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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 12 (10 self)
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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...
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 ..."
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Cited by 6 (4 self)
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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 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 1 (1 self)
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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