E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality ', J. of Comp. and Syst. Sci., 62 (2001), 356--366.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....lower bound on the size of any Sigma 2 3 circuit for prime testing now follows. It should be stressed that the significance of this bound is that it is strictly exponential. n, rather than c n or n c for some constant c 1, appears in the exponent. Allender, Saks, and Shparlinski [2] have already proved that all constant depth circuits for primality (including Sigma 2 3 circuits which have depth 3) have size at least 2 n c for a fractional constant c depending on the depth. Theorem 11 Consider a Boolean function f of the variables X 1 ; X 2 ; X n defined by a ....

....standard switching lemma of [6] via the technique of [18] Lemma 4.5. Even Ajtai s original paper, combined with the conjecture, is enough to show that primality is not in AC 0 . Of course, stronger lower bounds for primality have been obtained unconditionally by Allender, Saks, and Shparlinski [2] using their very elegant indirect argument. Applying the generalized switching lemma of Lipton and Viglas [12] to obtain a whole circuit appoximation also yields unconditional results. However it would be nice to have a direct proof based on the standard cylinder techniques. Finally one can ....

E. Allender, M. Saks, and I. Shparlinski. A lower bound for primality. In Proceedings Fourteenth Annual IEEE Conference on Computational Complexity, pages 10--14, 1999.

....on the degree of certain polynomial representations for g . The linear bound on the average sensitivity of g also provides an alternative proof for the statement proved in [3, 4] that g does not belong to the class AC 0 . On the other hand, even a stronger result has recently been obtained in [1]. 2 Basic Definitions Let B n = f0; 1g n denote the n dimensional Boolean cube. For a binary vector a 2 B n we denote by a (i) the vector obtained from a by flipping its ith coordinate. Now we introduce the main combinatorial parameters of Boolean functions f : B n f0; 1g considered in this ....

....mentioning that since the average sensitivity of Boolean functions of the class AC 0 does not exceed (log n) O(1) as it is shown in [14] Theorem 5 provides an alternative proof for the statement proved in [3, 4] that g does not belong to AC 0 . This result has recently been improved in [1], where it is shown that for any prime p, testing square free numbers as well as primality testing and testing co primality of two given integers cannot be computed by AC 0 [p] circuits, that is, AC 0 circuits enhanced by Mod p gates. Apparently the result of Lemma 4 can be improved by means ....

E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality', Proc. IEEE Conf. on Comp. Compl., IEEE, 1999 (to appear).

....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 ....

....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. There is a very common belief that problems for polynomials over function fields are usually simpler than their analogues for the integers. For example, this belief is supported by the dramatic di#erence in the di#culty of polynomial and integer factorization, ....

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E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality ', J. of Comp. and Syst. Sci., 62 (2001), 356--366.

....results do not seem to have any cryptographic implications it is still interesting to study complexity characteristics of such an attractive number theoretic function. Various complexity lower bounds for Boolean functions associated with other natural number theoretic problems can be found in [1, 5 7, 14, 30]. Our main tool is exponential sums including a new upper bound of double exponential sums S a ( X , Y) X x#X X y#Y e (ag xy ) where e(z) exp(2#i p) with a # IF p and arbitrary sets X , Y # B n . These sums are of independent number theoretic interest. In particular they ....

E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality', Proc. 14 IEEE Conf. on Comp. Compl., Atlanta, 1999, IEEE Press, 1999, 10--14.

.... 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 ....

....constant, then the size turns out to be superpolynomial M exp(cn fl ) for some constants c 0 and fl 0 . In particular, this means that testing square free numbers, and thus integer factorization, cannot be done by a circuit of the class AC o . This result has recently been improved in [1], where it is shown that for any prime p , testing square free numbers as well as primality testing and testing co primality of two given integers cannot be computed by AC o [p] circuits, that is, AC o circuits enhanced by Mod p gates. Apparently the result of Lemma 2 can be improved by means ....

E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality', Preprint , 1998, 1--11.

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