P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15(4):994--1003, November 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Hesse University of Massachusetts E mail: whesse cs.umass.edu and Eric Allender Rutgers University Piscataway, NJ 08854 8019 E mail: allender cs.rutgers.edu and David A. Mix Barrington University of Massachusetts E mail: barring cs.umass.edu It has been known since the mid 1980 s [16, 48, 49] that integer division can be performed by poly time uniform circuits of Majority gates; equivalently, the division problem lies in P uniform TC . Recently this was improved to L uniform TC [21] but it remained unknown whether division can be performed by DLOGTIMEuniform circuits. The ....

....follows that L NP. Key Words: computer arithmetic, division, iterated multiplication, threshold circuits, uniformity, circuit complexity, complexity classes, finite model theory, Chinese remainder representation, computation in abelian groups 1. INTRODUCTION In 1984 Beame, Cook, and Hoover [16] presented new parallel algorithms for division, powering, and iterated multiplication of integers. They showed that these problems can be solved by families of circuits with fan in two and O(log n) depth, placing them in the circuit class NC . As Reif [48, 49] observed soon after, their ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

....Special elliptic curves with ecient arithmetics has also been considered [11] 20] 21] In this paper we propose the application of parallel processing to speed up the exponentiation on elliptic curves. There already exists parallel algorithms for exponentiation of integers modulo a m bit integer [2, 1, 13] and also for exponentiation on the elds GF (2 ) 22] and GF (q ) 23] We present in this paper a parallel algorithm to compute scalar multiplication on elliptic curves. This algorithm would be specially ecient for anomalous binary curves, also known as Koblitz curves. The application of ....

P.W. Beame, S.A. Cook and H.J. Hoover, Log depth circuits for division and related problems, SIAM Journal on Computing, vol. 15, pp. 994-1003, 1986.

....are present on real computers. Instead we do allow them, but charge them more, using the well studied depth measure from circuit complexity. Note that in the Circuit RAM model, the two level hashing scheme of [FKS84] has query time O(log w= log log w) since multiplication and integer division [BCH86] are in NC and any function in NC has polynomial size circuits of depth O(log w= log log w) CSV84] We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is no Circuit RAM solution to the static dictionary problem using space and with query ....

P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

....we do not disallow expensive instructions such as multiplication, we just charge them more, using the well studied depth measure from circuit complexity. In the Circuit RAM model, the two level hashing scheme of [13] has query time O(log w= log log w) since multiplication and integer division [7] are in NC and any function in NC has polynomial size circuits of depth O(logw= log log w) 9] We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is no Circuit RAM solution to the static dictionary problem using space and with query time ....

P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

....is in Uniform TC 0 William Hesse # Department of Computer Science University of Massachusetts Amherst, MA 01002 FAX: 001) 413 545 1249 whesse cs.umass.edu Abstract. Integer division has been known since 1986 [4, 13, 12] to be in slightly non uniform TC 0 , i.e. computable by polynomial size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in ....

....complexity of integer division has been harder to pin down than the complexities of addition, subtraction, and multiplication. In 1986, Beame, Cook, and Hoover showed that iterated multiplication, and thus division, could be performed by Boolean circuits of logarithmic depth (NC 1 circuits) [4]. In 1987, Reif showed that these circuits could be implemented as constant depth circuits containing threshold gates (TC 0 circuits) 12, 13] Since then, the remaining issue has been the complexity of constructing these circuits. Division is the only prominent natural problem whose computation ....

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

....NP. # Supported in part by NSF grant CCR 9734918. Supported in part by NSF grant CCR 9988260. # Supported by NSF grant CCR 9877078. 1 Introduction The exact complexity of division, powering, and iterated multiplication of integers has been a major open problem since Beame, Cook, and Hoover [7] showed these problems to be in P uniform TC 0 in 1984 1 . TC 0 is the set of problems solvable by threshold circuits of constant depth and polynomial size, P uniform means that these circuits can be constructed by a poly time Turing machine. In a recent breakthrough, Chiu, Davida and ....

....This predicate calculates powers modulo a prime of O(log n) bits. We then consider various algorithms for powering modulo a small prime, and their consequences for the uniformity of division circuits. Following an argument of Chiu [10] powering modulo a small prime can be achieved by 1 [7] claimed only P uniform NC 1 , but it was observed later by Reif [23] that their algorithm is implementable in TC 0 . fully uniform circuits of logarithmic depth and fan in two ( Ruzzo uniform NC 1 ) and hence division is itself in uniform NC 1 , in fact in NC 1 uniform TC 0 . But ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994-- 1003, 1986.

....r n by dropping all but the low order n 3 bits. To raise the number ra j;i to the kth power, we will need the following lemma: Lemma 3. Finding the n O(1) th power of a number with n O(1) bits is in TC 1 uniform TC 0 Proof. A corollary to the result of Beame, Cook, and Hoover 3 [2] shows that we can multiply n numbers with n bits each with a TC 0 circuit that is constructible in logspace from the product of the rst n 2 primes [7] The product of the rst n 2 primes can be computed by a TC 1 circuit (a depth O(log n) binary tree of TC 0 circuits performing ....

Paul W. Beame, Stephen A. Cook, and H. James Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994-1003, 1986.

....other hard closure properties in [OH93] Consider, for example, bf(x) g(x)c for f 2 #P and nonzero g 2 #P. It is easy to design a #P function, say h(x) f(x)2 p(jxj) g(x) for some suitably large polynomial p, from which logarithmically depth bounded circuits can compute the division (see [BCH86]) But, combining the result of Furst, Saxe and Sipser [FSS84] with the easily provable fact that the parity function is AC 0 reducible to integer division, it is seen that no AC 0 circuit can compute the division from h above. Thus, studying the closure properties of #P in context AC 0 ....

P. Beame, S. Cook and H. Hoover, Log depth circuits for division and related problems, SIAM J. Comput. 15 (1986), 994--1003. 15

....breakthrough will be presented at ICALP 2001 by Bill Hesse, a student at the University of Massachusetts. He will be receiving the best paper award for Track A at ICALP 2001 (combined with the best student paper award) All of these results build on the earlier work of Beame, Cook, and Hoover ([9]) In the following sections, I will provide the necessary background about the complexity classes I ll be discussing, and then I ll present the history of these breakthroughs, and the main ideas involved. In a closing section, I ll discuss some of the applications that these advances have already ....

....also complete for TC 0 under AC 0 T reductions. It had been known for a while that multiplication reduces to division, and thus division was known to be hard for TC 0 . In fact, division had been known to be in P uniform TC 0 ever since it was observed in [27, 28] that the algorithm of [9] can be implemented in P uniform TC 0 . The breakthrough of [18] is that division is in DLOGTIME uniform TC 0 . 3 Background on Division All of the recent work on division builds on the work of Beame, Cook, and Hoover [9] Beame, Cook, and Hoover make use of the fact that, for small enough ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

....is very cheap compared with a multiple product or a modular exponentiation. As described in the Introduction and in Section 2.2, we are also interested in nding the parallel time complexity of the pseudo random functions. In order to do so, let us rst recall the result of Beame, Cook and Hoover [2] who showed that division and related operations including multiple product are computable in NC 1 . Based on this result Reif and Tate [62, 63] showed that these operations are also computable in TC 0 . The exact depth 6 In the case that Q is much smaller than P we have that the rst ....

P. W. Beame, S. A. Cook and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., vol. 15, 1986, pp. 994-1003.

....5 D(n) The proof of theorem 3.2 requires that we build up families of Threshold Circuits for the basic problems of multiplication, iterated sum, and iterated product. The most costly problem we encounter is iterated product, and this is solved using techniques introduced for integer division [2, 8, 18]. As a consequence of Theorem 3.2, we have Corollary 3.3. Suppose an analytic function f(x) has a convergent Taylor Series Expansion of form f(x) 1 X n=0 c n (x Gamma x 0 ) n over an interval jx Gamma x 0 j ffl where 0 ffl 1, and the coefficients are rationals c n = an bn ....

.... logarithm of all input values (f Gamma1 i (x) above) performing the iterated sum of these values modulo p i Gamma 1, then raising the generator to the resulting power in Z p i (this is just f i (x) above) This is a fairly common method of performing iterated product (see, for example, [2]) The only part we haven t examined here is the iterated sum. By Lemma 5.1, we can calculate the exact iterated sum of m ffi numbers, each of log p i bits, in size O(m ffi ffi 2 log p i ) and depth O( 1 ffi ) With an m ffi log p i bit approximation to (1= p i Gamma 1) we can reduce ....

P. W. Beame, S. A. Cook, and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., 15 (1986), pp. 994--1003.

....uniformity notion under which NC 1 becomes the class of regular languages. If our non uniformity resource is between DLOGTIME and NC 1 itself, we get a robust class, equal to ALOGTIME. And if we allow polynomial time to build our circuits, we can then do integer division and related problems [BCH] which (as far as we know) we couldn t do before. In general our techniques for proving lower bounds on circuit complexity are combinatorial and algebraic and apply to the totally non uniform versions of the circuit classes. A notable exception is the result by Allender and Gore [AG] that the ....

P. Beame, S. Cook, H.J. Hoover, "Log depth circuits for division and related problems," SIAM J. Comput.15:4 (1986), 994-1003.

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15(4):994--1003, November 1986.

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P. W. Beame, S. A. Cook, and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., 15 (1986), pp. 994--1003.

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P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

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P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

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P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. In Annual IEEE Symposium on Foundations of Computer Science, pages l--6, 1984.

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

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P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

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P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15(4):994--1003, 1986.

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Paul W. Beame, Stephen A. Cook, and H. James Hoover. Log depth circuits for division and related problems. SIAM J. Comput. , 15(4):994--1003, 1986.

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P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. S.I.A.M. Journal on Computing, 15(4), 1986, pp. 994-1003.

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Beame, P., Cook, S.A., and Hoover, J. (1986). Log depth circuits for division and related problems. SIAM Journal on Computing 15, 994--1003. 28

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P.W. Beam, S.A. Cook, and H.J. Hoover, Log Depth Circuits for Division and Related Problems, in Proc. of the 25th IEEE Symp. on the Foundations of Computer Science (1984), pp. 1-6.

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