## Fast Bounds on the Distribution of Smooth Numbers (2006)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Parsell06fastbounds,

author = {Scott T. Parsell and Jonathan P. Sorenson},

title = {Fast Bounds on the Distribution of Smooth Numbers },

year = {2006}

}

### OpenURL

### Abstract

Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.

### Citations

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(Show Context)
Citation Context ...t prime divisor of n. If P(n) ≤ y,thenn is said to be y-smooth. Smooth numbers are utilized by many integer factoring and discrete logarithm algorithms, and hence they are of interest in cryptography =-=[19,22]-=-. Define Ψ(x, y) to be the number of integers n ≤ x that are y-smooth. In this paper, we present improvements to an algorithm of Bernstein[4,5], based on discrete generalized power series, which gives... |

148 |
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(Show Context)
Citation Context ...more error, but the greatly improved running time allows us to choose a larger α to more than compensate. Assuming the Riemann Hypothesis, we have √ t log t |π(t) − li(t)| < (2) 8π when t ≥ 1451 (see =-=[23,9]-=-), so we require that z>1451. We note that a very good estimate for li(t) can be computed in O(log t) time (see equations 5.1.3 and 5.1.10, or even 5.1.56, in [1]). Define n ± i , our upper and lower ... |

80 |
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(Show Context)
Citation Context ...[17] for several more. All of these algorithms are far too slow for use in applications related to factoring and cryptography. There are a number of asymptotic estimates for Ψ(x, y) in the literature =-=[8,10,11,13,14,15,18,20,21]-=-, many of which lead to algorithms. Dickman’s function, ρ(u), is defined as the unique continuous solution to ρ(u)=1 (for0≤u≤1), ρ(u − 1) + uρ ′ (u)=0 (foru>1). It is well-known that the estimate Ψ(x,... |

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(Show Context)
Citation Context ...gorithm for t ≤ z, but use the fast-to-compute estimate √ t log t |π(t) − li(t)| < (t ≥ 1451) 8π for t>z, where li(t) is the logarithmic integral. The above inequality follows from work of Schoenfeld =-=[23]-=- under the assumption of the Riemann Hypothesis (see also [9, Exercise 1.36]). This improvement, Algorithm 4.1, leads to a running time of � O α z2/3 � + α log xlog αy log z operations, with a relativ... |

36 |
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Citation Context ...[17] for several more. All of these algorithms are far too slow for use in applications related to factoring and cryptography. There are a number of asymptotic estimates for Ψ(x, y) in the literature =-=[8,10,11,13,14,15,18,20,21]-=-, many of which lead to algorithms. Dickman’s function, ρ(u), is defined as the unique continuous solution to ρ(u)=1 (for0≤u≤1), ρ(u − 1) + uρ ′ (u)=0 (foru>1). It is well-known that the estimate Ψ(x,... |

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(Show Context)
Citation Context ...[17] for several more. All of these algorithms are far too slow for use in applications related to factoring and cryptography. There are a number of asymptotic estimates for Ψ(x, y) in the literature =-=[8,10,11,13,14,15,18,20,21]-=-, many of which lead to algorithms. Dickman’s function, ρ(u), is defined as the unique continuous solution to ρ(u)=1 (for0≤u≤1), ρ(u − 1) + uρ ′ (u)=0 (foru>1). It is well-known that the estimate Ψ(x,... |

29 |
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Citation Context |

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Citation Context |

10 | Prime sieves using binary quadratic forms
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Citation Context ...at x is sufficiently large and exp(− � log y) <εlog x<1/2. In view of (1), we can take ε =1/(α lg 3).s174 S.T. Parsell and J.P. Sorenson As for the running time, Step 2 can be done with a prime sieve =-=[2]-=-, taking O(y/log log y) operations. In Step 3, G(X) will have O(αh) nonzero terms, and so takes O(hy/(log y) 2 ) time to construct. The FFT-based exponentiation algorithm in Step 4 takes only O(αh log... |

10 | Approximating the number of integers free of large prime factors
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(Show Context)
Citation Context ...06. c○ Springer-Verlag Berlin Heidelberg 2006sFast Bounds on the Distribution of Smooth Numbers 169 leads to a simple recursive algorithm. Bernstein presents several algorithms in his thesis [3]. See =-=[17]-=- for several more. All of these algorithms are far too slow for use in applications related to factoring and cryptography. There are a number of asymptotic estimates for Ψ(x, y) in the literature [8,1... |

9 |
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(Show Context)
Citation Context ...es 1 ≤ u ≤ exp[(log y) 3/5−ε ]. This range can be extended if we assume the Riemann Hypothesis. Highly accurate estimates for ρ(u) can be computed quickly using numerical integration; see for example =-=[27]-=-. Hildebrand and Tenenbaum [14] gave a more complicated estimate for Ψ(x, y) using the saddle-point method. Define ζ(s, y):= � (1 − p −s ) −1 , p≤y φ(s, y):=logζ(s, y), φk(s, y):= dk φ(s, y) (k≥1). ds... |

7 |
Integers without large prime factors, Journal de Theorie des Nombres de Bordeaux 5
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(Show Context)
Citation Context |

6 |
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Citation Context |

4 |
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Citation Context ...f Smooth Numbers 175 Proof. The accuracy analysis of Algorithm 3.1 is identical to that of Algorithm 2.1, so we only need to perform a runtime analysis. We can use the algorithm of Deléglise and Rivat=-=[12]-=- to compute π(t) intimeO(t2/3 /(logt) 2 ). This means that it takes � y O α log y · 2/3 (log y) 2 � operations to compute all the ni values (Step 2). The time to construct G(X) or G(X) (Step 3) is the... |

3 |
Bounding smooth integers
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(Show Context)
Citation Context ...rithms, and hence they are of interest in cryptography [19,22]. Define Ψ(x, y) to be the number of integers n ≤ x that are y-smooth. In this paper, we present improvements to an algorithm of Bernstein=-=[4,5]-=-, based on discrete generalized power series, which gives rigorous upper and lower bounds for Ψ(x, y). 1.1 Previous Work To compute the exact value of Ψ(x, y), one could simply factor all the integers... |

3 | Arbitrarily tight bounds on the distribution of smooth integers
- Bernstein
(Show Context)
Citation Context ... those used in Algorithm 4.1, but they are much easier to work with, and the results we obtain still apply to Algorithm 4.1. It follows easily from (3) that n − i ≥ ni(1 − δi) and n + i ≤ ni(1 + δi), =-=(5)-=- where δi := 2Δi/ni. Moreover, it follows from (3) and (4) after some computation that √ � w log w π(w)−π(w/c) ≥ li(w)−li(w/c)− ≥ 1− 4π 1 � w log c li(w)− c c(log w) 2 −√w log w. Taking c =21/α and no... |

3 | Multiple precision zero-finding methods and the complexity of elementary function evaluation - Brent - 1976 |

3 | Approximating the number of integers without large prime numbers
- Suzuki
- 2006
(Show Context)
Citation Context ...log y) + u y uniformly for 2 ≤ y ≤ x. This theorem has led to a string of algorithms that, in practice, appear to give significantly better estimates to Ψ(x, y) than those based on Dickman’s function =-=[17,24,25]-=-. Recently, Suzuki [26] showed how to estimate Ψ(x, y) quite nicely in only O( √ log xlog y) operations using this approach. Bernstein’s algorithm [4,6] provides a very nice compromise between computi... |

2 |
Enumerating and counting smooth integers, chapter 2
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(Show Context)
Citation Context ...8–181, 2006. c○ Springer-Verlag Berlin Heidelberg 2006sFast Bounds on the Distribution of Smooth Numbers 169 leads to a simple recursive algorithm. Bernstein presents several algorithms in his thesis =-=[3]-=-. See [17] for several more. All of these algorithms are far too slow for use in applications related to factoring and cryptography. There are a number of asymptotic estimates for Ψ(x, y) in the liter... |

2 |
Proving primality in essentially quartic time
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- 2006
(Show Context)
Citation Context ... y) than those based on Dickman’s function [17,24,25]. Recently, Suzuki [26] showed how to estimate Ψ(x, y) quite nicely in only O( √ log xlog y) operations using this approach. Bernstein’s algorithm =-=[4,6]-=- provides a very nice compromise between computing an exact value of Ψ(x, y) (which is very slow) and computing an estimate (which is fast, but not as reliably accurate): compute rigorous upper and lo... |

2 |
A fast algorithm for approximately counting smooth numbers, ANTS-IV proceedings
- Sorenson
- 2000
(Show Context)
Citation Context ...log y) + u y uniformly for 2 ≤ y ≤ x. This theorem has led to a string of algorithms that, in practice, appear to give significantly better estimates to Ψ(x, y) than those based on Dickman’s function =-=[17,24,25]-=-. Recently, Suzuki [26] showed how to estimate Ψ(x, y) quite nicely in only O( √ log xlog y) operations using this approach. Bernstein’s algorithm [4,6] provides a very nice compromise between computi... |

1 |
On the local behavior of Ψ(x,y
- Hildebrand
- 1986
(Show Context)
Citation Context ... y log x + + α log xlog α (log y) 2 p≤y � and ε2 =max 1 − p≤y ≤ 1+2logx α lg 3 � lg p lg p (1 − X (1+ε)lgp ) −1 ≤ B − (x, y) Ψ(x 1/(1−ε) � ,y)=distrh (1 − X (1−ε)lgp ) −1 ≥ B + (x, y). p≤y Hildebrand =-=[16]-=- shows that Ψ(cx, y) ≤ cΨ(x, y) wheny is sufficiently large and c ≥ 1+exp(− √ log y). Taking c = x ε/(1±ε) , we find that B− (x, y) Ψ(x, y) ≥ x−ε/(1+ε) ≥ 1 − ε log x and B+ (x, y) Ψ(x, y) ≤ xε/(1−ε) ≤... |