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53
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
Ideal forms of Coppersmith’s theorem and GuruswamiSudan list decoding
"... Abstract: We develop a framework for solving polynomial equations with size constraints on solutions. We obtain our results by showing how to apply a technique of Coppersmith for finding small solutions of polynomial equations modulo integers to analogous problems over polynomial rings, number field ..."
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Abstract: We develop a framework for solving polynomial equations with size constraints on solutions. We obtain our results by showing how to apply a technique of Coppersmith for finding small solutions of polynomial equations modulo integers to analogous problems over polynomial rings, number fields, and function fields. This gives us a unified view of several problems arising naturally in cryptography, coding theory, and the study of lattices. We give (1) a polynomialtime algorithm for finding small solutions of polynomial equations modulo ideals over algebraic number fields, (2) a faster variant of the GuruswamiSudan algorithm for list decoding of ReedSolomon codes, and (3) an algorithm for list decoding of algebraicgeometric codes that handles both singlepoint and multipoint codes. Coppersmith’s algorithm uses lattice basis reduction to find a short vector in a carefully constructed lattice; powerful analogies from algebraic number theory allow us to identify the appropriate analogue of a lattice in each case and provide efficient algorithms to find a suitably short vector, thus allowing us to give completely parallel proofs of the above theorems.
Computing Arakelov class groups
, 2008
"... Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of ..."
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Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.
Factoring polynomials over special finite fields
, 2000
"... We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which Φ (p) is built up from small prime factors; here Φ denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In th ..."
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We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which Φ (p) is built up from small prime factors; here Φ denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k"1, when Φ (p)"p!1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which Φ (p) has a prime factor l with l,1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.
Computing automorphisms of abelian number fields
 Math. Comput
, 1999
"... Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x)ofαover L. In practice, the existing algorithms for factori ..."
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Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x)ofαover L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on padic techniques. Given mα(x) anda rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α. 1.
Algebraic Number Theory
 www.jmilne.org/math
, 2009
"... Version 3.06 May 28, 2014An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of int ..."
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Version 3.06 May 28, 2014An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. BibTeX information
Solvability by Radicals from an Algorithmic Point of View
, 2001
"... Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to nd suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extension ..."
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Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to nd suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extensions of prime degree and then work out the radicals, using the work of Girstmair. We give numerical examples of Abelian and nonAbelian solvable equations and apply the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.
Algorithms for Quadratic Orders
 PROCEEDINGS OF SYMPOSIUM ON MATHEMATICS OF COMPUTATION
, 1993
"... We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining ..."
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We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining discrete logarithms in the class group. We also prove upper bounds for the time and space complexity of the algorithms.
Computing Igusa class polynomials
, 2008
"... We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our alg ..."
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We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O( ∆ 7/2), where ∆ is the discriminant of K. 1