Algebraic methods for the distance of cyclic codes

In this thesis we provide known and new results which explain the relationship between the actual minimum distance of cyclic codes, bounds that use only information on the defining sets of cyclic codes to lower bound the distance (root bounds) and bounds that also need the knowledge of the defining sets of all cyclic subcodes (border bounds). We propose a new bound which is provably better of many known bounds and that can be computed in polynomial time with respect to the length of the code. We sketch how to use the generalized Newton identities to give alternative proofs of known bounds. Finally, we use Groebner bases to prove that the optimal root bound can be computed in finite time

Implementation and Evaluation of Algorithmic Skeletons: Parallelisation of Computer Algebra Algorithms

This thesis presents design and implementation approaches for the parallel algorithms of computer algebra. We use algorithmic skeletons and also further approaches, like data parallel arithmetic and actors. We have implemented skeletons for divide and conquer algorithms and some special parallel loops, that we call ‘repeated computation with a possibility of premature termination’. We introduce in this thesis a rational data parallel arithmetic. We focus on parallel symbolic computation algorithms, for these algorithms our arithmetic provides a generic parallelisation approach. The implementation is carried out in Eden, a parallel functional programming language based on Haskell. This choice enables us to encode both the skeletons and the programs in the same language. Moreover, it allows us to refrain from using two different languages—one for the implementation and one for the interface—for our implementation of computer algebra algorithms. Further, this thesis presents methods for evaluation and estimation of parallel execution times. We partition the parallel execution time into two components. One of them accounts for the quality of the parallelisation, we call it the ‘parallel penalty’. The other is the sequential execution time. For the estimation, we predict both components separately, using statistical methods. This enables very confident estimations, although using drastically less measurement points than other methods. We have applied both our evaluation and estimation approaches to the parallel programs presented in this thesis. We haven also used existing estimation methods. We developed divide and conquer skeletons for the implementation of fast parallel multiplication. We have implemented the Karatsuba algorithm, Strassen’s matrix multiplication algorithm and the fast Fourier transform. The latter was used to implement polynomial convolution that leads to a further fast multiplication algorithm. Specially for our implementation of Strassen algorithm we have designed and implemented a divide and conquer skeleton basing on actors. We have implemented the parallel fast Fourier transform, and not only did we use new divide and conquer skeletons, but also developed a map-and-transpose skeleton. It enables good parallelisation of the Fourier transform. The parallelisation of Karatsuba multiplication shows a very good performance. We have analysed the parallel penalty of our programs and compared it to the serial fraction—an approach, known from literature. We also performed execution time estimations of our divide and conquer programs. This thesis presents a parallel map+reduce skeleton scheme. It allows us to combine the usual parallel map skeletons, like parMap, farm, workpool, with a premature termination property. We use this to implement the so-called ‘parallel repeated computation’, a special form of a speculative parallel loop. We have implemented two probabilistic primality tests: the Rabin–Miller test and the Jacobi sum test. We parallelised both with our approach. We analysed the task distribution and stated the fitting configurations of the Jacobi sum test. We have shown formally that the Jacobi sum test can be implemented in parallel. Subsequently, we parallelised it, analysed the load balancing issues, and produced an optimisation. The latter enabled a good implementation, as verified using the parallel penalty. We have also estimated the performance of the tests for further input sizes and numbers of processing elements. Parallelisation of the Jacobi sum test and our generic parallelisation scheme for the repeated computation is our original contribution. The data parallel arithmetic was defined not only for integers, which is already known, but also for rationals. We handled the common factors of the numerator or denominator of the fraction with the modulus in a novel manner. This is required to obtain a true multiple-residue arithmetic, a novel result of our research. Using these mathematical advances, we have parallelised the determinant computation using the Gauß elimination. As always, we have performed task distribution analysis and estimation of the parallel execution time of our implementation. A similar computation in Maple emphasised the potential of our approach. Data parallel arithmetic enables parallelisation of entire classes of computer algebra algorithms. Summarising, this thesis presents and thoroughly evaluates new and existing design decisions for high-level parallelisations of computer algebra algorithms

The Design and Implementation of a High-Performance Polynomial System Solver

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation. Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical property of the particular input system’s solution set. Despite these challenges, we have effectively applied parallel computing to triangular decomposition through the layering and cooperation of many parallel code regions. This parallel computing is supported by our generic object-oriented framework based on the dynamic multithreading paradigm. Meanwhile, the required polynomial algebra is sup- ported by an object-oriented framework for algebraic types which allows type safety and mathematical correctness to be determined at compile-time. Our software is implemented in C/C++ and have extensively tested the implementation for correctness and performance on over 3000 polynomial systems that have arisen in practice. The parallel framework has been re-used in the implementation of Hensel factorization as a parallel pipeline to compute roots of a polynomial with multivariate power series coefficients. Hensel factorization is one step toward computing the non-trivial limit points of quasi-components

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Contributions to Confidentiality and Integrity Algorithms for 5G

The confidentiality and integrity algorithms in cellular networks protect the transmission of user and signaling data over the air between users and the network, e.g., the base stations. There are three standardised cryptographic suites for confidentiality and integrity protection in 4G, which are based on the AES, SNOW 3G, and ZUC primitives, respectively. These primitives are used for providing a 128-bit security level and are usually implemented in hardware, e.g., using IP (intellectual property) cores, thus can be quite efficient. When we come to 5G, the innovative network architecture and high-performance demands pose new challenges to security. For the confidentiality and integrity protection, there are some new requirements on the underlying cryptographic algorithms. Specifically, these algorithms should: 1) provide 256 bits of security to protect against attackers equipped with quantum computing capabilities; and 2) provide at least 20 Gbps (Gigabits per second) speed in pure software environments, which is the downlink peak data rate in 5G. The reason for considering software environments is that the encryption in 5G will likely be moved to the cloud and implemented in software. Therefore, it is crucial to investigate existing algorithms in 4G, checking if they can satisfy the 5G requirements in terms of security and speed, and possibly propose new dedicated algorithms targeting these goals. This is the motivation of this thesis, which focuses on the confidentiality and integrity algorithms for 5G. The results can be summarised as follows.1. We investigate the security of SNOW 3G under 256-bit keys and propose two linear attacks against it with complexities 2172 and 2177, respectively. These cryptanalysis results indicate that SNOW 3G cannot provide the full 256-bit security level. 2. We design some spectral tools for linear cryptanalysis and apply these tools to investigate the security of ZUC-256, the 256-bit version of ZUC. We propose a distinguishing attack against ZUC-256 with complexity 2236, which is 220 faster than exhaustive key search. 3. We design a new stream cipher called SNOW-V in response to the new requirements for 5G confidentiality and integrity protection, in terms of security and speed. SNOW-V can provide a 256-bit security level and achieve a speed as high as 58 Gbps in software based on our extensive evaluation. The cipher is currently under evaluation in ETSI SAGE (Security Algorithms Group of Experts) as a promising candidate for 5G confidentiality and integrity algorithms. 4. We perform deeper cryptanalysis of SNOW-V to ensure that two common cryptanalysis techniques, guess-and-determine attacks and linear cryptanalysis, do not apply to SNOW-V faster than exhaustive key search. 5. We introduce two minor modifications in SNOW-V and propose an extreme performance variant, called SNOW-Vi, in response to the feedback about SNOW-V that some use cases are not fully covered. SNOW-Vi covers more use cases, especially some platforms with less capabilities. The speeds in software are increased by 50% in average over SNOW-V and can be up to 92 Gbps.Besides these works on 5G confidentiality and integrity algorithms, the thesis is also devoted to local pseudorandom generators (PRGs). 6. We investigate the security of local PRGs and propose two attacks against some constructions instantiated on the P5 predicate. The attacks improve existing results with a large gap and narrow down the secure parameter regime. We also extend the attacks to other local PRGs instantiated on general XOR-AND and XOR-MAJ predicates and provide some insight in the choice of safe parameters

Parallel Arbitrary-precision Integer Arithmetic

Arbitrary-precision integer arithmetic computations are driven by applications in solving systems of polynomial equations and public-key cryptography. Such computations arise when high precision is required (with large input values that fit into multiple machine words), or to avoid coefficient overflow due to intermediate expression swell. Meanwhile, the growing demand for faster computation alongside the recent advances in the hardware technology have led to the development of a vast array of many-core and multi-core processors, accelerators, programming models, and language extensions (e.g. CUDA, OpenCL, and OpenACC for GPUs, and OpenMP and Cilk for multi-core CPUs). The massive computational power of parallel processors makes them attractive targets for carrying out arbitrary-precision integer arithmetic. At the same time, developing parallel algorithms, followed by implementing and optimizing them as multi-threaded parallel programs imposes a set of challenges. This work explains the current state of research on parallel arbitrary-precision integer arithmetic on GPUs and CPUs, and proposes a number of solutions for some of the challenging problems related to this subject

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics