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ªLowComplexity Bitparallel Canonical and Normal Basis Multipliers for a Class of Finite Fields,º
 IEEE Trans. Computers
, 1998
"... Abstract—We present a new lowcomplexity bitparallel canonical basis multiplier for the field GF(2 m) generated by an allonepolynomial. The proposed canonical basis multiplier requires m 2 1 XOR gates and m 2 AND gates. We also extend this canonical basis multiplier to obtain a new bitparallel n ..."
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Cited by 43 (8 self)
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Abstract—We present a new lowcomplexity bitparallel canonical basis multiplier for the field GF(2 m) generated by an allonepolynomial. The proposed canonical basis multiplier requires m 2 1 XOR gates and m 2 AND gates. We also extend this canonical basis multiplier to obtain a new bitparallel normal basis multiplier. Index Terms—Finite fields, multiplication, normal basis, canonical basis, allonepolynomial. 1
Optimal lefttoright binary signeddigit recoding
 IEEE Transactions on Computers
, 2000
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A new construction of MasseyOmura parallel multiplier over GF(2m
 IEEE Transactions on Computers
, 2001
"... AbstractÐThe MasseyOmura multiplier of GF
2m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducib ..."
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Cited by 24 (4 self)
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AbstractÐThe MasseyOmura multiplier of GF
2m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another. In the past, it was shown that, for a class of finite fields defined by irreducible allone polynomials, the parallel MasseyOmura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed. In this article, it is shown that, not only does this type of multipliers contain redundancy in that special class of finite fields, but it also has redundancy in fields GF
2m defined by any irreducible polynomial. By removing the redundancy, we propose a new architecture for the normal basis parallel multiplier, which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original MasseyOmura normal basis parallel multiplier. The proposed multiplier structure is also modular and, hence, suitable for VLSI realization. When applied to fields defined by the irreducible allone polynomials, the multiplier's circuit complexity matches the best result available in the open literature. Index TermsÐFinite field, MasseyOmura multiplier, allone polynomial, optimal normal bases. æ 1
A New Finite Field Multiplier Using Redundant Representation
 IEEE Trans. Computers
, 2008
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Mastrovito multiplier for general irreducible polynomials
 IEEE Transactions on Computers
, 2000
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Reconfigurable implementation of elliptic curve crypto algorithms
 Parallel and Distributed Processing Symposium., Proceedings International, IPDPS 2002, Abstracts and CDROM
, 2002
"... For FPGA based coprocessors for elliptic curve cryptography, a significant performance gain can be achieved when hybrid coordinates are used to represent points on the elliptic curve. We provide a new area/performance tradeoff analysis of different hybrid representations over fields of characteris ..."
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Cited by 16 (0 self)
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For FPGA based coprocessors for elliptic curve cryptography, a significant performance gain can be achieved when hybrid coordinates are used to represent points on the elliptic curve. We provide a new area/performance tradeoff analysis of different hybrid representations over fields of characteristic two. Moreover, we present a new generic cryptoprocessor architecture that can be adapted to various area/performance constraints and finite field sizes, and show how to apply high level synthesis techniques to the controller design. 1
Customizable elliptic curve cryptosystems
 IEEE Transactions on Very Large Scale Integration (VLSI) Systems
, 2005
"... Abstract—This paper presents a method for producing hardware designs for elliptic curve cryptography (ECC) systems over the finite field qp@P A, using the optimal normal basis for the representation of numbers. Our field multiplier design is based on a parallel architecture containing multiplebit s ..."
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Cited by 13 (1 self)
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Abstract—This paper presents a method for producing hardware designs for elliptic curve cryptography (ECC) systems over the finite field qp@P A, using the optimal normal basis for the representation of numbers. Our field multiplier design is based on a parallel architecture containing multiplebit serial multipliers; by changing the number of such serial multipliers, designers can obtain implementations with different tradeoffs in speed, size and level of security. A design generator has been developed which can automatically produce a customised ECC hardware design that meets userdefined requirements. To facilitate performance characterization, we have developed a parametric model for estimating the number of cycles for our generic ECC architecture. The resulting hardware implementations are among the fastest reported: for a key size of 270 bits, a point multiplication in a Xilinx XC2V6000 FPGA at 35 MHz can run over 1000 times faster
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Cited by 12 (0 self)
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...