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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 ..."
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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 Aspect of Dual Basis for Efficient Field Arithmetic
 IN PROCEEDINGS OF ESCOL 7
, 1990
"... In this manuscript we consider the special type of dual basis for finite fields, GF (2 m ), where the variants of m are presented in the following contents. Here we introduce our field representing method for its efficient arithmetic(of field multiplication and field inversion). It revealed a very ..."
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In this manuscript we consider the special type of dual basis for finite fields, GF (2 m ), where the variants of m are presented in the following contents. Here we introduce our field representing method for its efficient arithmetic(of field multiplication and field inversion). It revealed a very effective role for both software and VLSI implementations, but the aspect of hardware design for its structure is out of this manuscript and so, here, we deal only the case of its software implementation (the efficiency of hardware implementation is appeared in another article submitted to IEEE Transactions on Computers). A brief description of this advantageous characteristics is that (1) the field multiplication can be constructed only by k( m 2 ) rotations and the same amount of vector XOR processes, (2) there is needed no additional work load as basis changing(from standard to the dual basis or from the dual basis to standard basis as the conventional dual based arithmetic does), (3...
Normal and Selfdual Normal Bases from Factorization of cx
 SIAM J. Discrete Mathematics
, 1992
"... The present paper is interested in a family of normal bases, considered by V. M. Sidel'nikov, with the property that all the elements in a basis can be obtained from one element by repeatedly applying to it a linear fractional function of the form #(x)=(ax + b)/(cx + d), a, b, c, d # F q . Side ..."
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The present paper is interested in a family of normal bases, considered by V. M. Sidel'nikov, with the property that all the elements in a basis can be obtained from one element by repeatedly applying to it a linear fractional function of the form #(x)=(ax + b)/(cx + d), a, b, c, d # F q . Sidel'nikov proved that the cross products for such a basis {# i } are of the form # i # j = e ij # i + e ji # j +#, i #= j, where e k ,##F q . We will show that every such basis can be formed by the roots of an irreducible factor of F (x)=cx  ax  b. We will construct: (a) a normal basis of F q n over F q with complexity at most 3n  2 for each divisor n of q  1 and for n = p where p is the characteristic of F q ; (b) a selfdual normal basis of F q n over F q for n = p and for each odd divisor n of q  1orq+ 1. When n = p, the selfdual normal basis constructed of F q p over F q also has complexity at most 3p  2. In all cases, we will give the irreducible polynomials and the multiplication tables explicitly. Abbreviated title: Normal Bases. 1991 Mathematics subject classification: 11T30, 11T06. Key words: finite field, irreducible polynomial, normal basis. 1