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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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(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
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.
Counting numbers in multiplicative sets: Landau versus Ramanujan
 Math. Newsl
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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
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
NONZERO VALUES OF DIRICHLET LFUNCTIONS IN VERTICAL ARITHMETIC PROGRESSIONS
"... Every Dirichlet Lfunction L(s, χ) has infinitely many nontrivial zeros, and our knowledge about their vertical distribution is rather good. Our understanding of the algebraic nature of their imaginary parts, on the other hand, is extremely meager. We have no reason to doubt that these imaginary par ..."
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Every Dirichlet Lfunction L(s, χ) has infinitely many nontrivial zeros, and our knowledge about their vertical distribution is rather good. Our understanding of the algebraic nature of their imaginary parts, on the other hand, is extremely meager. We have no reason to doubt that these imaginary parts are haphazard transcendental numbers that have no algebraic relationships among