W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....elliptic curve method in [108] A uniform random choice of z 1 o(1) elliptic curves seems to nd every prime y in total time z 2 o(1) exp p (2 o(1) log y log log y with negligible error probability. For further discussion see [32] 119] 88] 33] 120] 10] 167] 151] [29], and [34] The other k (p) methods in [15] and the hyperelliptic curve method in [109] seem slower than the p 1 method. The hyperelliptic curve method has the virtue of provably nding every prime y in subexponential time with negligible error probability. Many more methods are available ....

Wieb Bosma, Arjen K. Lenstra, An implementation of the elliptic curve integer factorization method, in [30] (1995), 119-136. MR 96d:11134.

....standard continuation. 3.3. The birthday paradox continuation. The birthday paradox continuation is an alternative to the (improved) standard continuation. It was suggested 8 R. P. BRENT in [8] and has been implemented in several of our programs (see x5) and in the programs of A. Lenstra et al. [4, 31, 45]. There are several variations on the birthday paradox idea. We describe a version which is easy to implement and whose efficiency is comparable to that of the improved standard continuation. Following a suggestion of Suyama, we choose a positive integer parameter e. The choice of e is considered ....

....is proportional to the expected number of multiplications mod N : W = K 1 B 1 K 2 B 3 ) ff; fi) Recall that ff and fi are functions of B 1 and B 2 , so this is a simple problem of minimization in two variables. Suppose that the minimum is W opt . Tables of optimal parameters are given in [4, 8, 45, 51, 71], with each paper making slightly different assumptions. In Table 2 we give a small table of log 10 W opt for factors of D decimal digits. We assume that K 1 = 11= log 2, K 2 = 1, and log 10 p D Gamma 0:5. Some computed values of (p) are also shown in Table 2, where (p) log W opt ) 2 ....

W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136. MR MR96d:11134

....than for the standard continuation. 3.3. The birthday paradox continuation. The birthday paradox continuation is an alternative to the (improved) standard continuation. It was suggested in [7] and has been implemented in several of our programs (see x5) and in the programs of A. Lenstra et al. [3, 33, 54]. There are several variations on the birthday paradox idea. We describe a version which is easy to implement and whose efficiency is comparable to that of the improved standard continuation. Following a suggestion of Suyama, we choose a positive parameter e. The choice of e is considered below. ....

....is proportional to the expected number of multiplications mod N : W = K 1 B 1 K 2 B 3 ) ff; fi) Recall that ff and fi are functions of B 1 and B 2 , so this is a simple problem of minimization in two variables. Suppose that the minimum is W opt . Tables of optimal parameters are given in [3, 7, 54, 59, 79], with each paper making slightly different assumptions. In Table 2 we give a small table of log 10 W opt for factors of D decimal digits. We assume that K 1 = 11= log 2, K 2 = 1, and log 10 p D Gamma 0:5. Some computed values of (p) are also shown in Table 2, where (p) log W opt ) 2 ....

W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136.

....ecient than the group law for the projective case, assuming that the di erence P Q is available whenever the sum P Q of P and Q must be computed. This condition makes it impossible to use the ordinary square and multiply exponentiation (Remark 3.7.2) to compute scalar products. Refer 31 to [13, 79] for detailed descriptions of the Montgomery model and a suitable algorithm to compute a scalar multiplication in this case. Refer to [25] for a comparison of various elliptic curve point representations in cryptographic applications. 3.7.4 Elliptic curves modulo a composite. The eld F p in ....

....For composites without known properties and, in particular, a smallest factor of unknown size, one generally starts o with a relatively small k aimed at nding small factors. This k is gradually increased for each new attempt, until a factor is found or until the factoring attempt is aborted. See [13, 79] for implementation details of the elliptic curve method. The method is ideally parallelizable: any number of attempts can be run independently on any number of processors in parallel, until one of them is lucky. The reason that the runtime argument is heuristic is that #E p (F p ) is contained ....

W. Bosma, A.K. Lenstra, An implementation of the elliptic curve integer factorization method, chapter 9 in Computational algebra and number theory (W. Bosma, A. van der Poorten, eds.), Kluwer Academic Press (1995).

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W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136.

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W. Bosma and A. K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136. MR 96d:11134

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