This paper reports on the factorization of the 512–bit number RSA–155 by the Number Field Sieve factoring method (NFS) and discusses the implications for RSA.

Main results. A rigorous proof that the number of comparisons of keys performed in the worst case by Heapsort on any array of size N ≥ 2 is equal to: 2(N − 1) ( lg N − 1 2 + ε )− 2s2(N)− e2(N) + min(blg(N − 1)c, 2) + 6 + c, where ε, given by: ε = 1 + dlg (N − 1)e − lg (N − 1)− 2dlg (N−1)e−lg (N−1), is a function of N with the minimum value 0 and and the supremum value δ = 1− lg e+ lg lg e ≈ 0.0...

We describe an enhanced version of the TWINKLE factoring device and analyse to what extent it can be expected to speed up the sieving step of the Quadratic Sieve and Number Field Sieve factoring algorithms. The bottom line of our analysis is that the TWINKLE-assisted factorization of 768-bit numbers is difficult but doable in about 9 months (including the sieving and matrix parts) by a large or...

Steffen, T., Dixon, R., Goodall, Roger M. and Zolotas, Argyrios (2008) Multi-Variable Control of a High Redundancy Actuator. In: Actuator 08 : 11th International Conference on New Actuators & 5th International Exhibition on Smart Actuators and Drive Systems, 9-11 June 2008, Bremen, Germany : conference proceedings. HVG Hanseatische Veranstaltungs-GmbH Division Messe Bremen, pp. 473-476. ISBN 97...

We present normwise and componentwise perturbation bounds for the hyperbolic QR factorization by using a new approach. The explicit expressions of mixed and componentwise condition numbers for the hyperbolic QR factorization are derived.

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 selected polynomials can be modelled in terms of size and root properties. We propose a new kind of polynomials for GNFS: with a new degree of freedom, we further improve the size property. We demons...

In this paper, a new factorization algorithm is presented, which finds a prime factor p of an integer n in time (D log n)O(1), if 4p − 1 = Db2 where D and b are integers. Hence this algorithm will factor a number efficiently, if it has a prime factor p such that 4p−1 is a product of a small integer and a square. Such primes should be avoided when we select the RSA secret keys. Some generalizati...

Most of the discussion of the 2-solenoid did not depend upon the 2, apart from the fact that 2 is an integer larger than 1. We consider a larger family of solenoids, at first with 2 replaced by a larger integer N , suggesting that such limits factor according to the prime factorization of N . This factorization does occur, and requires a proof which can be given in a general context. Incidental...

Abstract. A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}ks=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}ks=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.

The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used...