The impact of finite-alphabet input on the secrecy-achievable rates for broadcast channel with confidential message

Abstract

This paper investigates the maximization of the secrecyachievable rate region for the Gaussian broadcast channel with confidential message (BCCM) using finite input constellations. The maximization is done jointly over symbol positions and their joint probabilities. The secrecy-achievable rate regions are given for various broadcast strategies which differ in their complexity of implementation. We compare these strategies in terms of improvement in achievable rates and we study the impact of finite input alphabet on the secrecyachievable rates. It is shown that finite alphabet constraints may change well known results holding in the Gaussian case.

Achievable Secrecy Rates for the Broadcast Channel with Confidential Message and Finite Constellation Inputs

“…is a concave function in D where D is the set of input alphabets with a minimum spacing between symbols greater than d and d is a function of the SNR and of the constellation size [15]. This condition was observed in experiments for most values of studied SNR (except when the value of s is very high such that s…”

“…Therefore, the problem is now to choose an appropriate initial point. It is observed in [15] that the size of the region T k,θ (P U X ) where the objective function in (10) is concave in P U X is larger when θ increases. Thus we have more chance that the algorithm converges from a random initial guess in this case.…”

“…Thus we have more chance that the algorithm converges from a random initial guess in this case. In our experiments, the initial guesses are chosen randomly (avoiding "Degenerate cases" such as uniform distribution, distribution with similarities [15] and distribution with null elements) and the Blahut-Arimoto type algorithm is observed to converge to reasonable solutions, since the resulting rate regions have a very smooth shape. In the case of general superposition coding, the algorithm converges to one of the M !…”

“…The achievable rate regions are also given for various broadcast transmission strategies which differ in their complexity of implementation. Preliminary and partial results were published in [15] by the same authors. The whole picture is given here.…”

“…The whole picture is given here. Additional contributions of this paper compared to [15] are specified hereafter. Regarding the achievable rate regions for the BCCM, comparisons between the various strategies are conducted, in this paper, in terms of SNR savings for target achievable rates and percentage of gain in achievable rates.…”