Results 1 
4 of
4
The Rectangle Complexity Of Functions On TwoDimensional Lattices
, 2001
"... Let X be a nonempty set. Let f : Z 2 ! X. All vectors which occur have integer coecients, and for ~a = (a 1 ; a 2 ), ~ b = (b 1 ; b 2 ) we write ~a ~ b or ~a < ~ b if a j b j or a j < b j for j = 1; 2, respectively. Let ~ b > ~ 0. A ~ bblock is a set of the form B ~ b (~c) := f~x 2 ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
Let X be a nonempty set. Let f : Z 2 ! X. All vectors which occur have integer coecients, and for ~a = (a 1 ; a 2 ), ~ b = (b 1 ; b 2 ) we write ~a ~ b or ~a < ~ b if a j b j or a j < b j for j = 1; 2, respectively. Let ~ b > ~ 0. A ~ bblock is a set of the form B ~ b (~c) := f~x 2 Z 2 j ~c ~x < ~c + ~ bg. A ~ bpattern is the restriction of f to some ~ bblock. The total number of distinct ~ bpatterns is called the ~ bcomplexity of f . A conjecture of the authors implies that f is periodic if there is a ~ b > ~ 0 such that the ~ bcomplexity of f does not exceed b 1 b 2 . In the paper we prove the statement for ~ b = (n; 2) where n is any positive integer. Keywords: complexity, lattices, patterns, periodicity, congurations. This work, done at Leiden University, was made possible by grants from the Stieltjes Institute for Mathematics, NUFFIC and the Hannoversche Hochschulgemeinschaft. The authors are grateful to these institutions for their support. 1 1
Strategies in Filtering in the Number Field Sieve
 In preparation
, 2000
"... A critical step when factoring large integers by the Number Field Sieve [8] consists of finding dependencies in a huge sparse matrix over the field F2 , using a Block Lanczos algorithm. Both size and weight (the number of nonzero elements) of the matrix critically affect the running time of Block ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
A critical step when factoring large integers by the Number Field Sieve [8] consists of finding dependencies in a huge sparse matrix over the field F2 , using a Block Lanczos algorithm. Both size and weight (the number of nonzero elements) of the matrix critically affect the running time of Block Lanczos. In order to keep size and weight small the relations coming out of the siever do not flow directly into the matrix, but are filtered first in order to reduce the matrix size. This paper discusses several possible filter strategies and their use in the recent record factorizations of RSA140, R211 and RSA155. 2000 Mathematics Subject Classification: Primary 11Y05. Secondary 11A51. 1999 ACM Computing Classification System: F.2.1. Keywords and Phrases: Number Field Sieve, factoring, filtering, Structured Gaussian elimination, Block Lanczos, RSA. Note: Work carried out under project MAS2.2 "Computational number theory and data security". This report will appear in the proceed...
The ThreeLargePrimes Variant of the Number Field Sieve
"... The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this method (but fortunately, also the easiest to parallelise). Pollard's original method allowed one large prime. After that the twolargeprimes variant led to substantial improvements [11]. In this paper we investigate whether the threelargeprimes variant may lead to any further improvement. We present theoretical expectations and experimental results. We assume the reader to be familiar with the NFS.
SIZE OPTIMIZATION OF SEXTIC POLYNOMIALS IN THE NUMBER FIELD SIEVE
"... Abstract. The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root propert ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some methods to optimize the size property of sextic polynomials.