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Institute
This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic 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 secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.
Flash-Speicher wurden ursprünglich als Speichermedium für Digitalkameras entwickelt, finden inzwischen aber in vielen Bereichen Anwendung.
Die in Konstanz ansässige Firma Hyperstone GmbH ist ein führender Anbieter von Flashcontrollern für Anwendungen mit erhöhten Anforderungen an Zuverlässigkeit und Datenintegrität. Bereits seit April 2011 kooperiert die Firma Hyperstone mit der HTWG Konstanz bei der Entwicklung von Fehlerkorrekturverfahren für einen zuverlässigen Einsatz von Flash-Speichern. Aufgrund der rasanten Entwicklung bei Flashspeicherbausteinen ist auch eine stetige Weiterentwicklung der Korrekturverfahren notwendig. Im Rahmen dieser Kooperation wurde inzwischen zwei Flashcontroller mit sehr leistungsfähiger Fehlerkorrektur entwickelt. Der folgende Artikel gibt Einblick in den Einsatz von Flash-Speichern und erläutert die Notwendigkeit für eine leistungsfähige Fehlerkorrektur.
Code-based cryptosystems are promising candidates for post-quantum cryptography. Recently, generalized concatenated codes over Gaussian and Eisenstein integers were proposed for those systems. For a channel model with errors of restricted weight, those q-ary codes lead to high error correction capabilities. Hence, these codes achieve high work factors for information set decoding attacks. In this work, we adapt this concept to codes for the weight-one error channel, i.e., a binary channel model where at most one bit-error occurs in each block of m bits. We also propose a low complexity decoding algorithm for the proposed codes. Compared to codes over Gaussian and Eisenstein integers, these codes achieve higher minimum Hamming distances for the dual codes of the inner component codes. This property increases the work factor for a structural attack on concatenated codes leading to higher overall security. For comparable security, the key size for the proposed code construction is significantly smaller than for the classic McEliece scheme based on Goppa codes.
The growing error rates of triple-level cell (TLC) and quadruple-level cell (QLC) NAND flash memories have led to the application of error correction coding with soft-input decoding techniques in flash-based storage systems. Typically, flash memory is organized in pages where the individual bits per cell are assigned to different pages and different codewords of the error-correcting code. This page-wise encoding minimizes the read latency with hard-input decoding. To increase the decoding capability, soft-input decoding is used eventually due to the aging of the cells. This soft-decoding requires multiple read operations. Hence, the soft-read operations reduce the achievable throughput, and increase the read latency and power consumption. In this work, we investigate a different encoding and decoding approach that improves the error correction performance without increasing the number of reference voltages. We consider TLC and QLC flashes where all bits are jointly encoded using a Gray labeling. This cell-wise encoding improves the achievable channel capacity compared with independent page-wise encoding. Errors with cell-wise read operations typically result in a single erroneous bit per cell. We present a coding approach based on generalized concatenated codes that utilizes this property.
Generalized Concatenated Codes over Gaussian and Eisenstein Integers for Code-Based Cryptography
(2021)
The code-based McEliece and Niederreiter cryptosystems are promising candidates for post-quantum public-key encryption. Recently, q-ary concatenated codes over Gaussian integers were proposed for the McEliece cryptosystem together with the one-Mannheim error channel, where the error values are limited to Mannheim weight one. Due to the limited error values, the codes over Gaussian integers achieve a higher error correction capability than maximum distance separable (MDS) codes with bounded minimum distance decoding. This higher error correction capability improves the work factor regarding decoding attacks based on information-set decoding. The codes also enable a low complexity decoding algorithm for decoding beyond the guaranteed error correction capability. In this work, we extend this coding scheme to codes over Eisenstein integers. These codes have advantages for the Niederreiter system. Additionally, we propose an improved code construction based on generalized concatenated codes. These codes extent the rate region where the work factor is beneficial compared to MDS codes. Moreover, generalized concatenated codes are more robust against structural attacks than ordinary concatenated codes.
This work introduces new signal constellations based on Eisenstein integers, i.e., the hexagonal lattice. These sets of Eisenstein integers have a cardinality which is an integer power of three. They are proposed as signal constellations for representation in the equivalent complex baseband model, especially for applications like physical-layer network coding or MIMO transmission where the constellation is required to be a subset of a lattice. It is shown that these constellations form additive groups where the addition over the complex plane corresponds to the addition with carry over ternary Galois fields. A ternary set partitioning is derived that enables multilevel coding based on ternary error-correcting codes. In the subsets, this partitioning achieves a gain of 4.77 dB, which results from an increased minimum squared Euclidean distance of the signal points. Furthermore, the constellation-constrained capacities over the AWGN channel and the related level capacities in case of ternary multilevel coding are investigated. Simulation results for multilevel coding based on ternary LDPC codes are presented which show that a performance close to the constellation-constrained capacities can be achieved.
Modular arithmetic over integers is required for many cryptography systems. Montgomeryreduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Mont-gomery reduction is used for rings of ordinary integers. In contrast, we investigate the modularreduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real andimaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings.In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Twoalgorithms for the precision reduction are presented. We demonstrate that the proposed Montgomeryreduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptogra-phy. In particular, we consider the elliptic curve point multiplication according to the randomizedinitial point method which is protected against side-channel attacks. The implementation of thisprotected point multiplication is significantly faster than comparable algorithms over ordinary primefields.
The computational complexity of the optimal maximum likelihood (ML) detector for spatial modulation increases rapidly as more transmit antennas or larger modulation orders are employed. Hence, ML detection may be infeasible for higher bit rates. This work proposes an improved suboptimal detection algorithm based on the Gaussian approximation method. It is demonstrated that the new method is closely related to the previously published signal vector based detection and the modified maximum ratio combiner, but can improve the detection performance compared to these methods. Furthermore, the performance of different signal constellations with suboptimal detection is investigated. Simulation results indicate that the performance loss compared to ML detection depends heavily on the signal constellation, where the recently proposed Eisenstein integer constellations are beneficial compared to classical QAM or PSK constellations.
This work proposes a decoder implementation for high-rate generalized concatenated (GC) codes. The proposed codes are well suited for error correction in flash memories for high reliability data storage. The GC codes are constructed from inner extended binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer Reed-Solomon (RS) codes. The extended BCH codes enable high-rate GC codes. Moreover, the decoder can take advantage of soft information. For the first three levels of inner codes we propose an optional Chase soft decoder. In this work, the code construction is explained and a decoder architecture is presented. Furthermore, area and throughput results are discussed.