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28
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 17 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Interpolation of ShiftedLacunary Polynomials [Extended Abstract]
"... Abstract. Given a “black box ” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shiftα∈Q, the exponents 0≤e1 ..."
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Cited by 9 (1 self)
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Abstract. Given a “black box ” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shiftα∈Q, the exponents 0≤e1< e2<···<et, and the coefficients c1,...,ct∈Q\{0} such that f (x)=c1(x−α) e1 + c2(x−α) e2 +···+ct(x−α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size and in particular is logarithmic in deg f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information. We give both an unconditional deterministic algorithm which is polynomialtime but has a rather high complexity, and a more practical probabilistic algorithm which relies on some unknown constants.
Group automorphisms with few and with many periodic points
 Proc. Amer. Math. Soc
, 2005
"... Abstract. For any C ∈ [0, ∞] a compact group automorphism T: X → X is constructed with the property that 1 n log {x ∈ X  T n (x) = x}  − → C. This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological en ..."
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Cited by 6 (5 self)
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Abstract. For any C ∈ [0, ∞] a compact group automorphism T: X → X is constructed with the property that 1 n log {x ∈ X  T n (x) = x}  − → C. This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy. 1.
Privacy amplification and nonmalleable extractors via character sums
 In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, 2011. [DO03] Y. Dodis and
"... In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a nonmalleable extractor. A nonmalleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weaklyrandom x and a uniform ..."
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Cited by 6 (2 self)
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In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a nonmalleable extractor. A nonmalleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weaklyrandom x and a uniformly random seed y, and outputs a string which appears uniform, even given y. For a nonmalleable extractor nmExt, the output nmExt(x,y) should appear uniform given y as well as nmExt(x, A(y)), where A is an arbitrary function with A(y) = y. We show that an extractor introduced by Chor and Goldreich is nonmalleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is 1/2 + α, for any α> 0. Previously, no nontrivial parameters were known for any nonmalleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widelybelieved conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves character sum estimates, which may be of independent interest. Using our nonmalleable extractor, we obtain protocols for “privacy amplification”: key agreement between two parties who share a weaklyrandom secret. Our protocols work in the
Graphs of Prescribed Girth and BiDegree
"... We say that a bipartite graph Γ(V1 ∪ V2, E) has bidegree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t> 3, very few examples of (r, s, t)–graphs were previously known. In this paper we gi ..."
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Cited by 5 (2 self)
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We say that a bipartite graph Γ(V1 ∪ V2, E) has bidegree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t> 3, very few examples of (r, s, t)–graphs were previously known. In this paper we give a recursive construction of (r, s, t)–graphs for all r, s, t ≥ 2, as well as an algebraic construction of such graphs for all r, s ≥ t ≥ 3.
Fast Integer Multiplication Using Modular Arithmetic
 In Fortieth Annual ACM Symposium on Theory of Computing
, 2008
"... We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses ..."
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Cited by 5 (0 self)
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We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer’s algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a padic version of Fürer’s algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar. 1
Average Multiplicative Orders of Elements Modulo n
 Acta Arith
"... We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary ..."
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Cited by 4 (1 self)
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We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary 20K01 1
On The Uniformity Of Distribution Of The NaorReingold PseudoRandom Number Generator
, 1999
"... We show that the new pseudorandom number generator, introduced recently by M. Naor and O. Reingold, possess one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds o ..."
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Cited by 4 (3 self)
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We show that the new pseudorandom number generator, introduced recently by M. Naor and O. Reingold, possess one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions.
LEAST TOTIENT IN A RESIDUE CLASS
 BULL. LONDON MATH. SOC. 39 (2007) 425–432
, 2007
"... For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler ..."
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Cited by 4 (2 self)
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For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler function in reduced residue classes modulo m. Some discussion and results are also given for classes with gcd(a, m)>1, in which case such n do not always exist, and also on the related problem for ‘cototients’.
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