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Improved NonCommitting Encryption with Applications to Adaptively Secure Protocols
"... Abstract. We present a new construction of noncommitting encryption schemes. Unlike the previous constructions of Canetti et al. (STOC ’96) and of Damg˚ard and Nielsen (Crypto ’00), our construction achieves all of the following properties: – Optimal round complexity. Our encryption scheme is a 2r ..."
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Abstract. We present a new construction of noncommitting encryption schemes. Unlike the previous constructions of Canetti et al. (STOC ’96) and of Damg˚ard and Nielsen (Crypto ’00), our construction achieves all of the following properties: – Optimal round complexity. Our encryption scheme is a 2round protocol, matching the round complexity of Canetti et al. and improving upon that in Damg˚ard and Nielsen. – Weaker assumptions. Our construction is based on trapdoor simulatable cryptosystems, a new primitive that we introduce as a relaxation of those used in previous works. We also show how to realize this primitive based on hardness of factoring. – Improved efficiency. The amortized complexity of encrypting a single bit is O(1) public key operations on a constantsized plaintext in the underlying cryptosystem. As a result, we obtain the first noncommitting publickey encryption schemes under hardness of factoring and worstcase lattice assumptions; previously, such schemes were only known under the CDH and RSA assumptions. Combined with existing work on secure multiparty computation, we obtain protocols for multiparty computation secure against a malicious adversary that may adaptively corrupt an arbitrary number of parties under weaker assumptions than were previously known. Specifically, we obtain the first adaptively secure multiparty protocols based on hardness of factoring in both the standalone setting and the UC setting with a common reference string. Key words: publickey encryption, adaptive corruption, noncommitting encryption, secure multiparty computation. 1
Finding Meaning in Error Terms
, 2007
"... (In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate ..."
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Cited by 9 (1 self)
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(In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate
The prime number race and zeros of Dirichlet Lfunctions off the critical line
 Duke Math. J
"... ABSTRACT. We show, for any q � 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet Lfunctions lying off the critical line implies that π(x;q,a) < π(x;q,b) for a set of real x of asymptotic density 1. 1 ..."
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ABSTRACT. We show, for any q � 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet Lfunctions lying off the critical line implies that π(x;q,a) < π(x;q,b) for a set of real x of asymptotic density 1. 1
THE DISTRIBUTION OF PRIME NUMBERS
, 2006
"... What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consec ..."
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What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the nth smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for
CHEBYSHEV’S BIAS FOR PRODUCTS OF TWO PRIMES
"... ABSTRACT. Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers � x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the Extended Riemann Hypothesis for Dirichlet Lfunction ..."
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ABSTRACT. Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers � x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the Extended Riemann Hypothesis for Dirichlet Lfunctions modulo q, and (ii) that the imaginary parts of the nontrivial zeros of these Lfunctions are linearly independent over the rationals. Our results are analogs of similar results proved for primes in arithmetic progressions by Rubinstein and Sarnak. 1.
AVERAGE PRIMEPAIR COUNTING FORMULA
, 2009
"... Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ 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) th ..."
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Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ 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.
FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES
"... Abstract. We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo m. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the “probability ” th ..."
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Abstract. We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo m. Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the “probability ” that a pair of consecutive prime numbers of size approximately x will be congruent to (a, a+d) modulo m. We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for x in various ranges. 1.
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
Article electronically published on September 25, 2009 AVERAGE PRIMEPAIR COUNTING FORMULA
"... Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ 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) th ..."
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Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ 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.
A RAPID GENERATION SCHEME OF LARGE PRIMES
"... RSA is by far the most widely adopted standard in public key cryptography algorithm. Its security depends on the Integer Factorization Problem, that it is very easy to calculate the product of two large prime numbers, but the decomposition the product and get the prime factors are very difficult. Th ..."
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RSA is by far the most widely adopted standard in public key cryptography algorithm. Its security depends on the Integer Factorization Problem, that it is very easy to calculate the product of two large prime numbers, but the decomposition the product and get the prime factors are very difficult. Therefore, the generations of large primes are important research field. This paper presented a rapid generation scheme of it, and the techniques for the related software implementation were presented. To speed up the modular multiplication and squaring, Montgomery's algorithms were used with sliding window method. Three pretreatments were also described in details. In view of prime generation of RSA, a series of design methods for software implementation was proposed and give the optimization programs.