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 so-called 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.

this paper, we shall use the following notations: p i for the i-th 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 FKFP-0139/1997 and by French-Hungarian APAPE-OMFB exchange program F5 /97

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 L-functions, 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 Ω-results 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. 1.

Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an r-th power residue for all small factors of p − 1. The corresponding Diffie-Hellman (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

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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 non-trivial 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

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

: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of S--unit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...

An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.

ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.