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Low Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation
, 2013
"... We study Dickson bases for binary field representation. Such a representation seems interesting when no optimal normal basis exists for the field. We express the product of two field elements as Toeplitz or Hankel matrixvector products. This provides a parallel multiplier which is subquadratic in s ..."
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We study Dickson bases for binary field representation. Such a representation seems interesting when no optimal normal basis exists for the field. We express the product of two field elements as Toeplitz or Hankel matrixvector products. This provides a parallel multiplier which is subquadratic in space and logarithmic in time. Using the matrixvector formulation of the field multiplication, we also present sequential multiplier structures with linear space complexity.
On Implementation of Quadratic and SubQuadratic Complexity Multipliers using Type II Optimal Normal Bases
"... Abstract. Finitefieldarithmetichasreceivedaconsiderableattentioninthecurrentcryptographic applications. Many researchers have focused on finite field multiplication due to its importance in various cryptographic operations. Moreover, finite field multiplication can be considered as a cornerstone for ..."
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Abstract. Finitefieldarithmetichasreceivedaconsiderableattentioninthecurrentcryptographic applications. Many researchers have focused on finite field multiplication due to its importance in various cryptographic operations. Moreover, finite field multiplication can be considered as a cornerstone for elliptic curve cryptosystems. Fan and Hasan [1] introduced a new subquadratic computational complexity approach for finite field multiplication. It is based on Toeplitz matrixvector products. In this paper we consider efficient implementation of this approach on general purpose processors usingType II Optimal Normal Basis (ONB II). To this end, a memory and time efficient implementation scheme is proposed for the Fan and Hasan approach. Also, in this paper we provide a modified version of the best quadratic complexity multiplication algorithm due to ReyhaniMasoleh [2]. The proposed modification reduces the number of OR and SHIFT instructions by 50% and the number of AND instructions by about 25%. We simulate the implementation on three different architectures and present the results. Furthermore, we present an idea to fully parallelize the implementation of the Fan and Hasan scheme.
New Complexity Results for Field Multiplication using Optimal Normal Bases
"... In this article, we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases. The schemes are based on a recently proposed method known as block recombination, which efficiently computes the sum of two products of Toeplitz matrices and vec ..."
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In this article, we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases. The schemes are based on a recently proposed method known as block recombination, which efficiently computes the sum of two products of Toeplitz matrices and vectors. Specifically, here we take advantage of some structural properties of the matrices and vectors involved in the formulation of field multiplication using optimal normal bases. This yields new space and time complexity results for corresponding bit parallel multipliers.