Herman J. J. te Riele

Factorization of a 768-bit RSA modulus

This paper reports on the factorization of the 768-bit number RSA-768 by the number field sieve factoring method and discusses some implications for RSA.

Factorization of RSA-140 Using the Number Field Sieve

We propose a mathematical problem, and show how to solve it elegantly. This problem is related with elliptic curve cryptosystems (ECC). The solving methods can be applied to a new paradigm of key generations of the ECC.

The mertens conjecture revisited

Let M(x)=∑1≤n≤xμ(n) where μ(n) is the Mobius function. The Mertens conjecture that

New Computations Concerning the Cohen-Lenstra Heuristics

|M(x)|/\sqrt{x} 1 was disproved in 1985 by Odlyzko and te Riele [13]. In the present paper, the known lower bound 1.06 for

Factoring Integers with Large-Prime Variations of the Quadratic Sieve

\limsup M(x)/\sqrt{x}

A Comparative Study of Algorithms for Computing Continued Fractions of Algebraic Numbers

is raised to 1.218, and the known upper bound –1.009 for

Average prime-pair counting formula

\liminf M(x)/\sqrt{x}

New Experimental Results Concerning the Goldbach Conjecture

is lowered to –1.229. In addition, the explicit upper bound of Pintz [14] on the smallest number for which the Mertens conjecture is false, is reduced from

Iterating the sum-of-divisors function

\exp(3.21\times10^{64})

Factoring with the quadratic sieve on large vector computers

to