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Fast Construction of Irreducible Polynomials over Finite Fields
 J. Symbolic Comput
, 1993
"... The main result of this paper is a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field F q . The algorithm is probabilistic, and is asymptotically faster than previously known algorithms for this problem. It uses an expected number of O~(n 2 + n log q) ..."
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Cited by 61 (6 self)
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The main result of this paper is a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field F q . The algorithm is probabilistic, and is asymptotically faster than previously known algorithms for this problem. It uses an expected number of O~(n 2 + n log q) operations in F q , where the "softO" O~ indicates an implicit factor of (log n) O(1) . In addition, two new polynomial irreducibility tests are described. 1 Introduction 1.1 Statement of main result Let F q be a finite field with q elements, where q is a primepower. A theorem due to Moore (1893) states that for every positive integer n, there exists a field extension F q n , unique up to isomorphism, with q n elements. Such extensions play an important role in coding theory (implementing error correcting codes), cryptography (implementing cryptosystems), and complexity theory (amplifying randomness). In this paper, we consider the algorithmic version of Moore's theorem: how to ...
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 22 (10 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
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 19 (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...
Cryptographic Counters and Applications to Electronic Voting
, 2001
"... We formalize the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly. ..."
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Cited by 11 (2 self)
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We formalize the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly.
Fast computation with two algebraic numbers
 September
, 2002
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 7 (3 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Fast arithmetic in unramified padic fields
 TO APPEAR IN FINITE FIELDS AND THEIR APPLICATIONS
, 2009
"... Let p be prime and Zpn a degree n unramified extension of the ring of padic integers Zp. In this paper we give an overview of some very fast deterministic algorithms for common operations in Zpn modulo pN. Combining existing methods with recent work of Kedlaya and Umans about modular composition of ..."
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Cited by 5 (2 self)
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Let p be prime and Zpn a degree n unramified extension of the ring of padic integers Zp. In this paper we give an overview of some very fast deterministic algorithms for common operations in Zpn modulo pN. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasilinear time algorithms in the parameters n and N, and quasilinear or quasiquadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmüller lifting and computing minimal polynomials.