The Hartley transform in a finite field

Souza, Ricardo M. Campello de; Oliveira, H. M. de; Kauffman, A.N.

doi:10.1109/its.1998.713126

Cited by 11 publications

(22 citation statements)

References 13 publications

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“…The increase in the size of the field also increases the output dynamic range, therefore, the FFFT offers no benefit in terms of PAPR reduction. Alternatively, a finite field Hartley transform (FFHT) is presented in [10] based on the traditional Hartley transform [11]. However, same as the FFFT, FFHT also requires the transformed data to the extension field, which will increase the dynamic range as the transform length increases.…”

Section: Basefield Hartley Transformmentioning

confidence: 99%

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A Pre-Equalized Transmission Based on Basefield Hartley Transform over Multi-Path Fading Channels

Liu

Luo

Fan

2011

2011 IEEE Vehicular Technology Conference (VTC Fall)

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This paper proposes a pre-equalized transmission scheme based on a finite field transform. Unlike orthogonal frequency-division multiplexing (OFDM) which transmits signals on orthogonal sub-carriers at the transmitter and explores postequalization to migrate the effect of multi-path fading at the receiver, in the proposed scheme, signals are processed with the finite field transform called basefield Hartley transform (BHT) before transmission. Since BHT has similar convolution property as discrete Fourier transform (DFT), the inter-symbol interference (ISI) can be mitigated by applying a finite field preequalizer and quantization pre-equalizer at the transmitter. By allowing the padded redundancy to take values in finite field, the proposed scheme has the inherent error correction capability with various coding rate. Since the received signals are equivalent to certain block error correction codes, only a decoding scheme is required to recover the source data without any DFT operations that are used in conventional OFDM receiver, which leads to a simple receiving structure with low process complexity. Our simulation results show that the proposed scheme reduces the peak-to-average power ratio (PAPR) and provides better bit error rate (BER) performance at the high signal-to-noise ratio (SNR) in comparison with the coded OFDM system.

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Section: Basefield Hartley Transformmentioning

confidence: 99%

“…In this paper, we choose a BHT [12] based on the structure of GI(p) [10]. A new structure of finite filed GI(p) is defined as follow, which forms an equivalent two-dimensional extension field of GF (p).…”

Section: Basefield Hartley Transformmentioning

confidence: 99%

A Pre-Equalized Transmission Based on Basefield Hartley Transform over Multi-Path Fading Channels

Liu

Luo

Fan

2011

2011 IEEE Vehicular Technology Conference (VTC Fall)

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“…Lemma 2: For each pair (x, T r (x)), the integer r satisfies T r (x) = T r (x) if and only if r = ± arccos ζ (T r (x)) (arccos ζ (x)) −1 (mod N ), (8) where N = ord(ζ).…”

Section: A Recovering the Plaintextmentioning

confidence: 99%

“…the discrete logarithmic form of the arccos ζ function, computed modulo p. Applying the above formula to Equation (8) and using some logarithm properties, we have…”

Section: From Definition 4mentioning

confidence: 99%

Security of public-key cryptosystems based on Chebyshev polynomials over prime finite fields

Lima

Souza

Panario

2008

2008 IEEE International Symposium on Information Theory

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In this paper, a new definition of Chebyshev polynomials over prime finite fields is introduced. Our approach uses a finite field trigonometry and reveals some aspects concerning the security of a recently proposed public-key encryption algorithm based on those polynomials. Particularly, we show that recovering the corresponding plaintext from a given ciphertext involves the discrete logarithm problem.

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“…Posteriormente, ela veio a ser utilizada em muitas outras aplicações, sobretudo nas áreas de Processamento Digital de Sinais, Teoria da Informação, Códigos Corretores de Erros e Criptografia [2][3][4][5][6]. Recentemente, a transformada de Hartley sobre corpos finitos foi introduzida em [7], [8], a qual apresenta propriedades de simetria que a tornam mais atraente, para diversas aplicações, que a transformada de Fourier de corpo finito, e tem importantes aplicações no campo da multiplexação digital [9], [10].…”

Section: Introductionunclassified

A Transformada Numérica de Hartley e Grupos de Inteiros Gaussianos

Silva¹,

Souza²,

Oliveira³

et al. 2002

JCIS

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48Resumo -Transformadas discretas desempenham um importante papel em Engenharia e suas aplicações devemse principalmente à existência das chamadas transformadas rápidas. Especificamente, transformadas discretas definidas sobre corpos finitos são atraentes por não introduzirem erros de truncagem ou arredondamento, e por permitirem aplicações com aritmética de baixa complexidade. Neste artigo, a Transformada Numérica de Hartley (TNH) é introduzida e a partir da mesma a Transformada Numérica de Hartley-Mersenne é definida e algumas transformadas sem multiplicações são apresentadas. Algumas estruturas algébricas relacionadas com a Transformada de Hartley de Corpo Finito são estabelecidas e, em particular, os grupos dos módulos e das fases de um corpo finito são introduzidos, o que leva a uma representação polar dos elementos do corpo finito GF(p 2 ). Aplicações envolvendo a TNH são discutidas. Palavras-Chave: Transformadas em corpos finitos, transformadas numéricas de Hartley, grupos de inteiros gaussianos.Abstract -Finite field transforms are attractive since they do not introduce roundoff errors and, in many cases, can be implemented with a low computational complexity. In this paper, the Hartley Number-Theoretic Transform (HNTT) is introduced. In particular, the Mersenne HNTT is defined and some multiplication free transforms are given. Some algebraic structures that are related to the HNTT are introduced and, in particular, the group of modules and the group of phases of a finite field are defined, which allows the construction of a polar representation for the elements of the Galois field GF(p 2 ). A few applications involving the TNH are discussed.

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The Hartley transform in a finite field

Cited by 11 publications

References 13 publications

A Pre-Equalized Transmission Based on Basefield Hartley Transform over Multi-Path Fading Channels

A Pre-Equalized Transmission Based on Basefield Hartley Transform over Multi-Path Fading Channels

Security of public-key cryptosystems based on Chebyshev polynomials over prime finite fields

A Transformada Numérica de Hartley e Grupos de Inteiros Gaussianos

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