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38
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 37 (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
, 2000
"... This paper describes new methods for producing optimal binary signeddigit representations. This can be useful in the fast computation of exponentiations. Contrary to existing algorithms, the digits are scanned from left to right (i.e., from the most significant position to the least significant ..."
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Cited by 34 (3 self)
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This paper describes new methods for producing optimal binary signeddigit representations. This can be useful in the fast computation of exponentiations. Contrary to existing algorithms, the digits are scanned from left to right (i.e., from the most significant position to the least significant position). This may lead to better performances in both hardware and software.
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.
Mastrovito multiplier for general irreducible polynomials
 IEEE Transactions on Computers
, 2000
"... We present a new formulation of the Mastrovito multiplication matrix for the field GF(2 m) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, allonepolynomials, and equallyspacedpolynomials, and obtain the tim ..."
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Cited by 20 (0 self)
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We present a new formulation of the Mastrovito multiplication matrix for the field GF(2 m) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, allonepolynomials, and equallyspacedpolynomials, and obtain the time and space complexity of these designs. Particular examples, illustrating the properties of the proposed architecture, are also given. The complexity results established in this paper match the best complexity results known to date. The most important new result is the space complexity of the Mastrovito multiplier for an equallyspacedpolynomial, which is found as (m 2 − ∆) XOR gates and m 2 AND gates, where ∆ is the spacing factor.
Fast Normal Basis Multiplication Using General Purpose Processors
 IEEE Transaction on Computers
, 2001
"... Abstract For cryptographic applications, normal bases have received considerable attention, especially for hardware implementation. In this document, we consider fast software algorithms for normal basis multiplication over the extended binary o/eld GF(2m). We present a vectorlevel algorithm which ..."
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Cited by 14 (2 self)
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Abstract For cryptographic applications, normal bases have received considerable attention, especially for hardware implementation. In this document, we consider fast software algorithms for normal basis multiplication over the extended binary o/eld GF(2m). We present a vectorlevel algorithm which essentially eliminates the bitwise inner products needed in the conventional approach to the normal basis multiplication. We then present another algorithm which signio/cantly reduces the dynamic instruction counts. Both algorithms utilize the full width of the datapath of the general purpose processor on which the software is to be executed. We also consider composite o/elds and present an algorithm which can provide further speedups and an added AEexibility toward hardwaresoftware codesign of processors for very large o/nite o/elds.
Reconfigurable Implementation of Elliptic Curve Crypto Algorithms
 RECONFIGURABLE ARCHITECTURES WORKSHOP, 16TH INTERNATIONAL PARALLEL AND DISTRIBUTED PROCESSING SYMPOSIUM
, 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 characteristi ..."
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Cited by 13 (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.
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 10 (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 9 (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...
A New Algorithm for Multiplication in Finite Fields
 IEEE Transactions on Computers
, 1989
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