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Parallel formulation of scalar multiplication on koblitz curves
 CACR Technical Reports
, 2007
"... Abstract: We present an algorithm that by using the τ and τ −1 Frobenius operators concurrently allows us to obtain a parallelized version of the classical τandadd scalar multiplication algorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to effi ..."
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Abstract: We present an algorithm that by using the τ and τ −1 Frobenius operators concurrently allows us to obtain a parallelized version of the classical τandadd scalar multiplication algorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to efficient implementations of both τ and τ −1, thus showing that our algorithm can be effectively applied on all the NISTrecommended curves. We also present design details of software and hardware implementations of our procedure. In a twoprocessor workstation software implementation, we report experimental data showing that our parallel algorithm is able to achieve a speedup factor of almost 2 when compared with the standard sequential point multiplication. In our hardware implementation, the parallel version yields a more modest acceleration of 17 % when compared with the traditional point multiplication algorithm. Although the focus is on Koblitz curves, analogous strategies are discussed for other curves, in particular for random curves over binary fields.
Specification For: Efficient implementation of Elliptic Curve Cryptosystems over binary Galois Fields, GF(2 m) in normal and polynomial bases
"... In this project, we will seek to efficiently implement an elliptic curve cryptographic library over a Galois field in normal and polynomial bases. Elliptic curve cryptosystems are particularly valuable as they are generally faster than other commonly used public key cryptosystems, and they require a ..."
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Cited by 2 (0 self)
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In this project, we will seek to efficiently implement an elliptic curve cryptographic library over a Galois field in normal and polynomial bases. Elliptic curve cryptosystems are particularly valuable as they are generally faster than other commonly used public key cryptosystems, and they require a smaller key size. Many performance studies have found that elliptic curve
Software implementation of arithmetic in F3 m
 International Workshop on the Arithmetic of Finite Fields (WAIFI 2007), volume 4547 of Lecture Notes in Computer Science
, 2007
"... Abstract. Fast arithmetic for characteristic three finite fields F3 m is desirable in pairingbased cryptography because there is a suitable family of elliptic curves over F3 m having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of F3 m, and use the r ..."
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Abstract. Fast arithmetic for characteristic three finite fields F3 m is desirable in pairingbased cryptography because there is a suitable family of elliptic curves over F3 m having embedding degree 6. In this paper we present some structure results for Gaussian normal bases of F3 m, and use the results to devise faster multiplication algorithms. We carefully compare multiplication in F3 m using polynomial bases and Gaussian normal bases. Finally, we compare the speed of encryption and decryption for the BonehFranklin and SakaiKasahara identitybased encryption schemes at the 128bit security level, in the case where supersingular elliptic curves with embedding degrees 2, 4 and 6 are employed. 1.
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.
1 A New Double Point Multiplication Method and its Implementation on Binary Elliptic Curves with
"... Efficient and highperformance implementation of point multiplication is crucial for elliptic curve cryptosystems. In this paper, we present a new double point multiplication algorithm based on differential addition chains. We use our scheme to implement single point multiplication on binary ellipti ..."
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Efficient and highperformance implementation of point multiplication is crucial for elliptic curve cryptosystems. In this paper, we present a new double point multiplication algorithm based on differential addition chains. We use our scheme to implement single point multiplication on binary elliptic curves with efficiently computable endomorphisms. Our proposed scheme has a uniform structure and has some degree of builtin resistance against side channel analysis attacks. We design a cryptoprocessor based on the proposed algorithm for double point multiplication and evaluate its area and time efficiency on FPGA. To the best of the authors ’ knowledge, this is the first hardware implementation of single point multiplication (using double point multiplication) on elliptic curves with efficiently computable endomorphisms. Our analysis and timing results show that the expected acceleration in point multiplication is considerable. Prototypes of the proposed architectures are implemented and experimental results are presented.