Asymptotic Analysis of Blind Reconstruction of BCH Codes Based on Consecutive Roots of Generator Polynomials

Abstract

In the military surveillance and security information systems, correct blind reconstruction of signal parameters from unknown signals is very important. Especially, many blind reconstruction methods of error-correcting codes have been proposed, and their theoretical performance analysis is essential for both the defender who wants to prevent information leakage and the challenger who wants to extract information from the intercepted signals. However, a proper performance analysis of most blind reconstruction methods has not been performed yet. Among many blind reconstruction methods of BCH codes proposed so far, the blind reconstruction method based on consecutive roots of generator polynomials proposed by Jo, Kwon, and Shin, called the JKS method, shows the best performance under the unknown channel information. However, the performance of the JKS method is only evaluated through simulation without performing theoretical analysis. In this paper, the JKS method is asymptotically analyzed under the binary symmetric channel with the cross-over probability p. Since the blind reconstruction performance heavily depends on how many and which received codewords are used even for the same channel environment, sufficiently many codewords are assumed to perform asymptotic analysis. More specifically, an asymptotic threshold on p, up to which blind reconstruction is successful, is derived when the number of received codewords is sufficiently large, which can be used as a new performance metric for blind reconstruction methods. Finally, the validity of the asymptotic analysis is confirmed through simulation.