E. Bach. Realistic analysis of some randomized algorithms. J. Comput. Syst. Sci., 42:30-53, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....practice a stream of random bits is produced by a pseudorandom function which has been seeded with a little nondeterminism 1 , say the digits from the computer s clock. It is not clear if the seed is truly random, and even less clear how good the commonly used pseudo random functions are. Bach [Bac91] studied a few number theoretic algorithms under the assumption of a truly random seed and a commonly used pseudo random func This work was supported in part by Texas Advanced Research Program Grant 003658 0029 1999 and NSF Grant CCR 9988160. Address: Dept. of Computer Sciences, The Univ. of ....

E. Bach. Realistic analysis of some randomized algorithms. J. Comput. Syst. Sci., 42:30-53, 1991.

.... random bits is produced by a pseudo random function which has been seeded with a little non determinism 1 , say the least signi cant digits from the computer s clock. It is not clear if the seed is truly random, and even less clear how good the commonly used pseudo random functions are. Bach [Bac91] studied a few number theoretic algorithms under the assumption of a truly random seed and a commonly used pseudo random function, and showed them to have a good probability of success, though not as good as guaranteed by using totally random bits. Karlo and Raghavan [KR93] assumed the same ....

E. Bach. Realistic analysis of some randomized algorithms. J. of Comput. and Syst. Sci., 42:30-53, 1991.

....Another reason is from the complexity theoretic point of view: to provide a full scale of options between an algorithm that is completely deterministic, and a randomized algorithm that consumes many bits. Examples of previous work in reducing the number of random bits in randomized algorithms are [2, 11, 20, 28, 32, 33, 43, 47, 49]. Karloff and Raghavan [33] for example, studied several randomized algorithms for selection and sorting and showed that they can be run successfully when only O(log n) random bits are available. In Section 7 we describe how to reduce the number of random bits for three problems. The first one is ....

E. Bach, Realistic analysis of some randomized algorithms, Proceedings of the 19th annual ACM Symposium on Theory of Computing, (1987), pp. 453-461.

....regardless of the distribution of x 1 ; x 2 . If x 1 ; x 2 are random numbers modulo N then we can set r i = i in Algorithm 1. We then use the heuristic assumption that the Jacobi symbols ( x i P ) i = 1; k) are approximately independent random variables 2 for random x. Recent work [2, 4, 12, 1] on the distribution of Jacobi symbols lends much theoretical support to this assumption. In particular, these sequences are ffl biased, a type of distribution which can be proven to closely approximate the distribution of fair independent coins [10, 12] Thus, we again expect the probability of ....

Bach, E.: Realistic analysis of some randomized algorithms. Journal of Computer and System Sciences 42 (1991) 30--53.

....pattern of quadratic residues and nonresidues. We give (exponentially low) upper bounds for the probability of failure achievable in polynomial time using, as a source of randomness, no more that one random number modulo P . 2 Notation and a result from algebraic geometry Following terminology in [3], we call the sequence x 1;x 2; x t the increment sequence of length t and seed x. Throughout this paper, ffl Theta will denote a set of integers. ffl f Theta will denote the polynomial Q i2 Theta (x i) ffl P will denote an odd prime number. ffl X P will denote the quadratic ....

Eric Bach. Realistic analysis of some randomized algorithms. Journal of Computer and System Sciences, 42(1):30--53, 1991.

....for the Solovay Strassen test of n Gamma1=2 1=c o(1) Indeed, if n is divisible by a prime less than (log n) c , then the algorithm will fail to find a witness with probability O( log n) c =n) otherwise, n) log n=c loglog n and the bound follows from Theorem 1.2. 1.2. Related Work. Bach (1991) examined the error probability of the Miller Rabin test using the sequence x; x 1; x k Gamma 1; where x 2 Z n is chosen at random and k = d 1 2 log 2 ne. Bach proved that the error probability is at most n Gamma1=4 o(1) in this case. Our Theorem 1.1 is a quantitative ....

....O( log n) c =n) otherwise, n) log n=c loglog n and the bound follows from Theorem 1.2. 1.2. Related Work. Bach (1991) examined the error probability of the Miller Rabin test using the sequence x; x 1; x k Gamma 1; where x 2 Z n is chosen at random and k = d 1 2 log 2 ne. Bach proved that the error probability is at most n Gamma1=4 o(1) in this case. Our Theorem 1.1 is a quantitative improvement of Bach s result, and the techniques we use are closely related to those used by Bach. However, the methods in Bach s paper do not appear to directly yield a similar result ....

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E. Bach. Realistic analysis of some randomized algorithms. J. Comput. System Sci. 42 (1991), 30--53.

....for generating the samples from the random bits can be designed so that the quantity P i=1 Y i can be computed directly and efficiently from the random bits, without the need to calculate the result of each trial explicitly. Similar ideas have been used successfully in other contexts [ACGS] [Bach], CG] CW] KR] Luby] 2) Is there a deterministic algorithm with running time o(2 n ) which accepts as 12 input A and ffl and which outputs Y such that (1 Gamma ffl)per(A) Y (1 ffl)per(A) ....

E. Bach, "Realistic Analysis of Some Randomized Algorithms", 19 th STOC, 1987, pp. 453-461.

....proof can be thought of as the output of a degenerate pseudorandom generator that takes no seed at all and does not affect the performance of the algorithm. The catch, of course, is that we do not know of any efficient way to generate g. LECTURE 11. BPP PSIZE; PSEUDORANDOM GENERATORS 33 Bach [5] was the first person to consider the effect of having pseudorandom generators on the performance of probabilistic algorithms. He confined himself to algorithms for number theoretic problems such as compositeness testing. We will discuss subsequent results of Karloff and Raghavan [23] who studied ....

E. Bach. Realistic analysis of some randomized algorithms. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 453--461, New York, NY, May 1987.

....that opening some of the blobs would give information about the bits hidden in other blobs, because the elements of Z N [ 1] are not chosen independently, or that joint information about several hidden bits could be computed efficiently by the Verifier. Nevertheless, recent results by Bach [4], Damgard [21] and Alon, Goldreich, Hastad and Peralta [38, 1] on the Legendre and Jacobi symbols of consecutive numbers provide a beginning of a theoretical justification for this scheme. 8 Statistical zero knowledge arguments Throughout this paper, we have used blobs that are unconditionally ....

Bach, E., "Realistic analysis of some randomized algorithms", Journal of Computer and System Sciences, Vol. 42, no. 1, pp. 30 -- 53, 1991.

....and Sipser [KPS85] They showed that the error probability of an RP algorithm can be made polynomially small, with no additional random bits. Chor and Goldreich [CG86] showed that the same amplification can be achieved for BPP algorithms, at the cost of doubling the amount of random bits. Bach [Bac87] showed that an exponentially small error can be attained for specific algorithms (e.g. primality testing) by increasing the number of random bits by only a constant factor. Using the sampling technique based on random walks on expander graphs we show that one can amplify a constant success ....

E. Bach. Realistic analysis of some randomized algorithms. In STOC, pages 453--461, 1987.

....over a ring. The worst case and average case analysis of the dependence on p in the running time of our algorithm makes use of estimates of the number of solutions to equations over finite fields; similar techniques have been previously used in the analysis of various probabilistic algorithms [4, 5, 6]. We also mention another deterministic factoring algorithm (conveyed to the author by Lenstra [29] that uses a baby step giant step method to obtain a p 1=2 Delta (n log p) O(1) running time bound; however, the algorithm requires time and space p 1=2 Delta (n log p) O(1) even in the ....

E. Bach. Realistic analysis of some randomized algorithms. In 19th Annual ACM Symposium on Theory of Computing, pages 453--461, 1987. Final version to appear, J. Comput. Sys. Sci.

....in n and log p, and upon termination, it either outputs an irreducible polynomial over F of degree n, or reports failure. Furthermore, the probability that it fails is no more than 1=p cn . This result is of value in a setting where random bits are viewed as a scarce resource. See Shoup [1987] Bach [1987], Bach and Shoup [1988] Karloff and Raghavan [1988] and Krizanc, Peleg and Upfal [1988] for other work along these lines. 2. Reduction to Constructing Cyclotomic Extensions and Finding Nonresidues This section is devoted to a proof of Theorem 2.1. Assume that for each prime q j n; q 6= p, we ....

E. Bach [1987]. "Realistic analysis of some randomized algorithms," in Proc. 19th Annual ACM Symp. on Theory of Computing, pp. 453-461.

....over a ring. The worst case and average case analysis of the dependence on p in the running time of our algorithm makes use of estimates of the number of solutions to equations over finite fields; similar techniques have been previously used in the analysis of various probabilistic algorithms [2, 3, 4]. The rest of this paper is organized as follows: Section 2 describes our new factoring algorithm, Section 3 analyzes its worst case complexity, and Section 4 analyzes its average case complexity. One last matter of notation: throughout this paper, log x denotes the logarithm of x to the base 2. ....

E. Bach. "Realistic analysis of some randomized algorithms," in Proc. 19th Annual ACM Symp. on Theory of Computing, pp. 453-461 (1987).

....paper was presented on November 3, 1995 at the AMS meeting at Kent State University, Kent, Ohio. Supported by the Butler Summer Institute. E mail: sorenson butler.edu, URL: http: www.butler.edu sorenson ; Supported by a Butler University Faculty Research Fellowship. a prime (see [4, 5, 21]) in writing a prime as a sum of two squares [31] and in several cryptography schemes that are based on the difficulty of computing square roots modulo a composite number (see, for example, 20, 37, 38] As there are (p Gamma 1) 2 quadratic nonresidues modulo any odd prime p, to find a ....

E. Bach. Realistic analysis of some randomized algorithms. Journal of Computer and System Sciences 42(1) (1991) 30--53.

....transformation of [4] mentioned in the Remark above) preserves security too. Preliminaries: selecting prime numbers Prime numbers play a key role in all our constructions and so efficient algorithms for selecting such numbers are of key importance to us. We will use two algorithms due to Bach [1, 2]. The first algorithm [2] is merely a very efficient (problem specific) deterministic amplification of the Miller Rabin primality tester [9, 13] The second algorithm [1] produces uniformly distributed integers together with their prime factorization. Theorem 1 (randonmess efficient primality ....

....in the Remark above) preserves security too. Preliminaries: selecting prime numbers Prime numbers play a key role in all our constructions and so efficient algorithms for selecting such numbers are of key importance to us. We will use two algorithms due to Bach [1, 2] The first algorithm [2] is merely a very efficient (problem specific) deterministic amplification of the Miller Rabin primality tester [9, 13] The second algorithm [1] produces uniformly distributed integers together with their prime factorization. Theorem 1 (randonmess efficient primality tester [2] There ....

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E. Bach, "Realistic Analysis of some Randomized Algorithms", 19th STOC, 1987, pp. 453-- 461.

....so is h(x) If ffl(x) 0, then ffl(x) 2 Gammab(x) so h(x) 1. Therefore, the best non zero half cost we could hope for is h(x) O(1) If h(x) H for all x, then ffl(x) 2 Gammab(x) H , i.e. the failure probability is exponentially small in the number of random bits used. Previously, Bach (1987) described a constant half cost Las Vegas algorithm for finding square roots modulo primes. In this paper we extend this result and give two constant half cost Las Vegas algorithms for polynomial factoring. The first is based on Berlekamp s algorithm and an extension described in Cantor ....

Bach, E. (1987). Realistic analysis of some randomized algorithms. Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pp. 453-461.

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Bach, E. Realistic Analysis of Some Randomized Algorithms, Proceedings of the 19th ACM Symposium on Theory of Computing (1987), 453--461.

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E. Bach, "Realistic Analysis of Some Randomized Algorithms", 19th STOC, 1987.

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