Reflecting the wide used algorithmic and number theory in computer science, cryptography, and medicine, these 20 survey articles cover such topics as the Pell equation, basic algorithms and number theory, the quadratic sieve, primary testing algorithms, lattices, elliptic curves, number theory as an element of computational theory (and beyond), discrete logarithms, the effects of the number field sieve on discreet logarithms, finite fields, reducing the lattice basis to examine univariate polynomials, computing Arakelov class groups, computational class field theory, the algorithm theory of zeta functions over finite fields, congruent number problems and their variants, and an introduction to computing modular forms using modular symbols.

To solve the factoring challenge, the consortium leveraged resources from around the world, including hardware from the Experimental Mathematics Institute in Essen, Germany, from the Bundesamt fur Sicherheit in der Informationstechnologie (BSI), and experts from the Number Field Sieve network of mathematicians throughout Canada, the United States and the United Kingdom.

Using a worldwide network of computers and a sophisticated mathematical technique called the Number Field Sieve, Herman te Riele of the National Research Institute for Mathematics and Computer Science (CWI) in Amsterdam and his coworkers succeeded in factoring a 155-digit number into two 78-digit primes.

Last week, a team of researchers using a relatively new method known as the number field sieve succeeded in factoring a 116-digit number.

New research reveals that the number field sieve may work significantly more efficiently than the quadratic sieve.