The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for this integer factorization is the General Number Field Sieve (GNFS). One important step within the GNFS is the factorization of mid-size numbers without small prime divisors. This can be done efficiently by the Elliptic Curve Method (ECM), e.g. in sp...

The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...

We have previously discussed the basic concepts and fundamental theorems on unique factorization in PIDs. We will provide a few examples of how unique factorization is used in applications to number theory. One cannot stress how important the concept of unique factorization is in many mathematical applications and definately in algebraic number theory. We unfortunately will only have the time t...

For sparse QR factorization, nding a good column ordering of the matrix to be factorized, is essential. Both the amount of ll in the resulting factors, and the number of oating-point operations required by the factorization, are highly dependent on this ordering. A suitable column ordering of the matrix A is usually obtained by minimum degree analysis on A T A. The objective of this analysis is...

A positive integer n is called a covering number if there are some distinct divisors n1, . . . , nk of n greater than one and some integers a1, . . . , ak such that Z is the union of the residue classes a1(mod n1), . . . , ak(mod nk). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (pri...

We present new factors of seventeen Mersenne numbers, obtained using a variant of the special number field sieve where sieving on the algebraic side is shared among the numbers. It reduced the overall factoring effort by more than 50%. As far as we know this is the first practical application of Coppersmith’s “factorization factory” idea. Most factorizations used a new double-product approach t...