Mersenne Factorization Factory

Abstract

Abstract. We present new factors of seventeen Mersenne numbers, obtained using a variant of the special number field sieve where sieving on the algebraic side is shared among the numbers. It reduced the overall factoring effort by more than 50%. As far as we know this is the first practical application of Coppersmith's "factorization factory" idea. Most factorizations used a new double-product approach that led to additional savings in the matrix step.

The Tower Number Field Sieve

“…d = 6. The current record, hold by Kleinjung et al [26], was obtained with a MNFS variant and targeted d-SNFS integers for d = 8. Their computations were much faster than the evaluated time to factor a 1024-bit RSA modulus, so it is safe to say that SNFS is the best option when log 2 Q ≈ 1024 and d = 6 or when d = 8 for slightly larger targets.…”

General Purpose Integer Factoring

“…Because in the preferred methods to find dependencies the same final matrix M is repeatedly multiplied by a (changing) vector (cf. Section 5.2.5 on page 7), it is advantageous to use this representation MT of M if W( M) + W(T ) < W(M); this approach was first used in [45].…”

“…These early experiments culminated in the summer of 1990 in the factorization of F 9 , reported by the author, Lenstra, Manasse and Pollard in [56]. Since then much more refined implementations have been used to obtain a string of factorization records for Cunningham numbers, most recently the shared factorizations from [45] mentioned in the introduction (which also uses one of the ideas from [22], described in Section 5. with f (α) = 0. Because |a| and |b| will be relatively small, the smoothness probabilities depend on the size of m, the degree d, and the sizes of the coefficients of f (X).…”

“…Overall, however, usage of large primes leads to a considerable speedup (which, as usual, disappears in the o(1) in the L-notation). Refer to [34] for the earliest results (which were, back then, found to be rather surprising) and to [62,45] for the most recent ones.…”

“…It required about two thousand core years of computation, accumulated over a period of almost three years on a wide variety of computers world-wide. The current record calculation of the second category is the shared factorization of 2 m − 1 for m ∈ {1007, 1009, 1081, 1111, 1129, 1151, 1153, 1159, 1177, 1193, 1199} which required about five thousand core years over a period of almost five years [22,45].…”