A. Bernasconi, C. Damm and I. E. Shparlinski, On the average sensitivity of testing square-free numbers,in"Proc.5thIntern.Com- puting and Combin. Conf.", Lect. Notes in Comp. Sci., SpringerVerlag, Berlin, 1627 (1999), 291--299.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77] but Supported in part by NSF grant CCR 9734918. y Supported in part by NSF grant CCR 9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS98b, BS99, Shp98], other than the work of [Med91, Man92] almost nothing has been published regarding lower bounds on the complexity of this set. To be sure, in the context of space bounded computation, it was shown in [HS68] that at least logarithmic space is required, in order to determine if a number is prime. ....

....an exponential lower bound on circuit size for this class of circuits. In order to simplify the exposition, we present only the superpolynomial lower bound in this note. Our proof applies equally well to other number theoretic problems, such as Square Free and GCD. It follows from the results of [BDS98a, BDS98b, BS99] that these problems do not belong to AC 0 . Here we extend their results to the more powerful complexity classes AC 0 [p] After presenting our definitions and the proofs of our main results, we close the paper with a discussion of related open problems. 2 Preliminaries A circuit family is a ....

A. Bernasconi, C. Damm and I. E. Shparlinski, On the average sensitivity of testing square-free numbers, in "Proc. 5th Intern. Computing and Combin. Conf.", Lect. Notes in Comp. Sci., SpringerVerlag, Berlin, (to appear).

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A. Bernasconi, C. Damm and I. E. Shparlinski, On the average sensitivity of testing square-free numbers,in"Proc.5thIntern.Com- puting and Combin. Conf.", Lect. Notes in Comp. Sci., SpringerVerlag, Berlin, 1627 (1999), 291--299.

....Boolean functions related to number theoretic problems are a natural object to study from the complexity viewpoint. Recently, lower bounds for several such functions have been obtained, for computational models such as unbounded fan in Boolean circuits, decision trees, and real polynomials (see [3, 9, 10, 11, 12, 17, 28, 29]) The two main ingredients of these papers are harmonic analysis and estimates based on number theoretic considerations. In this paper we extend some results of the aforementioned papers to some problems related to deciding arithmetic properties of polynomials over IF 2 . Our primary motivation ....

....proved. As one might expect, some of the same techniques that have proved useful in establishing lower bounds for number theoretic problems over the integers are also useful in proving lower bounds for analogous problems for polynomials over IF 2 ,and indeed our techniques are similar to those of [9, 10, 11, 12]. Nevertheless, some new di#culties and e#ects arise when working over IF 2 [T ] Forexample, some of our results are more precise than those known for analogous problems over the integers. On the other hand, we have not been able to extend some of the results of [3] to the case of polynomials. ....

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A. Bernasconi, C. Damm and I. E. Shparlinski, `On the average sensitivity of testing square-free numbers', Proc. 5th Intern. Computing and Combinatorics Conf., Tokyo, 1999, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 1999, v.1627, 291--299. 23

No context found.

A. Bernasconi, C. Damm and I. E. Shparlinski, `On the average sensitivity of testing square-free numbers', Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 1627 (1999), 291--299.

....part by ARC grant A69700294. 1 Introduction What is the computational complexity of the set of prime numbers There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77] but as was pointed out recently in [BDS98a, BDS98b, BS99, Shp98], other than the work of [Med91, Man92] almost nothing has been published regarding lower bounds on the complexity of this set. In the context of space bounded computation, it was shown in [HS68] that at least logarithmic space is required, in order to determine if a number is prime. This was ....

....of bounded depth of AND, OR, NOT and Mod p gates that computes Primes, GCD, or Square Free for n bit inputs has size exponential in n ffi for some ffi 0. In particular these three languages are not in AC 0 [p] For the case of Square Free and GCD our results strengthen and simplify those of [BDS98a, BDS98b, BS99] who showed that these two languages are not in AC 0 . 2 Preliminaries 2.1 Languages, functions and circuits For a string x 2 f0; 1g , jxj denotes the length of x. Since we are dealing with number theoretic functions and our strings often represent integers, it is convenient to index our ....

A. Bernasconi, C. Damm and I. E. Shparlinski, On the average sensitivity of testing square-free numbers, in "Proc. 5th Intern. Computing and Combin. Conf.", Lect. Notes in Comp. Sci., SpringerVerlag, Berlin, (to appear).

.... square free numbers is the only known problem, related to the integer factorization problem, for which an unconditional deterministic polynomial time algorithm is known, see [19] Some results of this paper have recently been generalized in [6] Several more relevant results can also be found in [1, 7]. 2 Basic Definitions Let Bn = f0; 1g n denote the n dimensional Boolean cube. We will use the notation jf j to denote the number of strings accepted by the function f , that is jf j = jfw 2 Bn j f(w) 1gj . Moreover, p f denotes the probability that the function f takes the value 1 (over the ....

....to the degree of the polynomial approximation of f over the reals, see [2, 5, 8, 22] Applying our results, one can derive the estimate oe av (f) c n 1=2 log Gamma2 n for the function f given by (1) where c 0 is an absolute constant. However, using a more direct approach, it is shown in [7] that in fact oe av (f) 4 9 2 n o(n) This bound implies several other results about various complexity characteristics of f , such as the formula size, the average decision tree depth and the degrees of exact and approximative polynomial representations of this function over the reals. ....

A. Bernasconi, C. Damm and I. E. Shparlinski, `On the average sensitivity of testing square-free numbers', Preprint , 1998, 1--11.

....of prime numbers There is a large body of work presenting important upper bounds on the complexity Supported in part by NSF grant CCR 9509603. y Supported in part by ARC grant A69700294. of the set of primes (including [AH87, APR83, Mil76, R80, SS77] but as was pointed out recently in [BDS98a, BDS98b, BS99, Shp98], other than the work of [Med91] almost nothing has been published regarding lower bounds on the complexity of this set. It was not even known if primality testing could be accomplished by constant depth, polynomial size circuits of AND and OR gates. That is, it was not known if Primes was in AC ....

....an exponential lower bound on circuit size for this class of circuits. In order to simplify the exposition, we present only the superpolynomial lower bound in this note. Our proof applies equally well to other number theoretic problems, such as Square Free and GCD. It follows from the results of [BDS98a, BDS98b, BS99] that these problems do not belong to AC 0 . Here we extend these results to more powerful complexity classes AC 0 [p] After presenting our definitions and the proofs of our main results, we close the paper with a discussion of related open problems. 2 Preliminaries A circuit family is a set ....

A. Bernasconi, C. Damm and I. E. Shparlinski, On the average sensitivity of testing square-free numbers, Preprint, 1998, 1--10.

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