Open Access

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.

Multi-dimensional spatial modulation is a multipleinput/ multiple-output wireless transmission technique, that uses only a few active antennas simultaneously. The computational complexity of the optimal maximum-likelihood (ML) detector at the receiver increases rapidly as more transmit antennas or larger modulation orders are employed. ML detection may be infeasible for higher bit rates. Many suboptimal detection algorithms for spatial modulation use two-stage detection schemes where the set of active antennas is detected in the first stage and the transmitted symbols in the second stage. Typically, these detection schemes use the ML strategy for the symbol detection. In this work, we consider a suboptimal detection algorithm for the second detection stage. This approach combines equalization and list decoding. We propose an algorithm for multi-dimensional signal constellations with a reduced search space in the second detection stage through set partitioning. In particular, we derive a set partitioning from the properties of Hurwitz integers. Simulation results demonstrate that the new algorithm achieves near-ML performance. It significantly reduces the complexity when compared with conventional two-stage detection schemes. Multi-dimensional constellations in combination with suboptimal detection can even outperform conventional signal constellations in combination with ML detection.

Many resource-constrained systems still rely on symmetric cryptography for verification and authentication. Asymmetric cryptographic systems provide higher security levels, but are very computational intensive. Hence, embedded systems can benefit from hardware assistance, i.e., coprocessors optimized for the required public key operations. In this work, we propose an elliptic curve cryptographic coprocessors design for resource-constrained systems. Many such coprocessor designs consider only special (Solinas) prime fields, which enable a low-complexity modulo arithmetic. Other implementations support arbitrary prime curves using the Montgomery reduction. These implementations typically require more time for the point multiplication. We present a coprocessor design that has low area requirements and enables a trade-off between performance and flexibility. The point multiplication can be performed either using a fast arithmetic based on Solinas primes or using a slower, but flexible Montgomery modular arithmetic.

NAND flash memory is widely used for data storage due to low power consumption, high throughput, short random access latency, and high density. The storage density of the NAND flash memory devices increases from one generation to the next, albeit at the expense of storage reliability. Our objective in this dissertation is to improve the reliability of the NAND flash memory with a low hard implementation cost. We investigate the error characteristic, i.e. the various noises of the NAND flash memory. Based on the error behavior at different life-aging stages, we develop offset calibration techniques that minimize the bit error rate (BER). Furthermore, we introduce data compression to reduce the write amplification effect and support the error correction codes (ECC) unit. In the first scenario, the numerical results show that the data compression can reduce the wear-out by minimizing the amount of data that is written to the flash. In the ECC scenario, the compression gain is used to improve the ECC capability. Based on the first scenario, the write amplification effect can be halved for the considered target flash and data model. By combining the ECC and data compression, the NAND flash memory lifetime improves three fold compared with uncompressed data for the same data model. In order to improve the data reliability of the NAND flash memory, we investigate different ECC schemes based on concatenated codes like product codes, half-product codes, and generalized concatenated codes (GCC). We propose a construction for high-rate GCC for hard-input decoding. ECC based on soft-input decoding can significantly improve the reliability of NAND flash memories. Therefore, we propose a low-complexity soft-input decoding algorithm for high-rate GCC.

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.