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Prime Numbers

Prime Numbers Prime Numbers

thales.doa.fmph.uniba.sk
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7.4 Elliptic curve method 345 if(1

346 Chapter 7 ELLIPTIC CURVE ARITHMETIC to obtain the factorization of 2 677 − 1as 1943118631 · 531132717139346021081 · 978146583988637765536217 · 53625112691923843508117942311516428173021903300344567 · P, where the final factor P is a proven prime. This beautiful example of serious ECM effort—which as of this writing involves one of the largest ECM factors yet found—looms even more beautiful when one looks at the group order #E(Fp) for the 53-digit p above (and for the given seed σ), which is 2 4 · 3 9 · 3079 · 152077 · 172259 · 1067063 · 3682177 · 3815423 · 8867563 · 15880351. Indeed, the largest prime factor here in #E is greater than B1, andsure enough, as Curry and Woltman reported, the 53-digit factor of M677 was found in stage two. Note that even though those investigators used detailed enhancements and algorithms, one should be able to find this particular factor—using the hindsight embodied in the above parameters—to factor M667 with the explicit Algorithm 7.4.4. Another success is the 54-digit factor of n = b 4 − b 2 +1,whereb =6 43 − 1, found in January 2000 by N. Lygeros and M. Mizony. Such a factorization can be given the same “tour” of group order and so on that we did above for the 53-digit discovery [Zimmermann 2000]. (See Chapter 1 for more recent ECM successes.) Other successes have accrued from the polynomial-evaluation method pioneered by Montgomery and touched upon previously. His method was used to discover a 47-digit factor of 5 · 2 256 + 1, and for a time this stood as an ECM record of sorts. Although requiring considerable memory, the polynomial-evaluation approach can radically speed up stage two, as we have explained. In case the reader wishes to embark on an ECM implementation—a practice that can be quite a satisfying one—we provide here some results consistent with the notation in Algorithm 7.4.4. The 33-decimal-digit Fermat factor listed in Section 1.3.2, namely 188981757975021318420037633 | F15, was found in 1997 by Crandall and C. van Halewyn, with the following parameters: B1 =10 7 for stage-one limit, and the choice B2 =50B1 for stagetwo limit, with the lucky choice σ = 253301772 determining the successful elliptic curve Eσ. After the 33-digit prime factor p was uncovered, Brent resolved the group order of Eσ(Fp) as #Eσ(Fp) =(2 5 · 3 · 1889 · 5701 · 9883 · 11777 · 5909317) · 91704181, where we have intentionally shown the “smooth” part of the order in parentheses, with outlying prime 91704181. It is clear that B1 “could have been” taken to be about 6 million, while B2 could have been about 100 million; but of course—in the words of C. Siegel—“one cannot guess the real

7.4 Elliptic curve method 345<br />

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