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103
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 78 (2 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].
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 26 (2 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.
Ramanujan’s ternary quadratic form
 Invent. Math
, 1997
"... In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and ..."
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Cited by 23 (3 self)
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In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and
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 ever ..."
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Cited by 16 (7 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.
CONJECTURES INVOLVING ARITHMETICAL SEQUENCES
"... Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further ..."
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Cited by 15 (6 self)
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Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further research. 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 15 (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 10 (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.