## On the period of the linear congruential and power generators

Venue: | Acta Arith |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Kurlberg_onthe,

author = {Pär Kurlberg and Carl Pomerance},

title = {On the period of the linear congruential and power generators},

journal = {Acta Arith},

year = {},

pages = {149--169}

}

### OpenURL

### Abstract

We consider two standard pseudorandom number generators from number theory: the linear congruential generator and the power generator. For the former, we are given integers e, b, n (with e, n> 1) and a seed u0, and we compute the sequence

### Citations

65 |
On Artin’s conjecture
- Hooley
- 1967
(Show Context)
Citation Context ...x : ord(e, p) ≤ p �� ��� ≪ y π(x) x log log x + y log 2 x , where the implied constant depends at most on the choice of e. Proof. Since the proof is rather similar to the proof of the main theorem in =-=[9]-=- and the proof of Theorem 2 in [11], we only give a brief outline. With ip = (p − 1)/ ord(e, p), we see that ord(e, p) ≤ p/y implies that ip ≥ y/2. First step: We first consider primes p such that ip ... |

34 |
Shifted primes without large prime factors
- Baker, Harman
- 1998
(Show Context)
Citation Context ...prime. As mentioned in the Introduction, one way of getting a fairly decent result here is to have a very large prime factor of p − 1 as afforded by a series of papers culminating in the recent paper =-=[2]-=-. Lemma 19 (Baker–Harman). For a positive proportion of the primes p, there is a prime q|p − 1 with q > p 0.677 . Note that this result follows from (7.1) in [2]. We use this result to immediately get... |

24 |
Carmichael’s lambda function
- Erdös, Pomerance, et al.
- 1991
(Show Context)
Citation Context ...≤x p∈L (log p a ) 2 the last inequality coming from the estimate for πL(x) in Lemma 6. Thus we immediately get the first assertion in the lemma. For the second assertion note that from (6) and (7) in =-=[6]-=- we have log(λ(n)/γ(λ(n))) ≪ log log x/ log log log x for all but o(x) choices of n ≤ x. Thus we have the second assertion. The third assertion follows from the fact that the normal order of ω(λ(n)) i... |

18 | Period of the power generator and small values of Carmichael’s function
- Friedlander, Pomerance, et al.
- 2001
(Show Context)
Citation Context ... is a positive constant γ such that ord(e, p) > p 1/2+γ for a positive proportion of the primes p. The period of the power generator uei (mod pl) was studied in Friedlander, Pomerance and Shparlinski =-=[7]-=-, where p, l are primes of the same magnitude. 1At the end of the paper we briefly consider the general case where this assumption is not made. 2More precisely, that the Riemann hypothesis holds for L... |

17 |
On the normal number of prime factors of φ(n
- Erdös, Pomerance
- 1984
(Show Context)
Citation Context ...) ≪ log log x/ log log log x for all but o(x) choices of n ≤ x. Thus we have the second assertion. The third assertion follows from the fact that the normal order of ω(λ(n)) is 1 2 (log log n)2 , see =-=[5]-=-. � Now we give the analog result to Lemmas 9 and 15. Lemma 18. Let ɛ(x) satisfy (1). Almost all numbers n have the property that λ(n)M < n 2/5 . Proof. Let M ′ = {p prime : (p − 1)M > p 1/3 }. Lemma ... |

15 |
The order of a (mod p
- Erdös, Murty
- 1999
(Show Context)
Citation Context ...ng rather slowly. A similar result with ψ(x) ≤ (log x) 1−ɛ is proved in the first author’s paper [11]. In Theorem 23 we obtain a small, yet for our purposes crucial, strengthening of this result.) In =-=[4]-=-, Erdős and Murty showed that if ɛ(x) is any decreasing function tending to zero as x tends to infinity, then ord(e, p) ≥ p 1/2+ɛ(p) for all but o(π(x)) primes p ≤ x, and in [10] Indlekofer and Timofe... |

12 |
Timofeev, Divisors of shifted primes
- Indlekofer, M
(Show Context)
Citation Context ...ng of this result.) In [4], Erdős and Murty showed that if ɛ(x) is any decreasing function tending to zero as x tends to infinity, then ord(e, p) ≥ p 1/2+ɛ(p) for all but o(π(x)) primes p ≤ x, and in =-=[10]-=- Indlekofer and Timofeev gave a similar lower bound with an explicit estimate on the number of exceptional primes. Further, it follows immediately from work of Goldfeld, Fouvry, and Baker–Harman that ... |

12 | On quantum ergodicity for linear maps of the torus
- Kurlberg, Rudnick
(Show Context)
Citation Context ...om work of Goldfeld, Fouvry, and Baker–Harman that there is a positive constant γ such that ord(e, p) > p1/2+γ for a positive proportion of the primes p. As for ord(e, n) for n a positive integer, in =-=[11]-=- Kurlberg and Rudnick proved that there exists δ > 0 such that ord(e, n) ≫ n1/2 exp((log n) δ )) for all but o(x) integers n ≤ x that are coprime to e. Further, in [10], Kurlberg showed that the GRH i... |

12 |
On the number of restricted prime factors of an integer, III, to appear. Printed by the St
- NORTON
(Show Context)
Citation Context ...d p a ) or p a+1 |n. As � q≤x q prime q ≡ 1 (mod d) 1 q = log log x + O(log d) φ(d) 1.s12 PÄR KURLBERG AND CARL POMERANCE uniformly for all integers d ≥ 2 (see [17], Theorem 1 and Remark 1, or Norton =-=[15]-=-), we have � 1 ≤ x + pa+1 � x q x log log x = φ(pa � a x log p + O ) pa � . Hence n≤x p a |λ(n) n≤x q≤x q prime q ≡ 1 (mod p a ) � log λ(n)L ≪ x log log x � log pa ≪ x log log x, p a ≤x p∈L p a � + x ... |

11 |
Sieve Methods, Academic Press (A subsidiary of Harcourt Brace
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ... the set of prime numbers, the set of integers n = pl where p, l are primes with p < l < 2p, and the set of all natural numbers. We will need the following form of the Brun–Titchmarsh inequality (see =-=[8]-=-, Theorem 3.8): Lemma 13. Suppose k, l are coprime integers with k > 0 and let π(x, k, l) be the number of primes p ≤ x such that p ≡ l (mod k). Then π(x, k, l) ≪ x uniformly for x > k. φ(k) log(x/k) ... |

10 |
On the order of finitely generated subgroups of Q ∗ (mod p) and divisors of p − 1
- Pappalardi
- 1996
(Show Context)
Citation Context ...ven, and in general, λ(n) is divisible by the fixed number e for a set of numbers n of asymptotic density 1. We begin by reviewing some of the literature on statistical properties of ord * (e, n). In =-=[16]-=- Pappalardi showed that there exist α, δ > 0 such that ord(e, p) ≥ p 1/2 exp((log p) δ ) for all but O(x/ log 1+α x) primes p ≤ x. He also asserted, assuming the Generalized Riemann Hypothesis 2 (GRH)... |

8 |
The Brun-Titchmarsh Theorem on Average
- Baker, Harmon
- 1996
(Show Context)
Citation Context ...f primes comprises a positive proportion of all primes. Consider primes p ≤ x where q|p − 1 for some q ∈ P and with x 0.52−ɛ < q ≤ x 0.52 . Here, ɛ > 0 is arbitrarily small but fixed. It follows from =-=[1]-=-, Theorem 1, that a positive proportion of primes p are so representable. Further, it follows from Lemma 14 that by neglecting only a relative density 0 of suchsPERIOD OF THE LINEAR CONGRUENTIAL AND P... |

8 |
On the distribution of amicable numbers
- Pomerance
- 1977
(Show Context)
Citation Context ...r n is divisible by some prime q ≡ 1 (mod p a ) or p a+1 |n. As � q≤x q prime q ≡ 1 (mod d) 1 q = log log x + O(log d) φ(d) 1.s12 PÄR KURLBERG AND CARL POMERANCE uniformly for all integers d ≥ 2 (see =-=[17]-=-, Theorem 1 and Remark 1, or Norton [15]), we have � 1 ≤ x + pa+1 � x q x log log x = φ(pa � a x log p + O ) pa � . Hence n≤x p a |λ(n) n≤x q≤x q prime q ≡ 1 (mod p a ) � log λ(n)L ≪ x log log x � log... |

7 |
On the normal number of prime factors of p − 1 and some other related problems concerning Euler’s φ-function, Quart
- ERDŐS
- 1935
(Show Context)
Citation Context ... 1)/γ(p − 1) is similar, namely that a trivial argument is used when (p − 1)/γ(p − 1) is large and the Brun–Titchmarsh inequality when it is small. The final assertion follows from the main result of =-=[3]-=- that the normal number of prime factors of p − 1 is log log p. � We now turn our attention to an analog of Lemma 9 for shifted primes. Lemma 15. With ɛ(x) as specified in (1), the number of primes p ... |

7 |
On generalizing Artin’s conjecture on primitive roots to composite moduli
- Li, Pomerance
(Show Context)
Citation Context ...tegers n ≤ x that are coprime to e. Further, in [11], Kurlberg showed that the GRH implies that for each ɛ > 0, we have ord(e, n) ≫ n 1−ɛ for all but o(x) integers n ≤ x that are coprime to n, and in =-=[13]-=- Li and Pomerance improved the lower bound to ord(e, n) ≥ n(log n) −(1+o(1)) log log log n , a result that is best possible. To complement these theorems we give some new results on ord(e, n) and ord ... |

6 | The iterated Carmichael λ-function and the number of cycles of the power generator, Acta Arith
- Martin, Pomerance
- 2005
(Show Context)
Citation Context ... set of primes has property P2ɛ almost always.) The third class of numbers in Theorem 12, namely, the set of all numbers n, is more difficult. We begin with a new result: Theorem 16 (Martin–Pomerance =-=[14]-=-). As n → ∞ through a certain set of integers of asymptotic density 1, we have λ(λ(n)) = n · exp(−(1 + o(1))(log log n) 2 log log log n) Thus, λ(λ(n)) > n/ exp((log log n) 3 ) almost always. We now gi... |

4 | On the order of unimodular matrices modulo integers
- Kurlberg
(Show Context)
Citation Context ...nbounded monotone function ψ(x), it appears that the proof only supports the case when ψ(x) is increasing rather slowly. A similar result with ψ(x) ≤ (log x) 1−ɛ is proved in the first author’s paper =-=[11]-=-. In Theorem 23 we obtain a small, yet for our purposes crucial, strengthening of this result.) In [4], Erdős and Murty showed that if ɛ(x) is any decreasing function tending to zero as x tends to inf... |