Quantum stabilizer codes and beyond

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.Comment: Ph.D. Dissertation, Texas A&M University, 200

Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Contemporary Coding Theory

Coding Theory naturally lies at the intersection of a large number of disciplines in pure and applied mathematics. A multitude of methods and means has been designed to construct, analyze, and decode the resulting codes for communication. This has suggested to bring together researchers in a variety of disciplines within Mathematics, Computer Science, and Electrical Engineering, in order to cross-fertilize generation of new ideas and force global advancement of the field. Areas to be covered are Network Coding, Subspace Designs, General Algebraic Coding Theory, Distributed Storage and Private Information Retrieval (PIR), as well as Code-Based Cryptography

Unified field multiplier for GF(p) and GF(2 n) with novel digit encoding

In recent years, there has been an increase in demand for unified field multipliers for Elliptic Curve Cryptography in the electronics industry because they provide flexibility for customers to choose between Prime (GF(p)) and Binary (GF(2")) Galois Fields. Also, having the ability to carry out arithmetic over both GF(p) and GF(2") in the same hardware provides the possibility of performing any cryptographic operation that requires the use of both fields. The unified field multiplier is relatively future proof compared with multipliers that only perform arithmetic over a single chosen field. The security provided by the architecture is also very important. It is known that the longer the key length, the more susceptible the system is to differential power attacks due to the increased amount of data leakage. Therefore, it is beneficial to design hardware that is scalable, so that more data can be processed per cycle. Another advantage of designing a multiplier that is capable of dealing with long word length is improvement in performance in terms of delay, because less cycles are needed. This is very important because typical elliptic curve cryptography involves key size of 160 bits. A novel unified field radix-4 multiplier using Montgomery Multiplication for the use of G(p) and GF(2") has been proposed. This design makes use of the unexploited state in number representation for operation in GF(2") where all carries are suppressed. The addition is carried out using a modified (4:2) redundant adder to accommodate the extra 1 * state. The proposed adder and the partial product generator design are capable of radix-4 operation, which reduces the number of computation cycles required. Also, the proposed adder is more scalable than existing designs.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Soft Processing Techniques for Quantum Key Distribution Applications

This thesis deals with soft-information based information reconciliation and data sifting for Quantum Key Distribution (QKD). A novel composite channel model for QKD is identified, which includes both a hard output quantum channel and a soft output classic channel. The Log-Likelihood Ratios, - also called soft-metrics - derived from the two channels are jointly processed at the receiver, exploiting capacity achieving soft-metric based iteratively decoded block codes. The performance of the proposed mixed-soft-metric algorithms are studied via simulations as a function of the system parameters. The core ideas of the thesis are employing Forward Error Correction (FEC) coding as opposed to two-way communication for information reconciliation in QKD schemes, exploiting all the available information for data processing at the receiver including information available from the quantum channel, since optimized use of this information can lead to significant performance improvement, and providing a security versus secret-key rate trade-off to the end-user within the context of QKD system

Hardware Implementations for Symmetric Key Cryptosystems

The utilization of global communications network for supporting new electronic applications is growing. Many applications provided over the global communications network involve exchange of security-sensitive information between different entities. Often, communicating entities are located at different locations around the globe. This demands deployment of certain mechanisms for providing secure communications channels between these entities. For this purpose, cryptographic algorithms are used by many of today\u27s electronic applications to maintain security. Cryptographic algorithms provide set of primitives for achieving different security goals such as: confidentiality, data integrity, authenticity, and non-repudiation. In general, two main categories of cryptographic algorithms can be used to accomplish any of these security goals, namely, asymmetric key algorithms and symmetric key algorithms. The security of asymmetric key algorithms is based on the hardness of the underlying computational problems, which usually require large overhead of space and time complexities. On the other hand, the security of symmetric key algorithms is based on non-linear transformations and permutations, which provide efficient implementations compared to the asymmetric key ones. Therefore, it is common to use asymmetric key algorithms for key exchange, while symmetric key counterparts are deployed in securing the communications sessions. This thesis focuses on finding efficient hardware implementations for symmetric key cryptosystems targeting mobile communications and resource constrained applications. First, efficient lightweight hardware implementations of two members of the Welch-Gong (WG) family of stream ciphers, the WG$\left(29,11\right)$ and WG-$16$, are considered for the mobile communications domain. Optimizations in the WG$\left(29,11\right)$ stream cipher are considered when the $GF\left(2^{29}\right)$ elements are represented in either the Optimal normal basis type-II (ONB-II) or the Polynomial basis (PB). For WG-$16$, optimizations are considered only for PB representations of the $GF\left(2^{16}\right)$ elements. In this regard, optimizations for both ciphers are accomplished mainly at the arithmetic level through reducing the number of field multipliers, based on novel trace properties. In addition, other optimization techniques such as serialization and pipelining, are also considered. After this, the thesis explores efficient hardware implementations for digit-level multiplication over binary extension fields $GF\left(2^{m}\right)$. Efficient digit-level $GF\left(2^{m}\right)$ multiplications are advantageous for ultra-lightweight implementations, not only in symmetric key algorithms, but also in asymmetric key algorithms. The thesis introduces new architectures for digit-level $GF\left(2^{m}\right)$ multipliers considering the Gaussian normal basis (GNB) and PB representations of the field elements. The new digit-level $GF\left(2^{m}\right)$ single multipliers do not require loading of the two input field elements in advance to computations. This feature results in high throughput fast multiplication in resource constrained applications with limited capacity of input data-paths. The new digit-level $GF\left(2^{m}\right)$ single multipliers are considered for both the GNB and PB. In addition, for the GNB representation, new architectures for digit-level $GF\left(2^{m}\right)$ hybrid-double and hybrid-triple multipliers are introduced. The new digit-level $GF\left(2^{m}\right)$ hybrid-double and hybrid-triple GNB multipliers, respectively, accomplish the multiplication of three and four field elements using the latency required for multiplying two field elements. Furthermore, a new hardware architecture for the eight-ary exponentiation scheme is proposed by utilizing the new digit-level $GF\left(2^{m}\right)$ hybrid-triple GNB multipliers