Polar coding for achieving the capacity of marginal channels in nonbinary-input setting

Abstract

Achieving information-theoretic security using explicit coding scheme in which unlimited computational power for eavesdropper is assumed, is one of the main topics is security consideration. It is shown that polar codes are capacity achieving codes and have a low complexity in encoding and decoding. It has been proven that polar codes reach to secrecy capacity in the binary-input wiretap channels in symmetric settings for which the wiretapper's channel is degraded with respect to the main channel. The first task of this paper is to propose a coding scheme to achieve secrecy capacity in asymmetric nonbinary-input channels while keeping reliability and security conditions satisfied. Our assumption is that the wiretap channel is stochastically degraded with respect to the main channel and message distribution is unspecified. The main idea is to send information set over good channels for Bob and bad channels for Eve and send random symbols for channels that are good for both. In this scheme the frozen vector is defined over all possible choices using polar codes ensemble concept. We proved that there exists a frozen vector for which the coding scheme satisfies reliability and security conditions. It is further shown that uniform distribution of the message is the necessary condition for achieving secrecy capacity.

A Novel Key Generation Scheme Using Quaternary PUF Responses and Wiretap Polar Coding

“…In a wiretap channel, Alice sends messages to Bob through main channels, while an eavesdropper Eve receives the messages from wiretap channels. Wiretap polar coding is proposed to protect the communications between Alice and Bob in [9]. If the wiretap channel is stochastically degraded with respect to the main channel, Bob can recover the sent messages with high probability, whereas the probability that Eve can recover the message approaches zero [9].…”

“…Wiretap polar coding is proposed to protect the communications between Alice and Bob in [9]. If the wiretap channel is stochastically degraded with respect to the main channel, Bob can recover the sent messages with high probability, whereas the probability that Eve can recover the message approaches zero [9]. Let the transition matrices of the wiretap and main channels be W w and W m , respectively.…”

“…Let the transition matrices of the wiretap and main channels be W w and W m , respectively. If there is a channel with a transition matrix W 3 that holds W w = W 3 W m , the wiretap channel will be stochastically degraded with respect to the main channel so that the channels that are good for Eve are good for Bob too and the channels that are bad for Bob are bad for Eve too [9]. Hence, in the wiretap polar coding, indices {0, 1, • • • , N − 1} is divided into three disjoint subsets: R: Indices of the channels good for both Bob and Eve, S: Indices of the channels good for Bob but bad for Eve, F: Indices of the channels bad for both Bob and Eve.…”

“…As long as the wiretap channel is stochastically degraded with respect to the main channel, the non-binary wiretap polar codes of [9] can be applied in our fuzzy extractor. The genieaided simulation of [8] is used to construct the polar codes.…”

“…The wiretap polar coding in [9] is used in our FE to prevent eavesdroppers from recovering secret seed s. The N × N generator matrix G of a quaternary polar code with length N is written as G R G S G F T , where G R , G S , and G F consist of rows of G that are indexed by R, S, and F, respectively. G R G S T and G R will be the generator matrix of an (N, r + s) linear code C 1 and an (N, r) linear code C 2 , respectively, where r and s are the size of R and S, respectively.…”