E. Bach, "Toward a Theory of Pollard's Rho Method", Information and Computation, vol. 90 (1991), pp. 139-155.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 quickly: in particular, it seems to nd at least one out of every z primes in time z 1 o(1) if n has z o(1) bits, where 2(log z) 2 = log y log log y, The same comment applies to ....

Eric Bach, Toward a theory of Pollard's rho method, Information and Computation 90 (1991), 139-155. MR 92a:11151.

....properties of polynomials that are relevant to basic factorization algorithms. 1. ANALYTIC COMBINATORIAL METHODS 1 For instance the design of Pollard s rho method for integer factoring is entirely based on similar heuristics regarding integers; these were later largely vindicated by Bach [2]. ANALYSIS OF POLYNOMIAL FACTORIZATION 9 This section gathers basic tools needed to analyse properties of random polynomials. It centres around the use of generating functions, either univariate or multivariate, whose functional relations reflect the algebraic decompositions of various classes ....

....FACTORIZATION 17 procedure DDF(a : polynomial) a is a monic squarefree polynomial] n : deg(a) g : a; h : x; for k : 1 to n do 1. h : h q mod g; 2. b[k] gcd(h x,g) 3. g : g b[k] a without factors of deg =k 4. if b[k] 1 then h : h mod g fi; od; return(b[1] b[2]. b[n] b[k] is prod. of irred. of deg. k end; FIG. 5. The basic strategy of the distinct degree factorization algorithm (DDF) p. 91, Theorem 3.20) For k 1, the polynomial k : x q k x 2 F q [x] is the product of all monic irreducible polynomials in F q [x] whose degree divides ....

[Article contains additional citation context not shown here]

Bach, E. Toward a theory of Pollard's rho method. Information and Computation 90 (1991), 139--155.

....; ae for the iteration of random function (see Flajolet and Odlyzko [7] for several such results and references) and in [7] it is postulated that the properties of quadratic functions modulo an integer should be asymptotically the same as those of the class of all functions. Indeed, Bach [1] has proved that in initial stages, at least, quadratic functions do behave asymptotically like random functions. Thus, the expected values of ; ae here should have values close to p q=8, p q=8 and p q=2, respectively. Furthermore, the expected maximum values of ; ae are respectively ....

E. Bach, Toward a theory of Pollard's rho method, Information and Computation 90 (1991), 139--155.

....; ae for the iteration of random function (see Flajolet and Odlyzko [7] for several such results and references) and in [7] it is postulated that the properties of quadratic functions modulo an integer should be asymptotically the same as those of the class of all functions. Indeed, Bach [1] has proved that in initial stages, at least, quadratic functions do behave asymptotically like random functions. Thus, the expected values of ; ae here should have values close to p q=8, p q=8 and p q=2, respectively. Furthermore, the expected maximum values of ; ae are respectively ....

E. Bach, Toward a theory of Pollard's rho method, Information and Computation 90 (1991), 139--155.

....detail. The reader interested in quantitative estimates on random mappings rather than methodology can proceed directly to the self contained statements of Theorems 2 8. 1 In the case of Pollard s algorithms and iteration of quadratic functions modulo integers, a notable advance is due to Bach [2] who proved recently that in initial stages quadratic functions behave asymptotically like random functions. Bach s result ultimately relies on the Weil Deligne theorem establishing the truth of the Riemann hypothesis for zeta functions of algebraic curves 2 The term statistics is to ....

....2. The function x 2 2 (mod n) is one of a restricted set of polynomial functions whose iteration structure can be precisely described. For general polynomials, essentially, the only known approach is heuristic where one postulates that a polynomial behaves like a random mapping. See however [2] for one of the very few rigorous results in this domain. 2 Methods Any element of F m n can be viewed as a word over an m ary alphabet of length m. Thus, there are m n mappings from an n set into an m set. Specializing this observation, we find that the cardinality of F n j F n n is ....

[Article contains additional citation context not shown here]

E. Bach. Toward a theory of Pollard's rho--method. Information and Computation, to appear, 1989.

No context found.

E. Bach, "Toward a Theory of Pollard's Rho Method", Information and Computation, vol. 90 (1991), pp. 139-155.

No context found.

E. Bach. Toward a theory of Pollard's rho--method. Information and Computation, to appear, 1989.

No context found.

E. Bach, \Toward a Theory of Pollard's Rho Method." Information and Computation, 90(2):139-155, 1991.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC