Subquadratic Binary Field Multiplier in Double Polynomial System

Giorgi, Pascal; Nègre, Christophe; Plantard, Thomas

doi:10.5220/0002126102290236

Proceedings of the Second International Conference on Security and Cryptography 2007

DOI: 10.5220/0002126102290236

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Subquadratic Binary Field Multiplier in Double Polynomial System

Pascal Giorgi¹,

Christophe Nègre²,

Thomas Plantard³

Abstract: We propose a new space efficient operator to multiply elements lying in a binary field F 2 k . Our approach is based on a novel system of representation called Double Polynomial System which set elements as a bivariate polynomials over F 2 . Thanks to this system of representation, we are able to use a Lagrange representation of the polynomials and then get a logarithmic time multiplier with a space complexity of O(k 1:31 ) improving previous best known method. Disciplines Physical Sciences and Mathematics Pub… Show more

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2010

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Subquadratic Space Complexity Binary Field Multiplier Using Double Polynomial Representation

Bajard

Nègre

Plantard

2010

IEEE Trans. Comput.

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This paper deals with binary field multiplication. We use the bivariate representation of binary field called Double Polynomial System (DPS) presented in . This concept generalizes the composite field representation to every finite field. As shown in , the main interest of DPS representation is that it enables to use Lagrange approach for multiplication, and in the best case, Fast Fourier Transform approach, which optimizes Lagrange approach. We use here a different strategy from to perform reduction, and we also propose in this paper, some new approaches for constructing DPS. We focus on DPS, which provides a simpler and more efficient method for coefficient reduction. This enables us to avoid a multiplication required in the Montgomery reduction approach of , and thus to improve the complexity of the DPS multiplier. The resulting algorithm proposed in the present paper is subquadratic in space O(n 1.31 ) and logarithmic in time. The space complexity is 33 percent better than in and 18 percent faster. It is asymptotically more efficient than the best known method (specifiably more efficient than when n ≥ 3,000). Furthermore, our proposal is available for every n and not only for n a power of two or three. Abstract-This paper deals with binary field multiplication. We use the bivariate representation of binary field called Double Polynomial System (DPS) presented in [11]. This concept generalizes the composite field representation to every finite field. As shown in [11], the main interest of DPS representation is that it enables to use Lagrange approach for multiplication, and in the best case, Fast Fourier Transform approach, which optimizes Lagrange approach. We use here a different strategy from [11] to perform reduction, and we also propose in this paper, some new approaches for constructing DPS. We focus on DPS, which provides a simpler and more efficient method for coefficient reduction. This enables us to avoid a multiplication required in the Montgomery reduction approach of [11], and thus to improve the complexity of the DPS multiplier. The resulting algorithm proposed in the present paper is subquadratic in space Oðn 1:31 Þ and logarithmic in time. The space complexity is 33 percent better than in [11] and 18 percent faster. It is asymptotically more efficient than the best known method [6] (specifiably more efficient than [6] when n ! 3;000). Furthermore, our proposal is available for every n and not only for n a power of two or three.

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Subquadratic Space Complexity Binary Field Multiplier Using Double Polynomial Representation

Bajard

Nègre

Plantard

2010

IEEE Trans. Comput.

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show abstract

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Subquadratic Binary Field Multiplier in Double Polynomial System

Cited by 1 publication

References 17 publications

Subquadratic Space Complexity Binary Field Multiplier Using Double Polynomial Representation

Subquadratic Space Complexity Binary Field Multiplier Using Double Polynomial Representation

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