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130
Signature Schemes Based on the Strong RSA Assumption
 ACM TRANSACTIONS ON INFORMATION AND SYSTEM SECURITY
, 1998
"... We describe and analyze a new digital signature scheme. The new scheme is quite efficient, does not require the the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the socalled Strong RSA Assumption. Moreove ..."
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Cited by 150 (8 self)
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We describe and analyze a new digital signature scheme. The new scheme is quite efficient, does not require the the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the socalled Strong RSA Assumption. Moreover, a hash function can be incorporated into the scheme in such a way that it is also secure in the random oracle model under the standard RSA Assumption.
On Finite Pseudorandom Binary Sequences, IV. (The Liouville Function, II)
, 2000
"... this paper, we shall use the following notations: p i for the ith prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) ( ..."
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Cited by 34 (10 self)
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this paper, we shall use the following notations: p i for the ith prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) (this is the Liouville function) and (n) = ( 1) !(n) so that (n) is completely multiplicative and (n) is multiplicative, and let LN = f(1); (2); : : : ; (N)g GN = f(1); (2); : : : ; (N)g: For y 1 let y (n) and y (n) denote the multiplicative functions de ned by y (p ( 1) (= (p +1 for p > y y (p 1 (= (p +1 for p > y; respectively, and write LN (y) = f y (1); y (2); : : : ; y (N)g GN (y) = f y (1); y (2); : : : ; y (N)g: Research partially supported by Hungarian National Foundation for Scienti c Research, Grant No. T017433 MKM fund FKFP0139/1997 and by FrenchHungarian APAPEOMFB exchange program F5 /97
On the statistical properties of Diffie–Hellman distributions
 MR 2001k:11258 Zbl 0997.11066
"... Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that giv ..."
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Cited by 29 (10 self)
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Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen
Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
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Cited by 20 (1 self)
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“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients
, 1996
"... This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involv ..."
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Cited by 15 (1 self)
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This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is
The distribution of totients
, 1998
"... This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(x), the numb ..."
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Cited by 15 (6 self)
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This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(x), the number of m ≤ x with A(m) =k. The first of the main results of the paper is a determination of the true order of V (x). It is also shown that for each k ≥ 1, if there is a totient with multiplicity k, thenVk(x)≫V(x). We further show that every multiplicity k ≥ 2 is possible, settling an old conjecture of Sierpiński. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of V (x) andVk(x) also provides a description of the “normal ” multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a preimage of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x is c log log x, wherec≈2.186. Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler’s function.
SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES
, 2006
"... An old question of Erdős asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if � n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirm ..."
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Cited by 11 (0 self)
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An old question of Erdős asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if � n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, for each fixed number K> 1, the complement in Z of any union of residue classes r(n) (mod n), for distinct n ∈ (N, KN], has density at least dK for N sufficiently large. Here dK is a positive number depending only on K. Either of these new results implies another conjecture of Erdős and Graham, that if S is a finite set of moduli greater than N, with a choice for residue classes r(n) (mod n) for n ∈ S which covers Z, then the largest member of S cannot be O(N). We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight. 1
Chains of large gaps between consecutive primes
 Adv. in Math
, 1981
"... ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with ..."
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Cited by 11 (3 self)
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ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4 / Co e4/co = 3. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus. 1.