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15
Searching for Primitive Roots in Finite Fields
, 1992
"... Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). ..."
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Cited by 51 (3 self)
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Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...
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...
Constructing Normal Bases in Finite Fields
 J. Symbolic Comput
, 1990
"... This paper addresses the question: how can we find a normal element efficiently? More generally, we consider how to find an element of any given additive order. Hensel (1888) pioneered the study of normal bases for finite fields and proved that they always exist. We use his algorithm in Section 2. E ..."
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Cited by 6 (0 self)
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This paper addresses the question: how can we find a normal element efficiently? More generally, we consider how to find an element of any given additive order. Hensel (1888) pioneered the study of normal bases for finite fields and proved that they always exist. We use his algorithm in Section 2. Eisenstein (1850) had already noted that normal bases always exist. Hensel, and also Ore (1934), determine exactly the number of these bases, and Ore develops the more general concept of additive order. Ore's approach is developed into more constructive proofs of the normal basis theorem in several textbooks (for example, van der Waerden 1966, Section 67, and Albert 1956, Section 4.15); these all use some linear algebra calculations. Schwarz (1988) has given a new proof along these lines, and several recent papers have translated this approach into algorithms. Sidel'nikov (1988) deals with the case where n divides one of p (the characteristic of F q ), q + 1, or
Primitive free cubics with specified norm and trace
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... The existence of a primitive free (normal) cubic x 3 − ax 2 + cx − b over a finite field F with arbitrary specified values of a (�=0)andb (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed. ..."
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Cited by 5 (0 self)
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The existence of a primitive free (normal) cubic x 3 − ax 2 + cx − b over a finite field F with arbitrary specified values of a (�=0)andb (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
Permutation Groups, ErrorCorrecting Codes and Uncoverings
, 2005
"... We replace the traditional setting for errorcorrecting codes (i.e. linear codes) with that of permutation groups, with permutations in list form as the codewords. We introduce a decoding algorithm for these codes, which uses the following notion. A base for a permutation group is a sequence of poin ..."
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Cited by 4 (4 self)
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We replace the traditional setting for errorcorrecting codes (i.e. linear codes) with that of permutation groups, with permutations in list form as the codewords. We introduce a decoding algorithm for these codes, which uses the following notion. A base for a permutation group is a sequence of points whose stabiliser is trivial. An uncoveringbybases (or UBB) is a set of bases such that any combination of error positions is avoided by at least one base in the set. In the case of sharply ktransitive groups, any ktuple of points forms a base, so a UBB can be formed from the complements of the blocks of a covering design. (In this case, we use the term uncovering.) A large part of the thesis (chapters 2 to 5) is concerned with constructing UBBs for groups which are basetransitive, i.e. which act transitively on their irredundant bases, which were classified by T. Maund. Various combinatorial, algebraic and numbertheoretic techniques are employed in this. Other topics include a case study of the Mathieu group M12, where we investigate ways in which
Approximate constructions in finite fields
 Proc. 3rd Conf. on Finite Fields and Appl
, 1995
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MATHEMATICS ON THE ALGEBRAIC CLOSURE OF TWO BY
"... (Communicated by Prof. J. H. van Lint at the meeting of January 29, 1977) ..."
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(Communicated by Prof. J. H. van Lint at the meeting of January 29, 1977)
GENERATORS OF ELLIPTIC CURVES OVER FINITE
"... Abstract. We prove estimates on character sums on the subset of points of an elliptic curve over IFq n with xcoordinate of the form α + t where t ∈ IFq varies and fixed α is such that IFq n = IFq(α). We deduce that, for a suitable choice of α, this subset has a point of maximal order in E(IFq n). T ..."
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Abstract. We prove estimates on character sums on the subset of points of an elliptic curve over IFq n with xcoordinate of the form α + t where t ∈ IFq varies and fixed α is such that IFq n = IFq(α). We deduce that, for a suitable choice of α, this subset has a point of maximal order in E(IFq n). This provides a deterministic algorithm for finding a point of maximal order which for a very wide class of finite fields is faster than other available algorithms. 1.