Быстрое вычисление циклических сверток и их приложения в кодовых схемах асимметричного шифрования

The development of fast algorithms for key generation, encryption and decryption not only increases the efficiency of related operations. Such fast algorithms, for example, for asymmetric cryptosystems on quasi-cyclic codes, make it possible to experimentally study the dependence of decoding failure rate on code parameters for small security levels and to extrapolate these results to large values of security levels. In this article, we explore efficient cyclic convolution algorithms, specifically designed, among other things, for use in encoding and decoding algorithms for quasi-cyclic LDPC and MDPC codes. Corresponding convolutions operate on binary vectors, which can be either sparse or dense. The proposed algorithms achieve high speed by compactly storing sparse vectors, using hardware-supported XOR instructions, and replacing modulo operations with specialized loop transformations. These fast algorithms have potential applications not only in cryptography, but also in other areas where convolutions are used.Разработка быстрых алгоритмов генерации ключей, шифрования и дешифрования не только повышает эффективность соответствующих операций. Такие быстрые алгоритмы, например, для асимметричных криптосистем на квазициклических кодах, позволяют экспериментально исследовать зависимость вероятности ошибочного расшифрования от параметров кода для малых параметров безопасности и экстраполировать эти результаты на большие значения параметров безопасности. В этой статье мы исследуем эффективные алгоритмы циклической свертки, специально разработанные, в том числе, для использования в алгоритмах кодирования и декодирования квазициклических LDPC и MDPC кодов. Соответствующие свертки работают с двоичными векторами, которые могут быть как разреженными, так и плотными. Предлагаемые алгоритмы достигают высокой скорости за счет компактного хранения разреженных векторов, использования аппаратно поддерживаемых инструкций XOR и замены операций по модулю специализированными преобразованиями цикла. Эти быстрые алгоритмы имеют потенциальное применение не только в криптографии, но и в других областях, где используются свертки

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2

Study and implementation of a low complexity receiver using TCH codes

The use of coding in telecommunications systems reveals to be a technique with an essential contribution to the improvement of the recovery of transmitted signals. Depending on the circumstances to which a signal is subjected at transmission, by recurring to coding, it is possible to attenuate the unfavorable effects that result from this process, obtaining a signal with superior quality in comparison with a scenario where the presence of coding is absent. This study aims to test the feasibility of a high-rate wireless communications system using TCH codes being applied to an OFDM signal, subjected to noise components introduced by a wireless AWGN channel, considering a free path propagation model. Due to their correlation properties, the use of TCH codes reveals to be adequate since they allow the same codewords to be used to realize both error correction and channel estimation, mitigating the channels effects, leading to the realization of a receiver with lower complexity. With the intent of performing a qualitative analysis to this system, a simple simulation is executed in MATLAB where an OFDM signal is generated, being therefore applied various TCH codes and, through channel estimation, obtain the BER for their respective code lengths and, consequently, coding gains. The results were obtained for the modulation indexes of 16, 64 and 256-QAM. These demonstrate that the implementation of TCH codes is a viable option to reduce the rate of recovered errors, enabling the reception of a signal with better reliability, especially for higher code lengths and modulation indexes.O uso de codificação em sistemas de telecomunicações revela-se uma técnica com um contributo essencial na melhoria da recuperação de sinais transmitidos. Dependendo das circunstâncias às quais um sinal é submetido aquando a sua transmissão, com recurso à codificação, é possível atenuar os efeitos adversos resultantes deste processo, obtendo uma qualidade no sinal recebido superior face a um cenário com ausência da mesma. Este estudo tem como objetivo testar a exequibilidade de um sistema de comunicações sem fios para alto débito usando códigos TCH e aplicando-os a um sinal OFDM, sujeito a componentes de ruído introduzidos por um canal sem fios AWGN, considerando um modelo de propagação em espaço livre. Devido às propriedades de correlação destes códigos, a sua utilização revela-se adequada pois permite que as mesmas palavras de código sejam utilizadas para efetuar correção de erros, minimizando os efeitos do canal, possibilitando o desenvolvimento de um recetor com menor complexidade. Visando efetuar uma análise qualitativa do sistema, é realizada uma simulação simples em MATLAB onde é gerado um sinal OFDM ao qual são aplicados diversos códigos TCH, realizando estimação de canal com a finalidade de obter o BER para os diferentes comprimentos dos códigos e, consequentemente, ganhos de codificação relativos aos mesmos. Os resultados obtidos foram realizados para os índices de modulação 16, 64 e 256-QAM. Estes demonstram que o uso destes códigos é uma opção viável para reduzir os erros detetados, permitindo recuperar o sinal com maior fiabilidade, especialmente para comprimentos de código e índices de modulação elevados

An Iteratively Decodable Tensor Product Code with Application to Data Storage

The error pattern correcting code (EPCC) can be constructed to provide a syndrome decoding table targeting the dominant error events of an inter-symbol interference channel at the output of the Viterbi detector. For the size of the syndrome table to be manageable and the list of possible error events to be reasonable in size, the codeword length of EPCC needs to be short enough. However, the rate of such a short length code will be too low for hard drive applications. To accommodate the required large redundancy, it is possible to record only a highly compressed function of the parity bits of EPCC's tensor product with a symbol correcting code. In this paper, we show that the proposed tensor error-pattern correcting code (T-EPCC) is linear time encodable and also devise a low-complexity soft iterative decoding algorithm for EPCC's tensor product with q-ary LDPC (T-EPCC-qLDPC). Simulation results show that T-EPCC-qLDPC achieves almost similar performance to single-level qLDPC with a 1/2 KB sector at 50% reduction in decoding complexity. Moreover, 1 KB T-EPCC-qLDPC surpasses the performance of 1/2 KB single-level qLDPC at the same decoder complexity.Comment: Hakim Alhussien, Jaekyun Moon, "An Iteratively Decodable Tensor Product Code with Application to Data Storage

Application of Module to Coding Theory: A Systematic Literature Review

A systematic literature review is a research process that identifies, evaluates, and interprets all relevant study findings connected to specific research questions, topics, or phenomena of interest. In this work, a thorough review of the literature on the issue of the link between module structure and coding theory was done. A literature search yielded 470 articles from the Google Scholar, Dimensions, and Science Direct databases. After further article selection process, 14 articles were chosen to be studied in further depth. The items retrieved were from the previous ten years, from 2012 to 2022. The PRISMA analytical approach and bibliometric analysis were employed in this investigation. A more detailed description of the PRISMA technique and the significance of the bibliometric analysis is provided. The findings of this study are presented in the form of brief summaries of the 14 articles and research recommendations. At the end of the study, recommendations for future development of the code structure utilized in the articles that are further investigated are made