# Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography

@inproceedings{Mrabet2009FiniteFM, title={Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography}, author={Nadia El Mrabet and Christophe N{\`e}gre}, booktitle={ACISP}, year={2009} }

Pairings over elliptic curves use fields $\mathbb{F}_{p^k}$ with p *** 2160 and 6 < k ≤ 32. In this paper we propose to represent elements in $\mathbb{F}_p$ with AMNS sytem of [1]. For well chosen AMNS we get roots of unity with sparse representation. The multiplication by these roots are thus really efficient in $\mathbb{F}_p$. The DFT/FFT approach for multiplication in extension field $F_{p^k}$ is thus optimized. The resulting complexity of a multiplication in $\mathbb{F}_{p^k}$ combining…

## 10 Citations

Efficient Multiplication over Extension Fields

- Computer Science, MathematicsWAIFI
- 2012

This paper improves the construction of an AMNS basis and provides a fast implementation of the multiplication over $\mathbb{F}_{q^{m}}$, which is faster than GMP and NTL.

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

- Mathematics, Computer ScienceAdvances in Mathematics of Communications
- 2022

This work states a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds and presents classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.

Delaying Mismatched Field Multiplications in Pairing Computations

- Computer Science, MathematicsWAIFI
- 2010

It is shown that significant speedups in pairing computations can be achieved by delaying these "mismatched" multiplications for an optimal number of iterations, and that this technique can be easily integrated into traditional pairing algorithms.

Efficient and secure modular operations using the Adapted Modular Number System

- Mathematics, Computer ScienceArXiv
- 2019

This paper proposes a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in [15].

Efficient modular operations using the adapted modular number system

- Mathematics, Computer ScienceJournal of Cryptographic Engineering
- 2020

A complete set of algorithms without conditional branching is proposed to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in Negre and Plantard and the implementation outperforms GNU MP and OpenSSL libraries.

Contributions à la cryptographie à base de couplage

- Computer Science, Mathematics
- 2017

A variant of Miller’s formula is proposed which gives rise to a generically faster algorithm for any pairing friendly curve and provides an improvement in cases little studied until now, in particular when denominator elimination is not available.

A Variant of Miller's Formula and Algorithm

- Mathematics, Computer SciencePairing
- 2010

A variant of Miller's formula is proposed which gives rise to a generically faster algorithm for any pairing friendly curve and provides an improvement in cases little studied until now, in particular when denominator elimination is not available.

A Generalized RNS Mclaughlin Modular Multiplication with Non-Coprime Moduli Sets

- Mathematics, Computer ScienceIEEE Transactions on Computers
- 2019

A set of moduli that are non-coprime for RNS in the algorithm to take both the advantage of the fewer multiplications required for a modular multiplication in McLaughlin modular multiplication and theadvantage of the moduli sets of similar sizes in classic Montgomery modularmultiplication in RNS.

Multiplication in Finite Fields and Elliptic Curves

- Physics, Computer Science
- 2016

Dans cette these d'HDR nous allons presenter quelques contributions concernant l'implantation sure and efficace de protocoles cryptographiques bases sur les courbes elliptiques, plus precisement, un multiplieur base sur un produit de matrice de Toeplitz avec un vecteur en utilisant une recombinaison des blocs qui supprime certains calculs redondants.

Arithmétique des couplages, performance et résistance aux attaques par canaux cachés. (Arithmetic of Pairings, Efficiency and Weakness of Pairing Based Cryptography with respect to Side Channel Attacks)

- Philosophy, Computer Science
- 2009

Mes premiers travaux ont porte sur l'arithmetique des couplages, et plus particulierement leur utilisation en cryptographie, a travers une comparaison des complexites en nombre d'operations des Couplages de Weil et Tate.

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