Computing special powers  in finite fields

Gathen, Joachim von zur; Nöcker, Michael

doi:10.1090/s0025-5718-03-01599-0

Cited by 10 publications

(6 citation statements)

References 51 publications

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“…This exponentiation can be done at the expense of O(log q + log d) multiplications in F q d using an addition chain for d − 1 and another addition chain for q − 2. This is [13, Theorem 2] of Itoh and Tsujii generalized in [20,Corollary 30] by von zur Gathen and Nöcker. The computation also requires O(log d) exponentiations by powers of q.…”

Section: Inversion Using Lagrange's Theoremmentioning

confidence: 62%

Elliptic periods for finite fields

Couveignes

Lercier

2009

Finite Fields and Their Applications

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We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind

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Section: Inversion Using Lagrange's Theoremmentioning

confidence: 62%

Elliptic periods for finite fields

Couveignes

Lercier

2009

Finite Fields and Their Applications

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“…As already shown in [38] and [39], addition chains can prove to be perfectly suited to raise elements of IF 3 m to particular powers, such as the radix-3 repunit ð3 mÀ1 À 1Þ=2 required by our inversion algorithm. In the following, we will restrict ourselves to Brauer-type addition chains, 3 whose definition follows.…”

Section: Inversion Over If 3 Mmentioning

confidence: 68%

Algorithms and Arithmetic Operators for Computing the ηT Pairing in Characteristic Three

Beuchat

Brisebarre

Detrey

et al. 2008

IEEE Trans. Comput.

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Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. With software implementations being rather slow, the study of hardware architectures became an active research area. In this paper, we discuss several algorithms to compute the T pairing in characteristic three and suggest further improvements. These algorithms involve addition, multiplication, cubing, inversion, and sometimes cube root extraction over IF 3 m. We propose a hardware accelerator based on a unified arithmetic operator able to perform the operations required by a given algorithm. We describe the implementation of a compact coprocessor for the field IF 3 97 given by IF 3 ½x=ðx 97 þ x 12 þ 2Þ, which compares favorably with other solutions described in the open literature.

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“…As already shown in [32] and [23], additions chains can prove to be perfectly suited to raise elements of F p m to particular powers, such as the radix-p repunit (p m−1 − 1)/(p− 1) required by our inversion algorithm.…”

Section: Inversion Over F P Mmentioning

confidence: 69%

Arithmetic Operators for Pairing-Based Cryptography

Beuchat

Brisebarre

Detrey

et al. 2007

Cryptographic Hardware and Embedded Systems - CHES 2007

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Abstract. Since their introduction in constructive cryptographic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. Software implementations being rather slow, the study of hardware architectures became an active research area. In this paper, we first study an accelerator for the ηT pairing over F3 [x]/(x 97 + x 12 + 2). Our architecture is based on a unified arithmetic operator which performs addition, multiplication, and cubing over F 3 97 . This design methodology allows us to design a compact coprocessor (1888 slices on a Virtex-II Pro 4 FPGA) which compares favorably with other solutions described in the open literature. We then describe ways to extend our approach to any characteristic and any extension field.

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Computing special powers in finite fields

Cited by 10 publications

References 51 publications

Elliptic periods for finite fields

Elliptic periods for finite fields

Algorithms and Arithmetic Operators for Computing the ηT Pairing in Characteristic Three

Arithmetic Operators for Pairing-Based Cryptography

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