Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra... more Abstract. 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.
The present paper settles two questions raised by the author's paper,[3]. First, results of ... more The present paper settles two questions raised by the author's paper,[3]. First, results of Shafarevich are used to show that every possible solvable group occurs (in every possible way) as the Galois group of some CM-field. Second, the degrees of all reflex fields are ...
We present data concerning the factorization of the 120-digit number RSA-120, which we factored o... more We present data concerning the factorization of the 120-digit number RSA-120, which we factored on July 9, 1993, using the quadratic sieve method. The factorization took approximately 825 MIPS years and was completed within three months real time. At the time of writing RSA-120 is the largest integer ever factored by a general purpose factoring algorithm. We also present some conservative extrapolations to estimate the difficulty of factoring even larger numbers, using either the quadratic sieve method or the number field sieve, and discuss the issue of the crossover point between these two methods.
We report on algorithmic aspects of the problem of explicitly computing the rate of growth of the... more We report on algorithmic aspects of the problem of explicitly computing the rate of growth of the field of N k -th division points on an n-dimensional simple Abelian variety with Complex Multiplication. Two new examples are discussed.
Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra... more Abstract. 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.
The present paper settles two questions raised by the author's paper,[3]. First, results of ... more The present paper settles two questions raised by the author's paper,[3]. First, results of Shafarevich are used to show that every possible solvable group occurs (in every possible way) as the Galois group of some CM-field. Second, the degrees of all reflex fields are ...
We present data concerning the factorization of the 120-digit number RSA-120, which we factored o... more We present data concerning the factorization of the 120-digit number RSA-120, which we factored on July 9, 1993, using the quadratic sieve method. The factorization took approximately 825 MIPS years and was completed within three months real time. At the time of writing RSA-120 is the largest integer ever factored by a general purpose factoring algorithm. We also present some conservative extrapolations to estimate the difficulty of factoring even larger numbers, using either the quadratic sieve method or the number field sieve, and discuss the issue of the crossover point between these two methods.
We report on algorithmic aspects of the problem of explicitly computing the rate of growth of the... more We report on algorithmic aspects of the problem of explicitly computing the rate of growth of the field of N k -th division points on an n-dimensional simple Abelian variety with Complex Multiplication. Two new examples are discussed.
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Papers by Bruce Dodson