Vishwambhar Rathi

Nested Polar Codes for Wiretap and Relay Channels

We show that polar codes asymptotically achieve the whole capacity-equivocation region for the wiretap channel when the wiretappers channel is degraded with respect to the main channel, and the weak secrecy notion is used. Our coding scheme also achieves the capacity of the physically degraded receiver-orthogonal relay channel. We show simulation results for moderate block length for the binary erasure wiretap channel, comparing polar codes and two edge type LDPC codes.

Rate-equivocation optimal spatially coupled LDPC codes for the BEC wiretap channel

We consider transmission over a wiretap channel where both the main channel and the wiretappers channel are Binary Erasure Channels (BEC). We use regular convolutional LDPC ensembles, introduced by Felström and Zigangirov, together with Wyners coset encoding scheme. We show that such a construction achieves the whole rate-equivocation region of the BEC wiretap channel. This result is based on the recent observation by Kudekar, Richardson, and Urbanke who proved that convolutional LDPC ensembles exhibit a “threshold saturation” phenomenon which converts the MAP threshold into the BP threshold for transmission over the BEC. Although our present result is less general (since we only consider the BEC) than the elegant code constructions based on polar codes which were recently introduced by several research groups, we see two potential advantages which we believe makes our construction worth considering. First, the proposed codes have a significantly better performance already for moderate lengths. Second, and perhaps more importantly, the proposed construction has the potential of being universal. More precisely, the phenomenon of spatial coupling has been observed empirically to hold for general binary memoryless symmetric channels as well. Hence, we conjecture that our construction is a universal rate-equivocation achieving construction when the main channel and wiretappers channel are binary memoryless symmetric channels, and the wiretappers channel is degraded with respect to the main channel.

Polar Codes for Cooperative Relaying

We consider the symmetric discrete memoryless relay channel with orthogonal receiver components and show that polar codes are suitable for decode-and-forward and compress-and-forward relaying. In the first case we prove that polar codes are capacity achieving for the physically degraded relay channel; for stochastically degraded relay channels our construction provides an achievable rate. In the second case we construct sequences of polar codes that achieve the compress-and-forward rate by nesting polar codes for source compression into polar codes for channel coding. In both cases our constructions inherit most of the properties of polar codes. In particular, the encoding and decoding algorithms and the bound on the block error probability O(2-Nβ) which holds for any 0<;β<;1/2.

Performance Analysis and Design of Two Edge-Type LDPC Codes for the BEC Wiretap Channel

We consider transmission over a wiretap channel where both the main channel and the wiretappers channel are binary erasure channels (BEC). A code construction method is proposed using two edge-type low-density parity-check (LDPC) codes based on the coset encoding scheme. Using a single edge-type LDPC ensemble with a given threshold over the BEC, we give a construction for a two edge-type LDPC ensemble with the same threshold. If the given single edge-type LDPC ensemble has degree two variable nodes, our construction gives rise to degree one variable nodes in the code used over the main channel. This results in zero threshold over the main channel. In order to circumvent this problem, the degree distribution of the two edge-type LDPC ensemble is numerically optimized. We find that the resulting ensembles are able to perform close to the boundary of the rate-equivocation region of the wiretap channel. Further, a method to compute the ensemble average equivocation of two edge-type LDPC ensembles is provided by generalizing a recently published approach to measure the equivocation of single edge-type ensembles for transmission over the BEC in the point-to-point setting. From this analysis, we find that relatively simple constructions give very good secrecy performance.

Two edge type LDPC codes for the wiretap channel

We consider transmission over a wiretap channel where both the main channel and the wiretappers channel are Binary Erasure Channels (BEC). We propose a code construction using two edge type LDPC codes based on the method of Thangaraj, Dihidar, Calderbank, McLaughlin and Merolla. The advantage of our construction is that we can easily calculate the threshold over the main channel. Using standard LDPC codes with a given threshold over the BEC we give a construction for a two edge type LDPC code with the same threshold. Since this construction gives a code for the main channel with threshold zero we also give numerical methods to find two edge type LDPC codes with non-zero threshold for the main channel.

Polar codes for compress-and-forward in binary relay channels

We construct polar codes for binary relay channels with orthogonal receiver components. We show that polar codes achieve the cut-set bound when the channels are symmetric and the relay-destination link supports compress-and-forward relaying based on Slepian-Wolf coding. More generally, we show that a particular version of the compress-and-forward rate is achievable using polar codes for Wyner-Ziv coding. In both cases the block error probability can be bounded as O(2−Nβ) for 0 &#60; β &#60; 1over2 and sufficiently large block length N.

Some Results on MAP Decoding of Non-Binary LDPC Codes Over the BEC

In this paper, the transmission over the binary erasure channel (BEC) using non-binary LDPC (NBLDPC) codes is considered. The concept of peeling decoder and stopping sets is generalized to NBLDPC codes. Using these generalizations, a combinatorial characterization of decoding failures of NBLDPC codes is given, under assumption that the Belief Propagation (BP) decoder is used. Then, the residual ensemble of codes resulted by the BP decoder is defined and the design rate and the expectation of total number of codewords of the residual ensemble are computed. The decoding failure criterion combined with the density evolution analysis helps us to compute the asymptotic residual degree distribution for NBLDPC codes. Our approach to compute the residual degree distribution on the check node side is not efficient as it is based on enumeration of all the possible connections on the check node side which satisfy the decoding failure criterion. So, the computation of the asymptotic check node side residual degree distribution and further part of our analysis is performed for NBLDPC codes over GF2m with m = 2 . In order to show that asymptotically almost every code in such LDPC ensemble has a rate equal to the design rate, we generalize the argument of the Maxwell construction to NBLDPC codes, defined over GF22. It is also observed that, like in the binary setting, the Maxwell construction, relating the performance of MAP and BP decoding holds in this setting.

Equivocation of eve using two edge type LDPC codes for the binary erasure wiretap channel

We consider transmission over a binary erasure wiretap channel using the code construction method introduced by Rathi et al. based on two edge type Low-Density Parity-Check (LDPC) codes and the coset encoding scheme. By generalizing the method of computing conditional entropy for standard LDPC ensembles introduced by Méasson, Montanari, and Urbanke to two edge type LDPC ensembles, we show how the equivocation for the wiretapper can be computed. We find that relatively simple constructions give very good secrecy performance and are close to the secrecy capacity.

Bounds on threshold of regular random k -SAT

We consider the regular model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad et. al. in [6]. In [6], it was shown that the threshold for regular random 2-SAT is equal to unity. Also, upper and lower bound on the threshold for regular random 3-SAT were derived. Using the first moment method, we derive an upper bound on the threshold for regular random k-SAT for any k≥3 and show that for large k the threshold is upper bounded by 2k ln (2). We also derive upper bounds on the threshold for Not-All-Equal (NAE) satisfiability for k≥3 and show that for large k, the NAE-satisfiability threshold is upper bounded by 2k−1 ln (2). For both satisfiability and NAE-satisfiability, the obtained upper bound matches with the corresponding bound for the uniform model of formula generation [9,1]. For the uniform model, in a series of break through papers Achlioptas, Moore, and Peres showed that a careful application of the second moment method yields a significantly better lower bound on threshold as compared to any rigorously proven algorithmic bound [3,1]. The second moment method shows the existence of a satisfying assignment with uniform positive probability (w.u.p.p.). Thanks to the result of Friedgut for uniform model [10], existence of a satisfying assignment w.u.p.p. translates to existence of a satisfying assignment with high probability (w.h.p.). Thus, the second moment method gives a lower bound on the threshold. As there is no known Friedgut type result for regular random model, we assume that for regular random model existence of a satisfying assignments w.u.p.p. translates to existence of a satisfying assignments w.h.p. We derive the second moment of the number of satisfying assignments for regular random k-SAT for k≥3. There are two aspects in deriving the lower bound using the second moment method. The first aspect is given any k, numerically evaluate the lower bound on the threshold. The second aspect is to derive the lower bound as a function of k for large enough k. We address the first aspect and evaluate the lower bound on threshold. The numerical evaluation suggests that as k increases the obtained lower bound on the satisfiability threshold of a regular random formula converges to the lower bound obtained for the uniform model. Similarly, we obtain lower bounds on the NAE-satisfiability threshold of the regular random formulas and observe that the obtained lower bound seems to converge to the corresponding lower bound for the uniform model as k increases.

Binary weight distribution of non-binary LDPC codes

This paper is the first part of an investigation if the capacity of a binary-input memoryless symmetric channel under ML decoding can be achieved asymptotically by using non-binary LDPC codes. We consider (l, r)-regular LDPC codes both over finite fields and over the general linear group and compute their asymptotic binary weight distributions in the limit of large blocklength and of large alphabet size. A surprising fact, the average binary weight distributions that we obtain do not tend to the binomial one for values of normalized binary weights ω smaller than 1−2−l=r. However, it does not mean that non-binary codes do not achieve the capacity asymptotically, but rather that there exists some exponentially small fraction of codes in the ensemble, which contains an exponentially large number of codewords of poor weight. The justification of this fact is beyond the scope of this paper and will be given in [1].