## Some integer factorization algorithms using elliptic curves (1986)

Venue: | Australian Computer Science Communications |

Citations: | 46 - 13 self |

### BibTeX

@ARTICLE{Brent86someinteger,

author = {Richard P. Brent},

title = {Some integer factorization algorithms using elliptic curves},

journal = {Australian Computer Science Communications},

year = {1986},

volume = {8},

pages = {149--163}

}

### Years of Citing Articles

### OpenURL

### Abstract

Lenstra’s integer factorization algorithm is asymptotically one of the fastest known algorithms, and is also ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible assumptions, the speedup is of order log(p), where p is the factor which is found. In practice the speedup is significant. We mention some refinements which give greater speedup, an alternative way of implementing a second phase, and the connection with Pollard’s “p − 1” factorization algorithm. 1

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Citation Context ...of work), a squaring then requires 12 units and a nonsquaring multiplication requires 9 units of work. The reader who is interested in learning more about the theory of elliptic curves should consult =-=[11]-=-, [12] or [15]. 5 Lenstra's algorithm The idea of Lenstra's algorithm is to perform a sequence of pseudo-random trials, where each trial uses a randomly chosen elliptic curve and has a nonzero probabi... |

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Citation Context ...deas we obtain a speedup of about 6.6 over the one-phase algorithm for p = 10 20 . 9.2 Other second phases Our birthday paradox idea can be used as a second phase for Pollard's "p \Gamma 1" =-=algorithm [23]. The only-=- change is that we work over a different group. Conversely, the conventional second phases for Pollard's "p \Gamma 1" algorithm can be adapted to give second phases for elliptic curve algori... |

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Citation Context ...n ln N j were known [27]. However, since p 2sN , Lenstra's algorithm is comparable in the worst case and often much better, since it often happens that 2 ln psln N . The Brent-Pollard "rho" =-=algorithm [5]-=- is similar to Lenstra's algorithm in that its expected running time depends on p, in fact it is of order p 1=2 . Asymptotically T 1 (p)sp 1=2 , but because of the overheads associated with Lenstra's ... |

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Citation Context ... ; x r and x 1 ; : : : ; x s in O((r + s)e) group operations. The values of x j do not need to be stored, so storage requirements are O(r) even if s AE r. Moreover, by use of rational preconditioning =-=[22, 29]-=- it is easy to evaluate (9.1) in (r + O(log r))s=2 multiplications. Using these ideas we obtain a speedup of about 6.6 over the one-phase algorithm for p = 10 20 . 9.2 Other second phases Our birthday... |

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Citation Context ...e definition of "a random integer close to M ", see [14]. It is sufficient to consider integers uniformly distributed in [1; M ]:) Several authors have considered the function ae(ff), see fo=-=r example [7, 9, 13, 14, 18]-=-. It satisfies a differential-difference equation ffae 0 (ff) + ae(ff \Gamma 1) = 0 and may be computed by numerical integration from ae(ff) = ( 1 if 0sffs1 1 ff R ff ff\Gamma1 ae(t) dt if ff ? 1: We ... |

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Citation Context ...uaring then requires 12 units and a nonsquaring multiplication requires 9 units of work. The reader who is interested in learning more about the theory of elliptic curves should consult [11], [12] or =-=[15]-=-. 5 Lenstra's algorithm The idea of Lenstra's algorithm is to perform a sequence of pseudo-random trials, where each trial uses a randomly chosen elliptic curve and has a nonzero probability of findin... |

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Citation Context ... algorithm can be speeded up by the addition of a second phase which is based on the same idea as the well-known "paradox" concerning the probability that two people at a party have the same=-= birthday [25]. The twop-=-hase algorithm has expected running time O(T 1 (p)= ln p). In practice, for p around 10 20 , the "birthday paradox algorithm" is about 4 times faster than Lenstra's (one-phase) algorithm. Th... |

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Citation Context .... Suppose that the computation of a \Gamma1 (mod N) by the extended GCD algorithm takes the same time as K multiplications (mod N ). Our first implementation gave K ' 30, but by using Lehmer's method =-=[16]-=- this was reduced to 6sKs10 (the precise value depending on the size of N ). It turns out that most computations of a \Gamma1 (mod N) can be avoided at the expense of about 8 multiplications (mod N ),... |

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Citation Context ...mma 1j ! 2 p p: (4:2) Lenstra's heuristic hypothesis is that, if a and b are chosen at random, then g will be essentially random in that the results of x3 will apply with M = p. Some results of Birch =-=[3]-=- suggest its plausibility. Nevertheless, the divisibility properties of g are not quite what would be expected for a randomly chosen integer near p, e.g. the probability that g is even is asymptotical... |

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Citation Context ...amma x 2 ) mod p; y 3 := ((x 1 \Gamma x 3 ) \Gamma y 1 ) mod p end: It is well-known that (G; ) forms an Abelian group with identity element I. Moreover, by the "Riemann hypothesis for finite fie=-=lds" [12]-=-, the group order g = jGj satisfies the inequality jg \Gamma p \Gamma 1j ! 2 p p: (4:2) Lenstra's heuristic hypothesis is that, if a and b are chosen at random, then g will be essentially random in th... |

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Citation Context ...: : . For ffs1, fis1, define ae(ff) = lim M!1 Prob i n 1 ! M 1=ff j and (ff; fi) = lim M!1 Prob i n 2 ! M 1=ff and n 1 ! M fi=ff j : 2 (For a precise definition of "a random integer close to M &q=-=uot;, see [14]-=-. It is sufficient to consider integers uniformly distributed in [1; M ]:) Several authors have considered the function ae(ff), see for example [7, 9, 13, 14, 18]. It satisfies a differential-differen... |

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Citation Context ...e definition of "a random integer close to M ", see [14]. It is sufficient to consider integers uniformly distributed in [1; M ]:) Several authors have considered the function ae(ff), see fo=-=r example [7, 9, 13, 14, 18]-=-. It satisfies a differential-difference equation ffae 0 (ff) + ae(ff \Gamma 1) = 0 and may be computed by numerical integration from ae(ff) = ( 1 if 0sffs1 1 ff R ff ff\Gamma1 ae(t) dt if ff ? 1: We ... |

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Citation Context ...e definition of "a random integer close to M ", see [14]. It is sufficient to consider integers uniformly distributed in [1; M ]:) Several authors have considered the function ae(ff), see fo=-=r example [7, 9, 13, 14, 18]-=-. It satisfies a differential-difference equation ffae 0 (ff) + ae(ff \Gamma 1) = 0 and may be computed by numerical integration from ae(ff) = ( 1 if 0sffs1 1 ff R ff ff\Gamma1 ae(t) dt if ff ? 1: We ... |

3 |
Informal preliminary report (8), personal communication
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Citation Context ... table (or on-line generation) of primes for the second phase, so it is easier to program and has lower storage requirements. 11 9.3 Better choice of random elliptic curves Montgomery [21] and Suyama =-=[28] have show-=-n that it is possible to choose "random" elliptic curves so that g is divisible by certain powers of 2 and/or 3. For example, we have implemented a suggestion of Suyama which ensures that g ... |

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Citation Context ... ; x r and x 1 ; : : : ; x s in O((r + s)e) group operations. The values of x j do not need to be stored, so storage requirements are O(r) even if s AE r. Moreover, by use of rational preconditioning =-=[22, 29]-=- it is easy to evaluate (9.1) in (r + O(log r))s=2 multiplications. Using these ideas we obtain a speedup of about 6.6 over the one-phase algorithm for p = 10 20 . 9.2 Other second phases Our birthday... |

1 | Lenstra's Algorithm for Factoring with Elliptic Curves (expos'e - Bach - 1985 |

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Elliptic Curve Factorization, personal communication via Samuel Wagstaff Jr
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Citation Context ...factorization algorithm. 1 Introduction Recently H.W. Lenstra Jr. proposed a new integer factorization algorithm, which we shall call "Lenstra's algorithm" or the "one-phase elliptic cu=-=rve algorithm" [17]. Under so-=-me plausible assumptions Lenstra's algorithm finds a prime factor p of a large composite integer N in expected time T 1 (p) = exp `q (2 + o(1)) ln p ln ln p ' ; (1:1) where "o(1)" means a te... |

1 |
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Citation Context ...up. Conversely, the conventional second phases for Pollard's "p \Gamma 1" algorithm can be adapted to give second phases for elliptic curve algorithms, and various tricks can be used to spee=-=d them up [19]-=-. Theoretically these algorithms give a speedup of the order log log(p) over the one-phase algorithms, which is not as good as the log(p) speedup for the birthday paradox algorithm [6]. However, in pr... |

1 | Factorization of the tenth Fermat number, Mathematics of Computation 68 - Brent - 1999 |