Algebraic and quantum theoretical approach to coding on wiretap channels

Abstract

Algebraic codes designed on a criterion that has arisen from quantum cryptography are analyzed. The primary purpose of this analysis is to demonstrate the constructibility of codes for wiretap channels. Specifically, it is shown that reliable conjugate code pairs, or Calderbank-Shor-Steane (CSS) quantum error-correcting codes, can immediately be converted into reliable and secure codes for wiretap channels (Wyner, 1975). Here, a pair of linear codes is called a conjugate code pair if either of the pair contains the dual of the other; A code is called reliable if it has exponentially small decoding error probability, and it is called secure if the mutual information between the transmitted confidential message and the leaked data is exponentially small. It is argued that conjugate code pairs are applicable to wiretap channels whether they are classical or quantum theoretical. In particular, this implies that recently obtained conjugate code pairs that are constructible with polynomial complexity achieve positive rates for wiretap channels.