Results 1  10
of
14
On the Divisibility of Fermat Quotients
"... We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
(Show Context)
We show that for a prime p the smallest a with a p−1 ̸ ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1). Keywords: sieve. Fermat quotients, smooth numbers, Heilbronn sums, large AMS Mathematics Subject Classification: 11A07, 11L40, 11N25 1
Circuit and Decision Tree Complexity of Some Number Theoretic Problems
, 1998
"... We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems. In particular, we prove that deciding if a given integer is squarefree and testing coprimality of two int ..."
Abstract

Cited by 13 (11 self)
 Add to MetaCart
We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems. In particular, we prove that deciding if a given integer is squarefree and testing coprimality of two integers by unbounded fanin circuits of bounded depth requires superpolynomial size. 1 Introduction In recent years spectral techniques based on the Abstract Harmonic Analysis on the hypercube have been shown to represent a very useful tool for obtaining lower complexity bounds. Various links between Fourier coefficients of Boolean functions and their complexity characteristics have been studied in a number of works, see [1, 2, 3, 4, 6, 8, 13, 19, 20, 22, 23]. In particular, these spectral techniques have been successfully applied to the parity function and to threshold functions. Institut fur Informatik, Technische Universitat Munchen, D80290 Munchen, Germany. bernasco@informatik.tumue...
Circuit complexity of testing squarefree numbers
 Proceedings of the 16th Ann. Symposium on Theoretical Aspects of Computer Science, Lecture
"... ..."
(Show Context)
On polynomial representations of Boolean functions related to some number theoretic problems
 Electronic Colloq. on Comp. Compl
, 1998
"... Abstract. We say a polynomial P over ZM strongly Mrepresents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P onesidedly Mrepresents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of pol ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Abstract. We say a polynomial P over ZM strongly Mrepresents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P onesidedly Mrepresents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of polynomials over Z M, which strongly or onesidedly Mrepresent the Boolean function deciding if a given nbit integer is squarefree. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function. 1
On vanishing Fermat quotients and a bound of the Ihara sum
 Kodai Math. J. (to appear
"... ar ..."
(Show Context)
Uses of Randomness in Computation
, 1994
"... Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worst ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worstcase behaviour of deterministic algorithms, but it can also consider averagecase behaviour if it is assumed that the input data is drawn randomly from a given distribution. Rabin popularised the idea of &quot;probabilistic &quot; algorithms, where randomness is incorporated into the algorithm instead of being assumed in the input data. Yao showed that there is a close connection between the complexity of probabilistic algorithms and the averagecase complexity of deterministic algorithms. We give examples of the uses of randomness in computation, discuss the contributions of Rabin, Yao and others, and mention some open questions.