Additive Fast Fourier Transforms Over Finite Fields

Gao, Shuhong; Mateer, Todd D.

doi:10.1109/tit.2010.2079016

Cited by 61 publications

(83 citation statements)

References 14 publications

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“…The subspace polynomial [5,7] is defined as Definition 1 (Subspace polynomial [5,7]). Given a subspace V k of F p m , the subspace polynomial is defined as…”

Section: A Subspace Polynomialmentioning

confidence: 99%

“…The second approach is only for m a power of two, and its complexity is O(n lg(n) lg lg(n)). The ideas of both algorithms are based on [7]. In particular, the FFTs in [7] are algebraically similar to the approach by combining the proposed transforms with the basis conversion algorithms in this section.…”

Section: Basis Conversionmentioning

confidence: 99%

“…Later, the faster approaches were proposed by [5] and [6]. Currently, the asymptotically fastest approach is proposed by Gao and Mateer [7]. They gave an O(h lg 2 (h)) approach for arbitrary m, as well as an O(h lg(h) lg lg(h)) approach for m a power of two.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Novel Polynomial Basis With Fast Fourier Transform and Its Application to Reed–Solomon Erasure Codes

Lin

Al-Naffouri

Han

et al. 2016

IEEE Trans. Inform. Theory

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“…The subspace polynomial [5,7] is defined as Definition 1 (Subspace polynomial [5,7]). Given a subspace V k of F p m , the subspace polynomial is defined as…”

Section: A Subspace Polynomialmentioning

confidence: 99%

Section: Basis Conversionmentioning

confidence: 99%

See 1 more Smart Citation

Novel Polynomial Basis With Fast Fourier Transform and Its Application to Reed–Solomon Erasure Codes

Lin

Al-Naffouri

Han

et al. 2016

IEEE Trans. Inform. Theory

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“…Our additive FFT is an improvement of the Bernstein-Chou-Schwabe [11] additive FFT, which in turn is an improvement of the Gao-Mateer [24] additive FFT. This section presents details of our size-4, size-8, and size-16 additive FFTs over F 2 8 .…”

Section: Faster Additive Fftsmentioning

confidence: 99%

Faster Binary-Field Multiplication and Faster Binary-Field MACs

Bernstein

Chou

2014

Lecture Notes in Computer Science

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Abstract. This paper shows how to securely authenticate messages using just 29 bit operations per authenticated bit, plus a constant overhead per message. The authenticator is a standard type of "universal" hash function providing information-theoretic security; what is new is computing this type of hash function at very high speed.At a lower level, this paper shows how to multiply two elements of a field of size 2 128 using just 9062 ≈ 71 · 128 bit operations, and how to multiply two elements of a field of size 2 256 using just 22164 ≈ 87 · 256 bit operations. This performance relies on a new representation of field elements and new FFT-based multiplication techniques. This paper's constant-time software uses just 1.89 Core 2 cycles per byte to authenticate very long messages. On a Sandy Bridge it takes 1.43 cycles per byte, without using Intel's PCLMULQDQ polynomialmultiplication hardware. This is much faster than the speed records for constant-time implementations of GHASH without PCLMULQDQ (over 10 cycles/byte), even faster than Intel's best Sandy Bridge implementation of GHASH with PCLMULQDQ (1.79 cycles/byte), and almost as fast as state-of-the-art 128-bit prime-field MACs using Intel's integermultiplication hardware (around 1 cycle/byte).

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“…For multipoint evaluation we use a characteristic-2 "additive FFT" algorithm introduced in 2010 [39] by Gao and Mateer (improving upon an algorithm by von zur Gathen and Gerhard in [40], which in turn improves upon an algorithm proposed by Wang and Zhu in [77] and independently by Cantor in [29]), together with some new improvements described below. This algorithm evaluates a polynomial at every element of F q , or more generally every element of an F 2 -linear subspace of F q .…”

Section: Finding Roots: the Gao-mateer Additive Fftmentioning

confidence: 99%

McBits: Fast Constant-Time Code-Based Cryptography

Bernstein

Chou

Schwabe

2013

Cryptographic Hardware and Embedded Systems - CHES 2013

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Abstract. This paper presents extremely fast algorithms for code-based public-key cryptography, including full protection against timing attacks. For example, at a 2 128 security level, this paper achieves a reciprocal decryption throughput of just 60493 cycles (plus cipher cost etc.) on a single Ivy Bridge core. These algorithms rely on an additive FFT for fast root computation, a transposed additive FFT for fast syndrome computation, and a sorting network to avoid cache-timing attacks.

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Additive Fast Fourier Transforms Over Finite Fields

Cited by 61 publications

References 14 publications

Novel Polynomial Basis With Fast Fourier Transform and Its Application to Reed–Solomon Erasure Codes

Novel Polynomial Basis With Fast Fourier Transform and Its Application to Reed–Solomon Erasure Codes

Faster Binary-Field Multiplication and Faster Binary-Field MACs

McBits: Fast Constant-Time Code-Based Cryptography

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