] to perform this computation. It is important to note that this algorithm performs the computation on one adjacency matrix.…”

“…To use an efficient protocol in solving such a problem, we propose a version of the privacy-preserving Bellman-Ford protocol with public edges, presented in Algorithm 5. In general, the Bellman-Ford protocol has the same algorithmic structure and functionality as Algorithm 1 in [1]. The difference between them is that we replaced PrefixMin2 by getMin, both functions have the same functionality with a difference that getMin-function deals with public edges m and vertices n. This is the reason why getMin-function is faster than the two versions of the prefixMin2-function (Algorithm 3 and Algorithm 4 in [1]).…”

“…In general, the Bellman-Ford protocol has the same algorithmic structure and functionality as Algorithm 1 in [1]. The difference between them is that we replaced PrefixMin2 by getMin, both functions have the same functionality with a difference that getMin-function deals with public edges m and vertices n. This is the reason why getMin-function is faster than the two versions of the prefixMin2-function (Algorithm 3 and Algorithm 4 in [1]). The second difference is that no use for Laud's protocol [26] with its functions, prepareRead and performRead, this will reduce the round complexity a bit.…”

“…Single-Instruction-Multiple-Data (SIMD) is a parallel framework used to perform parallel computation for multiple data using single instruction simultaneously [19]. Recently, SIMD principles have been used to reduce the round complexities in many privacy-preserving graph algorithms, including minimum spanning tree [3,26] and shortest path [1,4,2]. These privacy-preserving graph protocols are constructed on top of SMC protocols, and they are capable to process sizeable private data sets, where both the location and weight of edges are private.…”