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Abstract: Recent work by Bernasconi, Damm and Shparlinski showed that the set of square-free numbers is not in AC , and raised as an open question whether similar (or stronger) lower bounds could be proved for the set of prime numbers. In this note, we show that the Boolean majority function is AC -Turing reducible to the set of prime numbers (represented in binary). From known lower bounds on Maj (due to Razborov and Smolensky) we conclude that primality cannot be tested in AC [p] for... (Update)
Context of citations to this paper: More
...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...
.... 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...
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BibTeX entry: (Update)
E. Allender, M. Saks and I. E. Shparlinski, `A lower bound for primality', Proc. IEEE Conf. on Comp. Compl., IEEE, 1999 (to appear). http://citeseer.ist.psu.edu/allender99lower.html More
@article{ allender99lower,
author = "Eric Allender and Igor Shparlinski and Michael E. Saks",
title = "A Lower Bound for Primality",
journal = "Electronic Colloquium on Computational Complexity (ECCC)",
volume = "6",
number = "010",
year = "1999",
url = "citeseer.ist.psu.edu/allender99lower.html" }
Citations (may not include all citations):189 Algebraic methods in the theory of lower bounds for Boolean .. (context) - Smolensky - 1987
92 Riemann's hypothesis and tests for primality (context) - Miller - 1976
92 Constant depth reducibility (context) - Chandra, Stockmeyer et al. - 1984
70 Two theorems on random polynomial time (context) - Adleman - 1978
66 A fast Monte Carlo test for primality (context) - Solovay, Strassen
57 and Circuit Complexity (context) - Straubing, Automata et al. - 1994
54 Lower bounds on the size of bounded depth networks over a co.. (context) - Razborov
51 On distinguishing prime numbers from composite numbers (context) - Adleman, Pomerance et al. - 1987
36 Recognizing primes in random polynomial time (context) - Adleman, Huang - 1987
28 Springer-Verlag (context) - Prachar - 1957
18 Number theoretic methods in cryptography: Complexity lower b.. (context) - Shparlinski - 1999
15 One-way functions and circuit complexity (context) - Boppana, Lagarias - 1987
14 Reducing the complexity of reductions - Agrawal, Allender et al. - 1997
10 Circuit complexity of testing square-free numbers - Bernasconi, Shparlinski - 1999
9 On polynomial representations of Boolean functions related t.. - Shparlinski - 1998
9 Circuit and decision tree complexity of some number theoreti.. - Bernasconi, Damm et al. - 1998
8 On tape bounds for single letter alphabet language processin.. (context) - Hartmanis, Berman - 1976
8 Probabilistic algorithm for primality testing (context) - Rabin - 1980
8 Reductions in circuit complexity: An isomorphism theorem and.. - Agrawal, Allender et al. - 1998
7 the average sensitivity of testing square-free numbers - Bernasconi, Damm et al. - 1999
5 Computational Complexity Column - Allender, the et al. - 1998
5 Lower bounds for arithmetic problems (context) - Meidanis - 1991
5 the recognition of primes by automata (context) - Hartmanis, Shank - 1968
4 Two infinite sets of primes with fast primality tests (context) - Pintz, Steiger et al. - 1989
3 the circuit complexity of primality (context) - Lipton, Viglas
3 Circuits in bounded arithmetic (context) - Mantzivis - 1992
2 The average sensitivity of square-freeness (context) - Bernasconi, Damm et al. - 1999
2 the number of primes in an arithmetic progression (context) - Page - 1935
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Algebraic Methods for Proving Lower Bounds in Circuit Complexity - Allender (Correct)
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