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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 ..."
Abstract

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...
Efficient VLSI implementation for Montgomery multiplication in GF(2 m
 Journal of Science and Engineering
"... The Montgomery multiplication algorithm without division operations is popular both in prime field GF(p) and Finite field GF(2 m). However, the Montgomery multiplication algorithm has the timedependent problem. We will present a timeindependent Montgomery multiplication algorithm. The results show ..."
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Cited by 1 (1 self)
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The Montgomery multiplication algorithm without division operations is popular both in prime field GF(p) and Finite field GF(2 m). However, the Montgomery multiplication algorithm has the timedependent problem. We will present a timeindependent Montgomery multiplication algorithm. The results show that our proposed timeindependent Montgomery multiplication algorithm not only saves about 50 % time complexity but also saves about 11 % space complexity as compared to the traditional Montgomery multiplication algorithm. Our proposed systolic array Montgomery multiplier has simplicity, regularity, modularity, and concurrency, and is very suitable for VLSI implementation.
i Preface
"... This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k). ..."
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This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k).