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Software Multiplication Using Gaussian Normal Bases
 IEEE Transactions on Computers
, 2006
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Software multiplication using normal bases
 Dept. of Combinatorics and Optimization, Univ. of
, 2004
"... Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over the NIST recommended binary fields. In this paper we present new software algorithms for efficient multiplication over the binary fie ..."
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Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over the NIST recommended binary fields. In this paper we present new software algorithms for efficient multiplication over the binary field F2m that use a Gaussian normal basis representation. Two approaches are presented, direct normal basis multiplication, and a method that exploits a mapping to a ring where fast polynomialbased techniques can be employed. Our analysis including experimental results on an Intel Pentium family processor shows that the new algorithms are faster and can use memory more efficiently than previous methods. Despite significant improvements, we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems. Key words Multiplication in F2 m, normal basis, Gaussian normal basis, elliptic curve cryptography. 1
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.
On complexity of normal basis multiplier using modified Booth’s algorithm
"... Abstract: This paper purposed a Booth’s multiplier for normal basis multiplier. The architecture is simple and highly regular architecture for finite fields using a new modified Booth’s algorithm. The proposed multiplier for finite fields requires a significantly lower number of bit level operation ..."
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Abstract: This paper purposed a Booth’s multiplier for normal basis multiplier. The architecture is simple and highly regular architecture for finite fields using a new modified Booth’s algorithm. The proposed multiplier for finite fields requires a significantly lower number of bit level operations and, hence, can reduce the space complexity of cryptographic systems. It is shown that proposed multiplier for type2 normal basis of GF(2 m) saves approximately 10 % space complexity as compared to related parallel multipliers. Moreover, the proposed architecture is regularity and modularity; they are well suited to VLSI implementations. Keywords: Finite field multiplication, Normal basis, Gaussian normal basis, Cryptographic. 1
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"... Abstract. Two novel algorithms are presented for a typet Gaussian normal basis (GNB) of GF(2m), which are appropriate for VLSI implementation. The proposed algorithms are based on the Hankel matrixvector representation to obtain the scalable and systolic multipliers in a flexible manner that can ..."
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Abstract. Two novel algorithms are presented for a typet Gaussian normal basis (GNB) of GF(2m), which are appropriate for VLSI implementation. The proposed algorithms are based on the Hankel matrixvector representation to obtain the scalable and systolic multipliers in a flexible manner that can adapt to the required precision. This investigation depicts that our multipliers have a latency of d+ n2 + n, where n= ⌈(mt+ 1)/d ⌉ , and d denotes the selected digital size. The proposed architectures are suitable for implementing all typet GNB multiplications. A comparison of the latency indicates that the proposed multipliers for type1 and type2 GNBs of GF(2m) save around 60 % and 40 % latency, respectively over related systolic multipliers with unscalable architecture. Moreover, because the new architectures are regularity and modularity, they are well suited to VLSI implementations.