Variations of the McEliece Cryptosystem

Two variations of the McEliece cryptosystem are presented. The first one is based on a relaxation of the column permutation in the classical McEliece scrambling process. This is done in such a way that the Hamming weight of the error, added in the encryption process, can be controlled so that efficient decryption remains possible. The second variation is based on the use of spatially coupled moderate-density parity-check codes as secret codes. These codes are known for their excellent error-correction performance and allow for a relatively low key size in the cryptosystem. For both variants the security with respect to known attacks is discussed

Update-Efficiency and Local Repairability Limits for Capacity Approaching Codes

Motivated by distributed storage applications, we investigate the degree to which capacity achieving encodings can be efficiently updated when a single information bit changes, and the degree to which such encodings can be efficiently (i.e., locally) repaired when single encoded bit is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible, and establish that the number of encoded bits that must change in response to a change in a single information bit must scale logarithmically in the block-length of the code if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost bit of the encoding. In particular, we show that if the code-rate is $\epsilon$ less than the capacity, then for optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as $\log1/\epsilon$. Several variations on---and extensions of---these results are also developed.Comment: Accepted to appear in JSA

Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes

Communication system links that do not have the ability to retransmit generally rely on forward error correction (FEC) techniques that make use of error correcting codes (ECC) to detect and correct errors caused by the noise in the channel. There are several ECC’s in the literature that are used for the purpose. Among them, the low density parity check (LDPC) codes have become quite popular owing to the fact that they exhibit performance that is closest to the Shannon’s limit. This thesis proposes a novel code-construction method for constructing not only (3, k) regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC codes is made because it has low decoding complexity and has a Hamming distance, at least, 4. In this work, the proposed code-construction consists of information submatrix (Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design of the proposed code-construction utilizes expanded deterministic base matrices in three stages. Deterministic base matrix of parity part starts with triple diagonal matrix while deterministic base matrix of information part utilizes matrix having all elements of ones. The proposed matrix H is designed to generate various code rates (R) by maintaining the number of rows in matrix H while only changing the number of columns in matrix Hinf. All the codes designed and presented in this thesis are having no rank-deficiency, no pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of 4-cycles and low encoding complexity of the order of (N + g2) where g2«N. The proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB from Shannon limit at bit error rate (BER) of 10 −6 when the code rate greater than R = 0.875. They have comparable BER and block error rate (BLER) performance with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from Shannon limit at BER 10 −6 with encoding complexity (1.0225 N), for R = 0.928 and N = 14364 – a result that no other published techniques can reach

Homological Product Codes

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $\sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.Comment: 49 page