On Communication Complexity of Secure Message Transmission in Directed Networks

• Published in: Distributed Computing and Networking pp. 42 - 53
• Main Authors: Patra, Arpita, Choudhary, Ashish, Rangan, C. Pandu
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Communication Complexity; Directed Network; Length Vector; Secret Share Scheme; Undirected Network
• ISBN: 9783642113215; 3642113214
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-11322-2_9

We re-visit the problem of perfectly secure message transmission (PSMT) in a directed network under the presence of a threshold adaptive Byzantine adversary, having unbounded computing power. Specifically, we derive the lower bounds on communication complexity of (a) two phase PSMT protocols and (b) three or more phase PSMT protocols in directed networks. Moreover, we show that our lower bounds are asymptotically tight, by designing communication optimal PSMT protocols in directed networks, which are first of their kind. We re-visit the problem of perfectly reliable message transmission (PRMT) as well. Any PRMT protocol that sends a message containing ℓ field elements by communicating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal O}(\ell)$\end{document} field elements, is referred as communication optimal PRMT or PRMT with constant factor overhead. Here, we characterize the class of directed networks over which communication optimal PRMT or PRMT with constant factor overhead is possible. Moreover, we design a communication optimal PRMT over a directed network that satisfies the conditions stated in our characterization.