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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.
Efficient Software Implementation for Finite Field Multiplication in Normal Basis
 In Information and Communications Security (ICICS), SpringerVerlag LNCS 2229
, 2001
"... Abstract. Finite field arithmetic is becoming increasingly important in today's computer systems, particularly for implementing cryptographic operations. Among various arithmetic operations, finite field multiplication is of particular interest since it is a major building block for elliptic curve c ..."
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Cited by 16 (0 self)
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Abstract. Finite field arithmetic is becoming increasingly important in today's computer systems, particularly for implementing cryptographic operations. Among various arithmetic operations, finite field multiplication is of particular interest since it is a major building block for elliptic curve cryptosystems. In this paper, we present new techniques for efficient software implementation of binary field multiplication in normal basis. Our techniques are more efficient in terms of both speed and memory compared with alternative approaches. 1 Introduction Finite field arithmetic is becoming increasingly important in today's computer systems, particularly for implementing cryptographic operations. Among the more common finite fields in cryptography are oddcharacteristic finite fields of degree 1 and evencharacteristic finite fields of degree greater than 1. The latter is conventionally known as GF (2m) arithmetic or binary field arithmetic. GF (2m) arithmetic is further classified according to the choice of basis for representing elements of the finite field; two common choices are polynomial basis and normal basis. Fast implementation techniques for GF (2m) arithmetic have been studied intensively in the past twenty years. Among various arithmetic operations, GF (2m) multiplication has attracted most of the attention since it is a major building block for implementing elliptic curve cryptosystems. Depending on the choice of basis, the mathematical formula for a GF (2m) multiplication can be quite different, thus making major differences in practical implementation. Currently, it seems that normal basis representation (especially optimal normal basis) offers the best performance in hardware [911], while in software polynomial basis representation is more efficient [2, 3, 8].
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...