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Design and Analysis of Practical PublicKey Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack
 SIAM Journal on Computing
, 2001
"... A new public key encryption scheme, along with several variants, is proposed and analyzed. The scheme and its variants are quite practical, and are proved secure against adaptive chosen ciphertext attack under standard intractability assumptions. These appear to be the first publickey encryption sc ..."
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Cited by 205 (11 self)
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A new public key encryption scheme, along with several variants, is proposed and analyzed. The scheme and its variants are quite practical, and are proved secure against adaptive chosen ciphertext attack under standard intractability assumptions. These appear to be the first publickey encryption schemes in the literature that are simultaneously practical and provably secure.
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 163 (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.
Prime specialization in genus 0
 Trans. Amer. Math. Soc
"... Abstract. For a prime polynomial f(T) ∈ Z[T], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T) ∈ κ[u][T], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T)i ..."
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Cited by 5 (2 self)
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Abstract. For a prime polynomial f(T) ∈ Z[T], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T) ∈ κ[u][T], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T)isapolynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values µ(f(g)) as g varies. We prove the surprising fact that this “Möbius average, ” which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve f =0. The periodic Möbius average behavior implies in specific examples that a polynomial in κ[u][T] does not take prime values as often as analogies with Z[T] suggest, and it leads to a modified conjecture for how often prime values occur. 1.
Largest known twin primes and Sophie Germain primes
 Math. Comp
, 1999
"... Abstract. The numbers 242206083 · 2 38880 ± 1 are twin primes. The number ..."
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Cited by 4 (0 self)
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Abstract. The numbers 242206083 · 2 38880 ± 1 are twin primes. The number
A generalization of a conjecture of hardy and littlewood to algebraic number fields
 Rocky Mountain J. Math
"... Abstract. We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and ktuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and ..."
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Abstract. We generalize conjectures of Hardy and Littlewood concerning the density of twin primes and ktuples of primes to arbitrary algebraic number fields. In one of their great Partitio Numerorum papers [7], Hardy and Littlewood advance a number of conjectures involving the density of pairs and ktuples of primes separated by fixed gaps. For example, if d is even, we define Pd(x) = {0 < n < x: n, n + d are both prime}. They conjecture both that and that P2(x) is asymptotic to 2 ∏ p>2 Pd(x) ∏ p − 1 lim = x→ ∞ P2(x) p − 2 odd pd
with the BrezingWeng Method
, 2008
"... Abstract. The BrezingWeng method is a general framework to generate families of pairingfriendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endo ..."
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Abstract. The BrezingWeng method is a general framework to generate families of pairingfriendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number. Keywords: Pairingfriendly curve generation, BrezingWeng method. 1
More Discriminants with the BrezingWeng Method
"... Abstract. The BrezingWeng method is a general framework to generate families of pairingfriendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endo ..."
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Abstract. The BrezingWeng method is a general framework to generate families of pairingfriendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number. Keywords: Pairingfriendly curve generation, BrezingWeng method. 1
LOWER BOUND FOR THE REMAINDER IN THE PRIMEPAIR CONJECTURE
, 2008
"... Taking r> 0 let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. A heuristic argument indicates that the remainder e2r(x) in this approximation cannot b ..."
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Taking r> 0 let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. A heuristic argument indicates that the remainder e2r(x) in this approximation cannot be of lower order than x β, where β is the supremum of the real parts of zeta’s zeros. The argument also suggests an approximation for π2r(x) similar to one of Riemann for π(x).
A Note On Shanks's Chains Of Primes
, 2000
"... . For integers a and b we define the Shanks chain p 1 ; p 2 ; : : : ; p k of length k to be a sequence of k primes such that p i+1 = ap i 2 \Gamma b for i = 1; 2; : : : ; k \Gamma 1. While for Cunningham chains it is conjectured that infinitely long chains exist, this is, in general, not true f ..."
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. For integers a and b we define the Shanks chain p 1 ; p 2 ; : : : ; p k of length k to be a sequence of k primes such that p i+1 = ap i 2 \Gamma b for i = 1; 2; : : : ; k \Gamma 1. While for Cunningham chains it is conjectured that infinitely long chains exist, this is, in general, not true for Shanks chains. In fact, with s = ab we show that for all but 56 values of s 1000 any corresponding Shanks chain must have finite length. For this, we study certain properties of functional digraphs of quadratic functions over prime fields, both in theory and practice. We give efficient algorithms to investigate these properties and present a selection of our experimental results. 1. Introduction Let ffl 2 f+1; \Gamma1g be fixed. A Cunningham chain p 1 ; p 2 ; p 3 ; : : : ; p n of length k (see Guy [9], xA7) is a sequence of k primes such that p i+1 = 2p i + ffl (i = 1; 2; : : : ; k \Gamma 1) : For example, if ffl = 1, we say that 2; 5; 11; 23; 47 is a Cunningham chain of length...
AVERAGE PRIMEPAIR COUNTING FORMULA
, 902
"... Abstract. Taking r> 0, let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) = π2r(x)−2C2r ..."
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Abstract. Taking r> 0, let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) = π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x) −li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results. 1.