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Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1
Anatomy of Integers and Cryptography
, 2008
"... It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and under ..."
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It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and understanding of the multiplicative structure of “typical ” integers and also “typical ” terms of various sequences such as shifted primes, polynomials, totients and so on. Part I
SEQUENCES OF CONSECUTIVE SMOOTH POLYNOMIALS
"... (Communicated by WenChing Winnie Li) Abstract. Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n. 1. ..."
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(Communicated by WenChing Winnie Li) Abstract. Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n. 1.
The Wild Numbers
, 2004
"... This paper studies the integers that belong the multiplicative semigroup W generated by ..."
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This paper studies the integers that belong the multiplicative semigroup W generated by
Wild and Wooley Numbers
, 2005
"... The wild integer semigroup W(Z) consists of the integers in the multiplicative semigroup generated by { 3n+2 1 ..."
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The wild integer semigroup W(Z) consists of the integers in the multiplicative semigroup generated by { 3n+2 1
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 ARITHMETIC PROPERTIES OF POLYNOMIAL SPECIALIZATIONS OVER FINITE FIELDS
"... We present applications of some recent results that establish a partial finite field analogue of Schinzel’s Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polyn ..."
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We present applications of some recent results that establish a partial finite field analogue of Schinzel’s Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polynomial substitutions without prime factors of large degree (“smooth ” polynomial substitutions); this confirms a finite field analogue of a conjecture of Martin in certain ranges of the parameters. Other topics considered include an analogue of Brun’s constant for polynomials and “smooth ” values of neighboring polynomials. 1.
Examining Committee:
, 2008
"... The ring of univariate polynomials over a finite field shares many foundational arithmetic properties with the ring of rational integers. This similarity makes it possible for many problems in elementary number theory to be translated ‘through the looking glass ’ into the universe of polynomials. In ..."
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The ring of univariate polynomials over a finite field shares many foundational arithmetic properties with the ring of rational integers. This similarity makes it possible for many problems in elementary number theory to be translated ‘through the looking glass ’ into the universe of polynomials. In this thesis we look at polynomial analogues of Schinzel’s Hypothesis H and other problems related to the multiplicative structure of polynomial values. We obtain results both in the situation where the finite field Fq is fixed and in the more uniform situation where Fq is allowed to vary. The most important tool in these investigations is Weil’s Riemann Hypothesis for global function fields, which yields an explicit form of the Chebotarev density theorem for such fields. ii Acknowledgements Perhaps the best place to start these acknowledgements is with my family, who are responsible for so much of who I am today. I am grateful to them for all of their love and support, and for only occasionally asking “but what is all this good for anyway?”