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TypeII Optimal Polynomial Bases
"... Abstract. In the 1990s and early 2000s several papers investigated the relative merits of polynomialbasis and normalbasis computations for F2n. Even for particularly squaringfriendly applications, such as implementations of Koblitz curves, normal bases fell behind in performance unless a typeI n ..."
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Abstract. In the 1990s and early 2000s several papers investigated the relative merits of polynomialbasis and normalbasis computations for F2n. Even for particularly squaringfriendly applications, such as implementations of Koblitz curves, normal bases fell behind in performance unless a typeI normal basis existed for F2n. In 2007 Shokrollahi proposed a new method of multiplying in a typeII normal basis. Shokrollahi’s method efficiently transforms the normalbasis multiplication into a single multiplication of two size(n + 1) polynomials. This paper speeds up Shokrollahi’s method in several ways. It first presents a simpler algorithm that uses only sizen polynomials. It then explains how to reduce the transformation cost by dynamically switching to a ‘typeII optimal polynomial basis ’ and by using a new reduction strategy for multiplications that produce output in typeII polynomial basis. As an illustration of its improvements, this paper explains in detail how the multiplication overhead in Shokrollahi’s original method has been reduced by a factor of 1.4 in a major cryptanalytic computation, the ongoing attack on the ECC2K130 Certicom challenge. The resulting overhead is also considerably smaller than the overhead in a traditional lowweightpolynomialbasis approach. This is the first stateoftheart binaryellipticcurve computation in which typeII bases have been shown to outperform traditional lowweight polynomial bases.
Subquadratic Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation
"... We study Dickson bases for binary field representation. Such representation seems interesting when no optimal normal basis exists for the field. We express the product of two elements as Toeplitz or Hankel matrix vector product. This provides a parallel multiplier which is subquadratic in space and ..."
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We study Dickson bases for binary field representation. Such representation seems interesting when no optimal normal basis exists for the field. We express the product of two elements as Toeplitz or Hankel matrix vector product. This provides a parallel multiplier which is subquadratic in space and logarithmic in time. 1
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.
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.
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.
1A Versatile MultiInput Multiplier over Finite Fields
"... Abstract—Multiplication of three elements over finite fields is used extensively in multivariate public key cryptography and solving system of linear equations over finite fields. This contribution shows the enhancements of multiplication of three elements over finite fields by using specific archit ..."
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Abstract—Multiplication of three elements over finite fields is used extensively in multivariate public key cryptography and solving system of linear equations over finite fields. This contribution shows the enhancements of multiplication of three elements over finite fields by using specific architecture. We firstly propose a versatile multiinput multiplier over finite fields. The parameters of this multiplier can be changed according to the requirement of the users which makes it reusable in different applications. Our evaluation of this multiplier gives optimum choices for multiplication of three elements over finite fields. Implemented results show that we takes 22.062 ns and 16.354 ns to execute each multiplication of three elements over GF ((24)2) based on table lookup and polynomial basis on a FPGA respectively. Experimental results and mathematical proofs clearly demonstrate the improvement of the proposed versatile multiplier over finite fields.
Low complexity bitparallel GF (2 m) multiplier for allone polynomials
"... Abstract. This paper presents a new bitparallel multiplier for the finite field GF (2 m) generated with an irreducible allone polynomial. Redundant representation is used to reduce the time delay of the proposed multiplier, while a threeterm Karatsubalike formula is combined with this representa ..."
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Abstract. This paper presents a new bitparallel multiplier for the finite field GF (2 m) generated with an irreducible allone polynomial. Redundant representation is used to reduce the time delay of the proposed multiplier, while a threeterm Karatsubalike formula is combined with this representation to decrease the space complexity. As a result, the proposed multiplier requires about 10 percent fewer AND/XOR gates than the most efficient bitparallel multipliers using an allone polynomial, while it has almost the same time delay as the previously proposed ones. 1