arXiv:1001.1768v2 [cs.IT] 13 Jan 2010 1 On the Secure DoF of the Single-Antenna MAC Ghadamali Bagherikaram, Abolfazl S. Motahari, Amir K. Khandani Coding and Signal Transmission Laboratory, University of Waterloo, Ontario, Canada Emails: { gbagheri,abolfazl,khandani } @cst.uwaterloo.ca Abstract —A new achievability rate region for the secure dis- crete memoryless Multiple-Access-Channel (MAC) is presented. Thereafter, a novel secure coding scheme is proposed to achieve a positive Secure Degrees-of-Freedom (S-DoF) in the single- antenna MAC. This scheme converts the single-antenna system into a multiple-dimension system with fractional dimensions. The achievability scheme is based on the alignment of signals into a small sub-space at the eavesdropper, and the simultaneous separation of the signals at the intended receiver. Tools from the field of Diophantine Approximation in number theory are used to analyze the probability of error in the coding scheme. The notion of information theoretic secrecy in communi- cation systems was first introduced in [1]. The information theoretic secrecy requires that the received signal of the eaves- dropper does not provide any information about the transmitted messages. Following the pioneering works of [2] and [3] which studied the wiretap channel, many multi-user channel models have been considered from a perfect secrecy point of view. The secure Gaussian multiple-access-channel with an external eavesdropper is introduced in [4], [5]. This channel consists of an ordinary Gaussian MAC and an external eavesdropper. The capacity region of this channel is still an open problem in the information theory field. For this channel, an achievabl e rate scheme based on Gaussian codebooks is proposed in [4], where also the sum secrecy capacity of the degraded Gaussian channel is also found. On the other hand, it is shown that the random coding argument may be insufficient to prove capacity theorems for certain channels; instead, structure codes can be used to construct efficient channel codes for Gaussian channels. In reference [6], nested lattice codes are used to provide secrecy in two-user Gaussian channels. In [6] it is shown that structure codes can achieve a positive S-DoF in a two-user MAC. Especially, the achievability scheme of [6] provides a S-DoF of 1 2 for a small category of channel gains and for the other categories, it provides a S-DoF of strictly less than 1 $_{2}$. In reference [7], the concept of interference alignment is introduced and has illustrated its capability in achieving the full DoF of a class of two-user X channels. In reference [8], a novel coding scheme applicable in networks with single antenna nodes is proposed. This scheme converts a single antenna system into an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. In this work we use the notion of real alignment of [8] to prove that for almost all channel gains in the secure K user single- antenna Gaussian MAC, we can achieve the S-DoF of K − 1 K . Consider a secure K -user Gaussian single-antenna Multiple- Access-Channel (MAC). In this confidential setting, each us er k ( k ∈ K △ = { 1 , 2 , ..., K } ) wishes to send the message W$_{k}$ to the intended receiver in n uses of the channel, simultaneously and prevent the eavesdropper from having any information about the messages. At a specific time, the signals received by the intended receiver and the eavesdropper is given by The scheme of this work differs from that of [6], in the sense that our scheme achieves the S-DoF of 1 2 for almost all channel gains.Therefore, the carve of S-DoF versus channel gains is almost certainly constant. The rest of the paper is organized as follows: Section II provides some background and preliminaries. In section III, we present our results for the achievable S-DoF of the single- antenna MAC. Finally, section IV concludes the paper. where X$_{k}$ for k ∈ K is a real input scalar under an input average power constraint. We require that E [ X 2 $_{k}$] ≤ P , Y and Z be real output scalars which are received by the destination and the eavesdropper, respectively, h$_{k}$ and h$_{k,e}$ for k = 1 , 2 , ..., K are fixed, real scalars which model the channel gains between the transmitters and the intended receiver and the eavesdropper, respectively. The channel state information is assumed to be known perfectly at all the transmitters and at all receivers, and ˜ W$_{1}$ , ˜ W$_{2}$ are real Gaussian random variables with zero means and unit variances. Let X n $_{k}$, Y n and Z n denote the random channel inputs and random channel outputs over a block of n samples. Furthermore, let ˜ W n 1 , and ˜ W n 2 denote the additive noises of the channels. The elements of ˜ W n 1 and ˜ W n 2 are independent zero mean Gaussian random variables with unit variances. In addition, ˜ W n 1 and ˜ W n 2 are independent of X n $_{k}$’s and W$_{k}$ ’s. A ((2 $^{nR$_{1}$}$, 2 $^{nR$_{2}$}$, ..., 2 $^{nR$_{k}$}$) , n ) secret code for the above channel consists of the following components: 1 ) K secret message sets W$_{k}$ = { 1 , 2 , ..., 2 $^{nR$_{k}$}$} . 2 ) K stochastic encoding functions f$_{k}$ ( . ) which map the se- cret messages to the transmitted symbols, i.e., f$_{k}$ : w$_{k}$ → X n k for each w$_{k}$ ∈ W$_{k}$ . At encoder k , each codeword is designed according to the transmitter’s average power constraint P . 3 ) A decoding function φ ( . ) which maps the received symbols to estimate the messages: φ ( Y $^{n}$) → ( ˆ W$_{1}$, ..., ˆ W$_{K}$ ) . The reliability of the transmission is measured by the average probability of error, which is defined as the probabi lity Y = K ∑ k =1 h$_{k}$X$_{k}$ + ˜ W$_{1}$ (1) Z = K k =1 h$_{k,e}$X$_{k}$ + W$_{2}$, I. INTRODUCTION II. PRELIMINARIES $^{0}$Financial support provided by Nortel, and the corresponding matching funds by the Natural Sciences and Engineering Research Council of Canada (NSERC), and Ontario Ministry of Research & Innovation (ORF-RE) are gratefully acknowledged.