This paper reports record-setting performance for the ellipticcurve method of integer factorization: for example, 604.99 curves/second for ECM stage 1 with B1 = 8192 for 280-bit integers on a single PC. The state-of-the-art GMP-ECM software handles 171.42 curves/second for ECM stage 1 with B1 = 8192 for 280-bit integers using all four cores of a 2.4GHz Core 2 Quad Q6600. The extra speed takes a...

18.1 Context of the ECM ...................................................... 461 18.1.1 ECM – What’s in a Name? ............................................. 462 18.1.2 Old and New ECMs ........................................................ 462 18.1.3 Biophysical Considerations ............................................. 462 18.1.4 Jelly Bombs ................................................

This paper introduces EECM-MPFQ, a fast implementation of the elliptic-curve method of factoring integers. EECM-MPFQ uses fewer modular multiplications than the well-known GMP-ECM software, takes less time than GMP-ECM, and finds more primes than GMP-ECM. The main improvements above the modular-arithmetic level are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use extende...

Completion of genome sequences for many organisms allows a reasonably complete definition of the complement of extracellular matrix (ECM) proteins. In mammals this "core matrisome" comprises ∼300 proteins. In addition there are large numbers of ECM-modifying enzymes, ECM-binding growth factors, and other ECM-associated proteins. These different categories of ECM and ECM-associated proteins coop...

The performance of the elliptic curve method (ECM) for integer factorization plays an important role in the security assessment of RSA-based protocols as a cofactorization tool inside the number field sieve. The efficient arithmetic for Edwards curves found an application by speeding up ECM. We propose techniques based on generating and combining additionsubtracting chains to optimize Edwards E...

The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra, Jr., in 1985 [14]. In the past 20 years, many improvements of ECM were proposed on the mathematical, algorithmic, and implementation sides. This paper summarizes the current state-of-the-art, as implemented in the GMP-ECM software.

n Part 2 of this article (August 2007), we described multipoint (or common-type) supple mental grounding and bonding between the T(M)GB [notation used here to indicate that the bar may be either the TMGB or TGB, depending on the location within the building] and the telecommunications and information technology equipment (ITE) systems. This installment will focus on the single-point (or insulat...

Extracellular matrix (ECM) remodeling occurs during normal homeostasis and also plays an important role during development, tissue repair, and in various disease processes. ECM remodeling involves changes in the synthesis, deposition, and degradation of ECM molecules. ECM molecules can be degraded extracellularly, as well as intracellularly following endocytosis. Our data show that the ECM prot...