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19
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Short Representation of Quadratic Integers
 PROCEEDINGS OF CANT
, 1992
"... Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representa ..."
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Cited by 13 (3 self)
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Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representations we can in polynomial time compute norms, signs, products, and inverses of numbers in O and principal ideals generated by numbers in O. We also show how to compare numbers given in compact represention in polynomial time.
An Investigation of Bounds for the Regulator of Quadratic Fields
 Experimental Mathematics
, 1995
"... This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large ..."
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Cited by 12 (6 self)
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This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large
On Smooth Ideals in Number Fields
 J. Number Theory
, 1993
"... For y 2 IR ?0 an integral ideal of an algebraic number field F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. Assuming the generalized Riemann hypothesis we prove a lower bound for the number /F (x; y) of integral ysmooth ideals in F whose norms are bounded ..."
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For y 2 IR ?0 an integral ideal of an algebraic number field F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. Assuming the generalized Riemann hypothesis we prove a lower bound for the number /F (x; y) of integral ysmooth ideals in F whose norms are bounded by x 2 IR ?0 . Apart from x and y this bound only depends on the degree of F . 1 Introduction and result Let F be an algebraic number field of discriminant \Delta F . For y 2 IR ?0 an integral ideal of F is called ysmooth if the norms of all of its prime ideal factors are bounded by y. The number of integral ysmooth ideals of F whose norms do not exceed x 2 IR ?0 is denoted by /F (x; y). In this paper we prove the following result. 1. Theorem Assume that the Generalized Riemann Hypothesis (GRH) is correct. For any n 2 IN and for any ffl 2 IR ?0 there is an effectively computable constant x 0 (ffl; n) 2 IR ?0 such that for any x; y 2 IR ?0 with x ? x 0 (ffl; n) and for every algebra...
Computational techniques in quadratic fields
, 1995
"... c○Michael John Jacobson, Jr. 1995iiI hereby declare that I am the sole author of this thesis. I authorize the University of Manitoba to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Manitoba to reproduce this thesis ..."
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Cited by 3 (1 self)
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c○Michael John Jacobson, Jr. 1995iiI hereby declare that I am the sole author of this thesis. I authorize the University of Manitoba to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Manitoba to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii The University of Manitoba requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. Since Kummer’s work on Fermat’s Last Theorem, algebraic number theory has been a subject of interest for many mathematicians. In particular, a great amount of effort has been expended on the simplest algebraic extensions of the rationals, quadratic fields. These are intimately linked to binary quadratic forms and have proven to be a good testing ground for algebraic number theorists because, although computing with ideals and field elements is relatively easy, there are still many unsolved and difficult problems remaining.
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
 J. COMPUT. SYSTEM SCI., STOC ’99 SPECIAL ISSUE
, 1999
"... We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I Given a polynomial f 2Z[v;x; y], decide the sentence 9v 8x 9y f(v; x; y) ..."
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We consider the averagecase complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: I Given a polynomial f 2Z[v;x; y], decide the sentence 9v 8x 9y f(v; x; y)
Finiteness for Arithmetic Fewnomial Systems
, 2001
"... Suppose L is any finite algebraic extension of either the ordinary rational numbers or the padic rational numbers. Also let g 1,..., g k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g i is exactly m. We prove that the ..."
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Suppose L is any finite algebraic extension of either the ordinary rational numbers or the padic rational numbers. Also let g 1,..., g k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g i is exactly m. We prove that the maximum number of isolated roots of G := (g 1 ,... , g k ) in L n is finite and depends solely on (m; n; L), i.e., is independent of the degrees of the g i . We thus obtain an arithmetic analogue of Khovanski's Theorem on Fewnomials, extending earlier work of Denef, Van den Dries, Lipshitz, and Lenstra.
Close to Uniform Prime Number Generation With Fewer Random Bits
"... Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algor ..."
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Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algorithm studied by Brandt and Damg˚aard and its many variants implemented in numerous software packages, which reduce the number of random bits used at the price of a distribution easily distinguished from uniform. Our new method is also no more computationally expensive than the ones in current use, and opens up interesting options for prime number generation in constrained environments. Keywords: Publickey cryptography, prime number generation, RSA, efficient implementations, random bits. 1
THE EFFECTIVE CHEBOTAREV DENSITY THEOREM AND MODULAR FORMS MODULO m
"... Abstract. Suppose that f (resp. g) is a modular form of integral (resp. halfintegral) weight with coefficients in the ring of integers OK of a number field K. For any ideal m ⊂ OK, we bound the first prime p for which f  Tp (resp. g  Tp2) is zero (mod m). Applications include the solution to a qu ..."
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Abstract. Suppose that f (resp. g) is a modular form of integral (resp. halfintegral) weight with coefficients in the ring of integers OK of a number field K. For any ideal m ⊂ OK, we bound the first prime p for which f  Tp (resp. g  Tp2) is zero (mod m). Applications include the solution to a question of Ono [AO] concerning partitions. 1.