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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Deterministically Testing Sparse Polynomial Identities of Unbounded Degree
, 2008
"... We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representa ..."
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Cited by 5 (0 self)
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We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2 s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only blackbox access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linnik’s Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Cited by 4 (2 self)
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
Zeros of families of automorphic Lfunctions close to 1
 Pacific J. Math
"... For many Lfunctions of arithmetic interest, the values on or closeto theedgeof theregion of absoluteconvergenceare of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to nonvanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet Lfunctions ( ..."
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Cited by 4 (1 self)
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For many Lfunctions of arithmetic interest, the values on or closeto theedgeof theregion of absoluteconvergenceare of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to nonvanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet Lfunctions (e.g., because of the Dirichlet classnumber formula) and the symmetric square Lfunctions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upperbound, on average, for the number of zeros of the Lfunctions which are very close to 1. We prove a very general statement of this typefor forms on GL(n)/Q for any n � 1, comparableto the logfree density theorems for Dirichlet characters first proved by Linnik. 1. Introduction.
On the Least Prime in Certain Arithmetic Progressions
"... We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. gr ..."
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Cited by 2 (0 self)
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We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the kfold iterated logarithm, fl is Euler's constant, and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm...
PAPER Special Section on Cryptography and Information Security The Computational Difficulty of Solving Cryptographic Primitive Problems Related to the Discrete Logarithm Problem
, 2005
"... SUMMARY To the authors ’ knowledge, there are not many cryptosystems proven to be as difficult as or more difficult than the discrete logarithm problem. Concerning problems related to the discrete logarithm problem, there are problems called the double discrete logarithm problem and the eth root of ..."
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SUMMARY To the authors ’ knowledge, there are not many cryptosystems proven to be as difficult as or more difficult than the discrete logarithm problem. Concerning problems related to the discrete logarithm problem, there are problems called the double discrete logarithm problem and the eth root of the discrete logarithm problem. These two problems are likely to be difficult and they have been utilized in cryptographic protocols such as verifiable secret sharing scheme and group signature scheme. However, their exact complexity has not been clarified, yet. Related to the eth root of the discrete logarithm problem, we can consider a square root of the discrete logarithm problem. Again, the exact complexity of this problem has not been clarified, yet. The security of cryptosystems using these underlying problems deeply depends on the difficulty of these underlying problems. Hence it is important to clarify their difficulty. In this paper we prove reductions among these fundamental problems and show that under certain conditions, these problems are as difficult as or more difficult than the discrete logarithm problem modulo a prime. key words: discrete logarithm problem, double discrete logarithm problem, square root of discrete logarithm problem, eth root of discrete logarithm problem 1.
Groups of cubefree order by
"... q, and r denote positive integers with p, q, and r signifying primes, and take x to be a positive real number. We call the number n cubefree if n is not divisible by the cube of a prime. Denote the Euler phifunction of n by φ(n), and the natural logarithm of x by log x. Put L2x = log log x, L3x = l ..."
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q, and r denote positive integers with p, q, and r signifying primes, and take x to be a positive real number. We call the number n cubefree if n is not divisible by the cube of a prime. Denote the Euler phifunction of n by φ(n), and the natural logarithm of x by log x. Put L2x = log log x, L3x = logL2x,