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Finite Field Multiplier Using Redundant Representation
 IEEE Transactions on Computers
, 2002
"... This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our ar ..."
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Cited by 21 (1 self)
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This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide areatime tradeoffs which enable us to implement the multipliers in a partialparallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that when there is an optimal normal basis, the proposed bitserial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have advantage in using the redundant representation. Index terms: Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
Software multiplication using Gaussian normal bases
 IEEE Trans. Comput
, 2006
"... Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over binary fields F2m. In this paper we present new software algorithms for efficient multiplication over F2m that use a Gaussian normal ..."
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Cited by 6 (2 self)
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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 binary fields F2m. In this paper we present new software algorithms for efficient multiplication over 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, Gaussian normal basis, elliptic curve cryptography. 1
Efficient algorithms and architectures for field multiplication using Gaussian normal bases
 IEEE Transactions on Computers
, 2004
"... Abstract—Recently, implementations of normal basis multiplication over the extended binary field GFð2 m Þ have received considerable attention. A class of low complexity normal bases called Gaussian normal bases has been included in a number of standards, such as IEEE [1] and NIST [2] for an ellipti ..."
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Cited by 5 (0 self)
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Abstract—Recently, implementations of normal basis multiplication over the extended binary field GFð2 m Þ have received considerable attention. A class of low complexity normal bases called Gaussian normal bases has been included in a number of standards, such as IEEE [1] and NIST [2] for an elliptic curve digital signature algorithm. The multiplication algorithms presented there are slow in software since they rely on bitwise inner product operations. In this paper, we present two vectorlevel software algorithms which essentially eliminate such bitwise operations for Gaussian normal bases. Our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms. The proposed algorithm is also more memory efficient compared with its lookup tablebased counterpart. Moreover, two new digitlevel multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures. As compared with similar digitlevel normal basis multipliers, the proposed multiplier with serial output requires the fewest number of XOR gates and the one with parallel output is the fastest multiplier. Index Terms—Finite field multiplication, normal basis, Gaussian normal basis, software algorithms, ECDSA. 1
Efficient multiplication using type 2 optimal normal bases
"... Abstract. In this paper we propose a new structure for multiplication using optimal normal bases of type 2. The multiplier uses an efficient linear transformation to convert the normal basis representations of elements of Fqn to suitable polynomials of degree at most n over Fq. These polynomials ar ..."
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Cited by 3 (0 self)
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Abstract. In this paper we propose a new structure for multiplication using optimal normal bases of type 2. The multiplier uses an efficient linear transformation to convert the normal basis representations of elements of Fqn to suitable polynomials of degree at most n over Fq. These polynomials are multiplied using any method which is suitable for the implementation platform, then the product is converted back to the normal basis using the inverse of the above transformation. The efficiency of the transformation arises from a special factorization of its matrix into sparse matrices. This factorization — which resembles the FFT factorization of the DFT matrix — allows to compute the transformation and its inverse using O(n log n) operations in Fq, rather than O(n 2) operations needed for a general change of basis. Using this technique we can reduce the asymptotic cost of multiplication in optimal normal bases of type 2 from 2M(n) + O(n) reported by Gao et al. (2000) to M(n) + O(n log n) operations in Fq, where M(n) is the number of Fqoperations to multiply two polynomials of degree n − 1 over Fq. We show that this cost is also smaller than other proposed multipliers for n> 160, values which are used in elliptic curve cryptography.
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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Cited by 3 (0 self)
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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
Elliptic periods for finite fields ∗
, 2008
"... We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal ..."
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We construct two new families of basis for finite field extensions. Bases in the first family, the socalled elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the socalled normal elliptic bases are normal bases and allow fast (quasilinear) arithmetic. We prove that all extensions admit models of this kind. 1
Fast Encoding and Decoding of Gabidulin Codes
, 901
"... Abstract—Gabidulin codes are the rankmetric analogs of ReedSolomon codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on lowcomplexity normal bases. In addition, a new decoding algorithm is p ..."
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Abstract—Gabidulin codes are the rankmetric analogs of ReedSolomon codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on lowcomplexity normal bases. In addition, a new decoding algorithm is proposed based on a transformdomain approach. Together, these represent the fastest known algorithms for encoding and decoding Gabidulin codes. I.
Research Summary
"... Normal bases and efficient arithmetic in finite fields Efficient arithmetic of finite fields is important in implementing cryptosystems, errorcorrecting codes and computer algebra systems. Normal bases offer considerable advantages. Optimal normal bases in finite fields were introduced at the Unive ..."
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Normal bases and efficient arithmetic in finite fields Efficient arithmetic of finite fields is important in implementing cryptosystems, errorcorrecting codes and computer algebra systems. Normal bases offer considerable advantages. Optimal normal bases in finite fields were introduced at the University of Waterloo by Mullin et al., and are used in practical hardware implementation of publickey cryptosystems. In the same paper, Mullin et al. constructed two families of optimal normal bases and, based on a computer experiment, they conjectured no more exist. This conjecture had remained open for several years before H. W. Lenstra, Jr. proved it for finite fields over F2. Lenstra’s method, however, is not applicable to other fields. In [4], we confirmed the conjecture for all finite fields by using a substantially different argument. Together with Lenstra, we proved the conjecture holds even for any finite Galois extension of an arbitrary field and the proof for the general case is again different but simpler. The final result is published in [5]. By our classification in [5], not all finite fields have optimal normal bases. For fields without optimal normal bases, it is desirable to have a normal basis of low complexity. In [10], we construct several families of such bases, which come from an explicit factorization of cxq+1 +