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Flash memories are non-volatile memory devices. The rapid development of flash technologies leads to higher storage density, but also to higher error rates. This dissertation considers this reliability problem of flash memories and investigates suitable error correction codes, e.g. BCH-codes and concatenated codes. First, the flash cells, their functionality and error characteristics are explained. Next, the mathematics of the employed algebraic code are discussed. Subsequently, generalized concatenated codes (GCC) are presented. Compared to the commonly used BCH codes, concatenated codes promise higher code rates and lower implementation complexity. This complexity reduction is achieved by dividing a long code into smaller components, which require smaller Galois-Field sizes. The algebraic decoding algorithms enable analytical determination of the block error rate. Thus, it is possible to guarantee very low residual error rates for flash memories. Besides the complexity reduction, general concatenated codes can exploit soft information. This so-called soft decoding is not practicable for long BCH-codes. In this dissertation, two soft decoding methods for GCC are presented and analyzed. These methods are based on the Chase decoding and the stack algorithm. The last method explicitly uses the generalized concatenated code structure, where the component codes are nested subcodes. This property supports the complexity reduction. Moreover, the two-dimensional structure of GCC enables the correction of error patterns with statistical dependencies. One chapter of the thesis demonstrates how the concatenated codes can be used to correct two-dimensional cluster errors. Therefore, a two-dimensional interleaver is designed with the help of Gaussian integers. This design achieves the correction of cluster errors with the best possible radius. Large parts of this works are dedicated to the question, how the decoding algorithms can be implemented in hardware. These hardware architectures, their throughput and logic size are presented for long BCH-codes and generalized concatenated codes. The results show that generalized concatenated codes are suitable for error correction in flash memories, especially for three-dimensional NAND memory systems used in industrial applications, where low residual errors must be guaranteed.
Autonomous moving systems require very detailed information about their environment and potential colliding objects. Thus, the systems are equipped with high resolution sensors. These sensors have the property to generate more than one detection per object per time step. This results in an additional complexity for the target tracking algorithm, since standard tracking filters assume that an object generates at most one detection per object. This requires new methods for data association and system state filtering.
As new data association methods, in this thesis two different extensions of the Joint Integrated Probabilistic Data Association (JIPDA) filter to assign more than one detection to tracks are proposed.
The first method that is introduced, is a generalization of the JIPDA to assign a variable number of measurements to each track based on some predefined statistical models, which will be called Multi Detection - Joint Integrated Probabilistic Data Association (MD-JIPDA).
Since this scheme suffers from exponential increase of association hypotheses, also a new approximation scheme is presented. The second method is an extension for the special case, when the number and locations of measurements are a priori known. In preparation of this method, a new notation and computation scheme for the standard Joint Integrated Data Association is outlined, which also enables the derivation of a new fast approximation scheme called balanced permanent-JIPDA.
For state filtering, also two different concepts are applied: the Random Matrix Framework and the Measurement Generating Points. For the Random Matrix framework, first an alternative prediction method is proposed to account for kinematic state changes in the extension state prediction as well. Secondly, various update methods are investigated to account for the polar to Cartesian noise transformation problem. The filtering concepts are connected with the new MD-JIPDA and their characteristics analyzed with various Monte Carlo simulations.
In case an object can be modeled by a finite number of fixed Measurement Generating Points (MGP), also a proposition to track these object via a JIPDA filter is made. In this context, a fast Track-to-Track fusion algorithm is proposed as well and compared against the MGP-JIPDA.
The proposed algorithms are evaluated in two applications where scanning is done using radar sensors only. The first application is a typical automotive scenario, where a passenger car is equipped with six radar sensors to cover its complete environment.
In this application, the location of the measurements on an object can be considered stationary and that is has a rectangular shape. Thus, the MGP based algorithms are applied here. The filters are evaluated by tracking especially vehicles on nearside lanes.
The second application covers the tracking of vessels on inland waters. Here, two different kind of Radar systems are applied, but for both sensors a uniform distribution of the measurements over the target's extent can be assumed. Further, the assumption that the targets have elliptical shape holds, and so the Random Matrix Framework in combination with the MD-JIPDA is evaluated.
Exemplary test scenarios also illustrate the performance of this tracking algorithm.
Side Channel Attack Resistance of the Elliptic Curve Point Multiplication using Gaussian Integers
(2020)
Elliptic curve cryptography is a cornerstone of embedded security. However, hardware implementations of the elliptic curve point multiplication are prone to side channel attacks. In this work, we present a new key expansion algorithm which improves the resistance against timing and simple power analysis attacks. Furthermore, we consider a new concept for calculating the point multiplication, where the points of the curve are represented as Gaussian integers. Gaussian integers are subset of the complex numbers, such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this concept is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of a secure hardware implementation.
Digitale Signaturen zum Überprüfen der Integrität von Daten, beispielsweise von Software-Updates, gewinnen zunehmend an Bedeutung. Im Bereich der eingebetteten Systeme kommen derzeit wegen der geringen Komplexität noch überwiegend symmetri-sche Verschlüsselungsverfahren zur Berechnung eines Authentifizierungscodes zum Einsatz. Asym-metrische Kryptosysteme sind rechenaufwendiger, bieten aber mehr Sicherheit, weil der Schlüssel zur Authentifizierung nicht geheim gehalten werden muss. Asymmetrische Signaturverfahren werden typischerweise zweistufig berechnet. Der Schlüssel wird nicht direkt auf die Daten angewendet, sondern auf deren Hash-Wert, der mit Hilfe einer Hash-funktion zuvor berechnet wurde. Zum Einsatz dieser Verfahren in eingebetteten Systemen ist es erforder-lich, dass die Hashfunktion einen hinreichend gro-ßen Datendurchsatz ermöglicht. In diesem Beitrag wird eine effiziente Hardware-Implementierung der SHA-256 Hashfunktion vorgestellt.
The Montgomery multiplication is an efficient method for modular arithmetic. Typically, it is used for modular arithmetic over integer rings to prevent the expensive inversion for the modulo reduction. In this work, we consider modular arithmetic over rings of Gaussian integers. Gaussian integers are subset of the complex numbers such that the real and imaginary parts are integers. In many cases Gaussian integer rings are isomorphic to ordinary integer rings. We demonstrate that the concept of the Montgomery multiplication can be extended to Gaussian integers. Due to independent calculation of the real and imaginary parts, the computation complexity of the multiplication is reduced compared with ordinary integer modular arithmetic. This concept is suitable for coding applications as well as for asymmetric key cryptographic systems, such as elliptic curve cryptography or the Rivest-Shamir-Adleman system.
The Lempel-Ziv-Welch (LZW) algorithm is an important dictionary-based data compression approach that is used in many communication and storage systems. The parallel dictionary LZW (PDLZW) algorithm speeds up the LZW encoding by using multiple dictionaries. The PDLZW algorithm applies different dictionaries to store strings of different lengths, where each dictionary stores only strings of the same length. This simplifies the parallel search in the dictionaries for hardware implementations. The compression gain of the PDLZW depends on the partitioning of the address space, i.e. on the sizes of the parallel dictionaries. However, there is no universal partitioning that is optimal for all data sources. This work proposes an address space partitioning technique that optimizes the compression rate of the PDLZW using a Markov model for the data. Numerical results for address spaces with 512, 1024, and 2048 entries demonstrate that the proposed partitioning improves the performance of the PDLZW compared with the original proposal.
The Lempel–Ziv–Welch (LZW) algorithm is an important dictionary-based data compression approach that is used in many communication and storage systems. The parallel dictionary LZW (PDLZW) algorithm speeds up the LZW encoding by using multiple dictionaries. This simplifies the parallel search in the dictionaries. However, the compression gain of the PDLZW depends on the partitioning of the address space, i.e. on the sizes of the parallel dictionaries. This work proposes an address space partitioning technique that optimises the compression rate of the PDLZW. Numerical results for address spaces with 512, 1024, and 2048 entries demonstrate that the proposed address partitioning improves the performance of the PDLZW compared with the original proposal. These address space sizes are suitable for flash storage systems. Moreover, the PDLZW has relative high memory requirements which dominate the costs of a hardware implementation. This work proposes a recursive dictionary structure and a word partitioning technique that significantly reduce the memory size of the parallel dictionaries.
The Burrows–Wheeler transformation (BWT) is a reversible block sorting transform that is an integral part of many data compression algorithms. This work proposes a memory-efficient pipelined decoder for the BWT. In particular, the authors consider the limited context order BWT that has low memory requirements and enable fast encoding. However, the decoding of the limited context order BWT is typically much slower than the encoding. The proposed decoder pipeline provides a fast inverse BWT by splitting the decoding into several processing stages which are executed in parallel.
In this work, we investigate a hybrid decoding approach that combines algebraic hard-input decoding of binary block codes with soft-input decoding. In particular, an acceptance criterion is proposed which determines the reliability of a candidate codeword. For many received codewords the stopping criterion indicates that the hard-decoding result is sufficiently reliable, and the costly soft-input decoding can be omitted. The proposed acceptance criterion significantly reduces the decoding complexity. For simulations we combine the algebraic hard-input decoding with ordered statistics decoding, which enables near maximum likelihood soft-input decoding for codes of small to medium block lengths.
Algorithms and Architectures for Cryptography and Source Coding in Non-Volatile Flash Memories
(2021)
In this work, algorithms and architectures for cryptography and source coding are developed, which are suitable for many resource-constrained embedded systems such as non-volatile flash memories. A new concept for elliptic curve cryptography is presented, which uses an arithmetic over Gaussian integers. Gaussian integers are a subset of the complex numbers with integers as real and imaginary parts. Ordinary modular arithmetic over Gaussian integers is computational expensive. To reduce the complexity, a new arithmetic based on the Montgomery reduction is presented. For the elliptic curve point multiplication, this arithmetic over Gaussian integers improves the computational efficiency, the resistance against side channel attacks, and reduces the memory requirements. Furthermore, an efficient variant of the Lempel-Ziv-Welch (LZW) algorithm for universal lossless data compression is investigated. Instead of one LZW dictionary, this algorithm applies several dictionaries to speed up the encoding process. Two dictionary partitioning techniques are introduced that improve the compression rate and reduce the memory size of this parallel dictionary LZW algorithm.