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International Symposium on Information Theory and its Applications

Coding Theoretic Study of Secure Network Coding and Quantum Secret Sharing

Ryutaroh Matsumoto

Published Dec 11, 2014• 0 citations
OpenAlex

Abstract

The common goal of (classical) secure net- work coding and quantum secret sharing is to encode secret so that an adversary has as little information of the secret as possible. Both can be described by a nested pair of classical linear codes, while the strategies available to the adversary are di erent. The security properties of both schemes are closely related to combinatorial properties of the underlying linear codes. We survey connections among them.

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The common goal of (classical) secure net- work coding and quantum secret sharing is to encode secret so that an adversary has as little information of the secret as possible. Both can be described by a nested pair of classical linear codes, while the strategies available to the adversary are di erent. The security properties of both schemes are closely related to combinatorial properties of the underlying linear codes. We survey connections among them.
Abstract Quantum secret sharing has wide application prospects in quantum communication, quantum network and other fields. It can provide a secure and reliable information sharing method for multi-party communication. In this paper, we mainly study the connection mechanism between the quantum secret-sharing scheme and quantum error-correcting codes. Firstly, we give the necessary and sufficient conditions for the existence of quantum error-correcting codes based on group algebra theory. Secondly, we give the corresponding quantum secret sharing scheme model with local quantum operation and classical communication capability can be constructed by any given quantum error-correcting codes. The advantage of this model is that the corresponding quantum secret-sharing scheme can be found for any quantum error-correcting code.
We explore the conversion of classical secret-sharing schemes to quantum ones, and how this can be used to give efficient QSS schemes for general adversary structures. Our first result is that quantum secret-sharing is possible for any structure for which no two disjoint sets can reconstruct the secret (this was also proved, somewhat differently, by D. Gottesman). To obtain this we show that a large class of linear classical SS schemes can be converted into quantum schemes of the same efficiency. We also give a necessary and sufficient condiion for the direct conversion of classical schemes into quantum ones, and show that all group homomorphic schemes satisfy it.
We consider the secure quantum communication over a network with the presence of a malicious adversary who can eavesdrop and contaminate the states. The network consists of noiseless quantum channels with the unit capacity and the nodes which applies noiseless quantum operations. As the main result, when the maximum number m1of the attacked channels over the entire network uses is less than a half of the network transmission rate m0(i.e., m10/2), our code implements secret and correctable quantum communication of the rate m0- 2m1by using the network asymptotic number of times. Our code is universal in the sense that the code is constructed without the knowledge of the specific node operations and the network topology, but instead, every node operation is constrained to the application of an invertible matrix to the basis states. Moreover, our code requires no classical communication. Our code can be thought of as a generalization of the quantum secret sharing.
Helper quantum secret sharing is a form of secret sharing defined by its unique access structure. One special fixed party, called the helper, can work together with any other party to fully decode the secret. A blind helper is one who can provide this assistance while not holding any local information about the encoded secret. In this work, we analyze the general structure of QSS helper codes and present new code constructions. We fully characterize the structure of blind helper stabilizer codes and show that for the encoding of a single qubit, recovery is always possible using one-way local operations and classical communication (LOCC) from the helper to the targeted party. Furthermore, we demonstrate how such codes also allow for the helper to target larger subsets of parties by one-way LOCC, enabling them to be authorized to recover the secret. Finally, when each party only holds a qubit, we identify the general form of all helper codes (including non-stabilizer codes) and find that LOCC recovery is only possible in special cases.