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43
SingleDatabase Private Information Retrieval with Constant Communication Rate
 In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming
, 2005
"... Abstract. We present a singledatabase private information retrieval (PIR) scheme with communication complexity O(k +d), where k ≥ log n is a security parameter that depends on the database size n and d is the bitlength of the retrieved database block. This communication complexity is better asympt ..."
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Cited by 53 (1 self)
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Abstract. We present a singledatabase private information retrieval (PIR) scheme with communication complexity O(k +d), where k ≥ log n is a security parameter that depends on the database size n and d is the bitlength of the retrieved database block. This communication complexity is better asymptotically than previous singledatabase PIR schemes. The scheme also gives improved performance for practical parameter settings whether the user is retrieving a single bit or very large blocks. For large blocks, our scheme achieves a constant “rate ” (e.g., 0.2), even when the userside communication is very low (e.g., two 1024bit numbers). Our scheme and security analysis is presented using general groups with hidden smooth subgroups; the scheme can be instantiated using composite moduli, in which case the security of our scheme is based on a simple variant of the “Φhiding ” assumption by Cachin, Micali and Stadler [2].
Primes in arithmetic progressions
 Math. Comp
, 1996
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 29 (2 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
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Cited by 22 (1 self)
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
Chebyshev’s bias for composite numbers with restricted prime divisors
 Math. Comp
, 2005
"... Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x ..."
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Cited by 13 (6 self)
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Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ a(mod d). In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x;4, 3) ≥ N(x;4, 1) for every x. In the process we express the socalled second order LandauRamanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants. 1.
Security of cryptosystems based on class groups of imaginary quadratic orders
 Okamoto (Ed.): Advances in Cryptology ASIACRYPT 2000, SpringerVerlag LNCS
, 1976
"... In this work we investigate the di culty of the discrete logarithm problem in class groups of imaginary quadratic orders. In particular, we discuss several strategies to compute discrete logarithms in those class groups. Based on heuristic reasoning, we give advice for selecting the cryptographic pa ..."
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Cited by 7 (1 self)
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In this work we investigate the di culty of the discrete logarithm problem in class groups of imaginary quadratic orders. In particular, we discuss several strategies to compute discrete logarithms in those class groups. Based on heuristic reasoning, we give advice for selecting the cryptographic parameter, i.e. the discriminant, such that cryptosystems based on class groups of imaginary quadratic orders would o er a similar security as commonly used cryptosystems. 1
A complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
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Cited by 6 (1 self)
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Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Laguerre polynomials with Galois group Am for each m
, 2008
"... In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m, there exists a polynomial of degree m with rational coefficients and associated Galois group Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational c ..."
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Cited by 5 (1 self)
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In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m, there exists a polynomial of degree m with rational coefficients and associated Galois group Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’s and early 1930’s, I. Schur found concrete examples of such polynomials among the classical Laguerre polynomials except in the case of polynomials with Galois group Am where m ≡ 2 (mod 4). Following up on work of R. Gow from 1989, this paper complements the work of Schur by showing that for every positive integer m ≡ 2 (mod 4), there is in fact a Laguerre polynomial of degree m with associated Galois group Am.
Prime factors of consecutive integers
 Math. Comp
"... A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, ..."
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Cited by 3 (1 self)
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A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing
An explicit zerofree region for the Dirichlet Lfunctions, ArXiv : math.NT/0510570
"... Abstract. Let Lq(s) be the product of Dirichlet Lfunctions modulo q. Then Lq(s) has at most one zero in the region ..."
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Cited by 3 (1 self)
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Abstract. Let Lq(s) be the product of Dirichlet Lfunctions modulo q. Then Lq(s) has at most one zero in the region