J. M. Pollard, A monte carlo method for factorization, BIT Nord. Tid. f. Inf. 15 (1975), no. 3, 331--334.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Pollard s fast factorial method in [136] achieves the same result as trial division in time only y 1=2 o(1) The o(1) can be reduced by Sch onhage s technique in [162] Pomerance showed in [140] that early abort fast factorial takes average time only y 1=4 o(1) Pollard s method in [137] seems to achieve the same result as trial division in time y 1=2 o(1) with the o(1) not quite as large as in the fast factorial method. See [31] and [35] for improvements, and [14] for analysis of a randomized version of the method. Pollard s p 1 method in [136] nds certain primes p ....

John M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), 331-334. MR 52 #13611.

....(it should require roughly 2 n=2 evaluations) Another interesting question is whether it is possible to construct pseudo random functions that can be iterated. The need for such functions (or the k wise independent version of them) arises in algorithmic applications such as Pollard s rho method [7] and Hellman s time space tradeo for inverting function ( 4] see also [1] Here, again, the approach of this paper tells us that it is sucient to come up with a family F of functions that has the correct distribution on the tree structure as well as the ability to compute iterations, but is ....

J. M. Pollard, A Monte Carlo method for factorization, BIT 15, 1975, pp. 331-334.

....modulo N , where m = b p Pc. Use Algorithm 4 to find a pair of x i ; x j such that GCD(x i Gamma x j ; N ) P . This algorithm finds P , with probability about 1 2 , using b p Pcblg p P c Jacobi symbol calculations. 2 Algorithm 5 is mostly a curiosity, since Pollard s ae algorithm [13] would find P in O( p P ) operations modulo N . However, things get much better if P Gamma 1 happens to be semi smooth [3] Definition3. Let z y be positive integers. An integer n is semi smooth with respect to y and z if all the prime factors of n are bounded above by z, with the possible ....

Pollard, J. M.: A Monte Carlo method for factorization. BIT 15 (1975) 331--334.

....hardware, may be competitive with the results presented here. Furthermore, the above variants do not require access to a partially oblivious function. Factorization Using a Partially Oblivious Function 3 on will repeat an element after O( p V ) steps. As in Pollard s rho method [11], we make the heuristic assumption that the same is true of a pseudo random walk on . We implement such a walk using an R oblivious function f . If is a point in then a step in the pseudo random walk consists of adding either = or = Which of the two paths to take is ....

J. M. Pollard, A Monte Carlo method for factorization, BIT, 15 (1975), 331-334.

....multiplications) in such a way that intermediate results are highly composite but nonzero. Suppose N is a number to be factored and p j N . If r is one of the intermediate results and p j r, then p j gcd(r; N ) we hope that this GCD does not equal N itself. The Pollard Rho algorithm [12] iterates a function. Let f 2 Z[X] be a univariate polynomial with integer coefficients. Select x 0 arbitrarily and define xn 1 = f(xn ) n 0) 6.1) If p is prime, then the sequence fxn mod pgn0 must eventually repeat, say xn1 j xn2 (mod p) where 0 n 1 n 2 . Since f is a polynomial function, ....

J.M. Pollard. A Monte Carlo method for factorization. BIT, 15(3):331--334, 1975.

....whether the cycle in the component beginning with x contains a zero node. For this, we compute the sequences (g i (0; s) i0 and (g 2i (0; s) i0 , and for each i = 1; 2; we check whether g i (0; s) j g 2i (0; s) mod q) This is expected to happen for i 1:0308 p q (Floyd s method, see [15]) When this is the case, we know that g i (0; s) is in the cycle for that i, so that we compute g i 1 (0; s) g i 2 (0; s) until we find a minimal j such that g i j (0; s) j 0 (mod q) or g i j (0; s) j g i (0; s) mod q) If the latter happens first, we know that the zero node is not in ....

J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), no. 3, 331--335.

....that certain Extended Riemann Hypotheses are true as is analyzed in [25] There are several other deterministic factoring methods as in [15] but the problem to re ne the run time has made not so much progress. One of the simplest probabilistic factoring methods is the Pollard method in [21] which computes the greatest common divisors of n and gaps of a sequence generated by a polynomial. If we assume that the polynomial used to generate the sequence behave like a random map, then the run time of this method is Ln [1; 1=4 o(1) Expecting a smooth prime factor p of n, several ....

J. Pollard, A Monte-Carlo method for factorization, BIT 15 (1975), 331-334.

....results The six first numbers of the series were easy to check for primality. Amongst them only A 0 = 7, A 1 = 73, and A 2 = 262657 appears to be prime. A 3 splits easily in three factors on general purposes symbolic computation software. A 4 splits into four factors by Pollard s rho method [4] in less than 10 minutes on a Sparc Station (assuming Pollard s conjecture[4] The unfactored part of A 5 counts 130 decimal digits and its factorization is unlikely for the time being: a trial during two weeks reveals unsuccessful. The table 1 summarizes the set of non trivial divisors for the ....

....Amongst them only A 0 = 7, A 1 = 73, and A 2 = 262657 appears to be prime. A 3 splits easily in three factors on general purposes symbolic computation software. A 4 splits into four factors by Pollard s rho method [4] in less than 10 minutes on a Sparc Station (assuming Pollard s conjecture[4]) The unfactored part of A 5 counts 130 decimal digits and its factorization is unlikely for the time being: a trial during two weeks reveals unsuccessful. The table 1 summarizes the set of non trivial divisors for the integers An with n 40. This table, let apart the cases where n 4 was ....

J.M. Pollard. -- A Monte-Carlo method for factorization. -- BIT, 15:331--334, 1975.

....order as well) contains several primes of medium size whose product is greater than some large bound ( BBHM00, Th. 4] This prevents the success of the following discrete log algorithm: First determine a multiple of ord ff using a method similar to Pollard s p Gamma 1 factorization method ([Pol75], see also [LL90] Then apply the PohligHellman algorithm. The first step does not succeed in our situation because of a combinatorical explosion since ord ff has several primes of medium size. Namely, for finding a multiple of the B smooth integer ord ff one needs O(B Delta ln m= ln B) group ....

J.M. Pollard. A Monte Carlo method for factorization. BIT, 15:331--334, 1975.

....quite time consuming, only those hash values were factored whose most significant 20 bits were all zero and which therefore are more likely to contain only short factors. Factoring was done in two steps, first trial division with all prime factors smaller than 2 20 and then Pollard s rho method [18] with 2 15 steps in the main loop (using LIP s zpollardrho function [15] to find the remaining (small) prime factors. On a SUN Enterprise Server this program did run for approximately one day to find the following result. RIPEMD 160 ( Check #00002228615476: value 1 000 000.00 ) ....

J. Pollard, "A Monte Carlo Method for Factorization", BIT, Vol. 15, (1975), pp. 331--334.

....but the numbers used for the modulus in the RSA system do not have any small factors. Therefore, general purpose factoring algorithms are the more important ones in the context of cryptographic systems and their security. Special purpose factoring algorithms include the Pollard rho method [66], with expected running time O( p p) and the Pollard p Gamma 1 method [67] with running time O(p 0 ) where p 0 is the largest prime factor of p Gamma 1. Both of these 28 take an amount of time that is exponential in the size of p, the prime factor that they find; thus these algorithms ....

J. Pollard. Monte Carlo method for factorization. BIT, 15:331--334, 1975.

....2m with Floyd s algorithm. Then, we iteratively test if x i = x i m for increasing values of i. The time complexity is always O( p Card(S) and the memory needed is still constant. Pollard s rho factoring algorithm Floyd s algorithm can be used to factor integers. The Pollard s rho algorithm [22] consists in choosing S = Zn and f(x) x 2 1 mod n. We do not search collisions x i = x j mod n but only indexes i and j such that gcd(x i Gamma x j ; n) 1, i.e. a collision modulo a prime factor of n. Since this gcd is equal to n with negligible probability, we obtain a non trivial factor ....

J. M. Pollard. A Monte Carlo Methods for Factorization. BIT, 15:331--334, 1975.

....) of two terms of the sequence is called a match if w i = w j and i j. Because of the picture one obtains when drawing the terms of (w k ) starting at the bottom and ending in a cycle, the method of solving computational problems by using sequences as in (2.1) is called the rho method . Pollard [Pol75] first applied this result to obtain an efficient and simple algorithm for factoring. Then in [Pol78] he found an algorithm that uses the rho method to compute discrete logarithms in the multiplicative group (Z=pZ) p prime) in the expected run time of O( p p ) group operations. This algorithm ....

J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), no. 3, 331--335. MR 52:13611

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J. M. Pollard, A monte carlo method for factorization, BIT Nord. Tid. f. Inf. 15 (1975), no. 3, 331--334.

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J.M. Pollard, "A Monte Carlo method for factorization", BIT, vol. 15 (1975), pp. 331-334.

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J.M. Pollard. A Monte Carlo method for factorization. BIT 15, 331-334, 1975.

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John M. Pollard. A monte carlo method for factorization. BIT, 15:331--334, 1975.

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J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), 331--334.

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J. M. Pollard. A Monte Carlo method for factorization. BIT, 15:331--334, 1975.

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J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), 331--334. MR 52:13611

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J. M. Pollard. A Monte Carlo method for factorization. BIT, 15(3):331--334, 1975.

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J. M. Pollard. A Monte Carlo method for factorization. BIT 15 (1975), 331-334.

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J. Pollard. Monte Carlo method for factorization. BIT, 15:331--334, 1975.

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J. Pollard. Monte Carlo method for factorization. BIT, 15:331--334, 1975.

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J. M. Pollard, A Monte Carlo method for factorization, BIT 15 (1975), 331--334.

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