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Smooth Orders and Cryptographic Applications
 Lecture Notes in Comptuer Science
, 2002
"... We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker re ..."
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Cited by 5 (1 self)
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We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.
A Survey of GcdSum Functions
, 2010
"... We survey properties of the gcdsum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcdsum function and for the function defined by the harmonic mean of the gcd’s. ..."
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Cited by 4 (2 self)
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We survey properties of the gcdsum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcdsum function and for the function defined by the harmonic mean of the gcd’s.
Simultaneous prime specializations of polynomials over finite fields
"... Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in par ..."
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Cited by 2 (0 self)
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Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in particular ranges of the parameters. We give some applications including the solution of a problem posed by C. Hall.
On the biunitary analogues of Euler’s arithmetical function and the gcdsum function
 J. Integer Sequences 12 (2009), Article 09.5.2
"... We give combinatorialtype formulae for the biunitary analogues of Euler’s arithmetical function and the gcdsum function and prove asymptotic formulae for the latter one and for another related function. 1 ..."
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Cited by 1 (1 self)
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We give combinatorialtype formulae for the biunitary analogues of Euler’s arithmetical function and the gcdsum function and prove asymptotic formulae for the latter one and for another related function. 1
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
ERDŐSTURÁN WITH A MOVING TARGET, EQUIDISTRIBUTION OF ROOTS OF REDUCIBLE QUADRATICS, AND DIOPHANTINE QUADRUPLES
, 903
"... ABSTRACT. A Diophantine mtuple is a set A of m positive integers such that ab + 1 is a perfect square for every pair a, b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ..."
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ABSTRACT. A Diophantine mtuple is a set A of m positive integers such that ab + 1 is a perfect square for every pair a, b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The ErdősTurán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials. 1.
ON COMMON VALUES OF φ(n) AND σ(m), I
"... Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1. ..."
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Abstract. We show, conditional on a uniform version of the prime ktuples conjecture, that there are x/(log x) 1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sumofdivisors function. 1.
SETS OF MONOTONICITY FOR EULER’S TOTIENT FUNCTION
"... Abstract. We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author. 1. ..."
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Abstract. We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author. 1.
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"... ErdősTurán with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples. (English summary) Mathematika 57 (2011), no. 1, 1–29.20417942 A Diophantine quadruple is an increasing sequence of four positive integers such that the product of any two is one less tha ..."
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ErdősTurán with a moving target, equidistribution of roots of reducible quadratics, and Diophantine quadruples. (English summary) Mathematika 57 (2011), no. 1, 1–29.20417942 A Diophantine quadruple is an increasing sequence of four positive integers such that the product of any two is one less than a square. Let N(x) be the number of such quadruples in the interval [1, x]. Then the principal result of the paper is that N(x) ∼ Cx 1/3 (log x) with C = 24/33−1Γ ( 2 3)−3. Indeed, an explicit error term is provided. It was shown by A. Dujella [Ramanujan J. 15 (2008), no. 1, 37–46; MR2372791 (2008j:11135)]