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Split-radix FFT algorithm
About: Split-radix FFT algorithm is a research topic. Over the lifetime, 1845 publications have been published within this topic receiving 41398 citations.
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19 Aug 2016
TL;DR: This work develops a hexagonal FFT in ASA coordinates that uses only the standard Fourier transform, allowing the user to implement the hexagonally sampled FFT using standard FFT routines.
Abstract: The discrete Fourier transform is an important tool for processing digital images. Efficient algorithms for computing the Fourier transform are known as fast Fourier transforms (FFTs). One of the most common of these is the Cooley-Tukey radix-2 decimation algorithm that efficiently transforms one-dimensional data into its frequency domain representation. The orthogonality of rectangular sampling allows the separability of the Fourier kernel which enables the use of the Cooley-Tukey algorithm on two-dimensional digital images that have been sampled rectangularly. Hexagonal sampling provides many benefits over rectangular sampling, but it does not result in the orthogonal rows and columns that can be transformed independently as is done with rectangular samples. Use of the Array Set Addressing (ASA) coordinate system for hexagonally sampled images has been shown to provide a separable Fourier kernel, leading to an efficient FFT, however its implementation is composed of nonstandard transforms that require custom routines to evaluate. This work develops a hexagonal FFT in ASA coordinates that uses only the standard Fourier transform, allowing the user to implement the hexagonal FFT using standard FFT routines.
15 citations
TL;DR: This paper proposes a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit- reversal is required.
Abstract: The necessity for an efficient bit-reversal routine in the implementation of fast discrete Fourier transform algorithms is well known. In this paper, we propose a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit-reversal is required. >
15 citations
22 Oct 2014
TL;DR: It is shown that FFT can reduce the time complexity of a linear attack and the complexity of the integral attack, and the estimation of the complexity is very simple.
Abstract: An integral attack is one of the most powerful attacks against block ciphers. We propose a new technique for the integral attack called the Fast Fourier Transform FFT key recovery. When the integral distinguisher uses N chosen plaintexts and the guessed key is k bits, a straightforward key recovery requires the time complexity of ON 2 k . However, the FFT key recovery method requires only the time complexity of ON+k 2 k . As a previous result using FFT, at ICISC 2007, Collard et al.proposed that FFT can reduce the time complexity of a linear attack. We show that FFT can also reduce the complexity of the integral attack. Moreover, the estimation of the complexity is very simple. We first show the complexity of the FFT key recovery against three structures, the Even-Mansour scheme, a key-alternating cipher, and the Feistel cipher. As examples of these structures, we show integral attacks against PrOst, CLEFIA, and AES. As a result, 8-round PrOst $\tilde{P}_{128,K}$ can be attacked with about an approximate time complexity of 280. Moreover, a 6-round AES and 12-round CLEFIA can be attacked with approximate time complexities of 252.6 and 287.5, respectively.
15 citations
TL;DR: The number of multiplications necessary to compute the proposed algorithm is significantly reduced while the number of additions remains almost identical to that of conventional Multidimensional FFT's (MFFT).
Abstract: In this paper, we propose a new approach for computing multidimensional Cooley-Tukey FFT‘s that is suitable for implementation on a variety of multiprocessor architectures. Our algorithm is derived in this paper from a Cooley decimation-in-time algorithm by using an appropriate indexing process and the tensor product properties. It is proved that the number of multiplications necessary to compute our proposed algorithm is significantly reduced while the number of additions remains almost identical to that of conventional Multidimensional FFT‘s (MFFT). Comparison results show the powerful performance of the proposed MFFT algorithm against the row-column FFT transform when data dimension M is large. Furthermore, this algorithm, presented in a simple matrix form, will be much easier to implement in practice. Connections of the proposed approach with well-known DFT algorithms are included in this paper and many variations of the proposed algorithm are also pointed out.
14 citations
TL;DR: A third possibility for hardware implementation of the fast Fourier transform of 2m samples is considered, in which in each pass the multipliers are generated from the values of the multiplier coefficient used in the previous pass.
Abstract: One possible hardware implementation for the fast Fourier transform (FFT) of 2m samples is to have 2m-1 cells, each of which performs two of the necessary computations during each of the m passes through the processor. But in each of these m passes, each of the 2m-1cells may require a different multiplier coefficient for its computations. The two most obvious solutions are costly. The multipliers could be stored in a central memory and sent to each cell when needed; however, it takes time to transmit them and uses many pins, or interconnections between cells. Alternatively, the multipliers could be stored in a ROM in each cell. This makes each cell bigger, and the cells are no longer identical copies of one another. We consider a third possibility in this note. In each pass the multipliers are generated from the values of the multipliers used in the previous pass. This technique requires no increase in the number of pins per cell and little increase in the time required to perform the Fourier transformation.
14 citations