Turning Euler's Factoring Method into a Factoring Algorithm

An algorithm is presented which, given a positive integer n,will either factor n or prove it to be prime. The algorithmtakes O(n^1/3+) steps.     

Integer Factoring

Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.     

Factoring Factor Maps

A noninjective bounded-to-one factor map from an irreducibleshift of finite type onto a sofic system can be factored asa composition of other such maps in only finitely many ways(up to isomorphism). This generalizes to factor maps from systemswith canonical coordinates to finitely presented dynamical systems.     

Factoring groups and tiling space

The problem of tiling space by translates of certain star bodies, called crosses and semicrosses, is intimately connected with finding a subsetA of a finite abelian groupG such that for a particular subset of the integersS each non-zero element ofG is uniquely expressible in the forms·g withs inS andg inA. This paper examines some of the algebraic questions raised; in particular it obtains bounds on the number of elements inS, constructs factorizations ofZ _p ⁿ , and presents an example of a setS that factors no group.     

Factoring multi-sublinear maps

Using Rademacher type, maximal estimates are established for k-sublinear operators with values in the space of measurable functions. Maurey–Nikishin factorization implies that such operators factor through a weak-type Lebesgue space. This extends known results for sublinear operators and improves some results for bilinear operators. For example, any continuous bilinear operator from a product of type 2 spaces into the space of measurable functions factors through a Banach space. Also included are applications for multilinear translation invariant operators.     

Factoring cartesian-product graphs

In a fundamental paper, G. Sabidussi [“Graph Multiplication,” Mathematische Zeitschrift, Vol. 72 (1960), pp. 446–457] used a tower of equivalence relations on the edge set E(G) of a connected graph G to decompose G into a Cartesian product of prime graphs. Later, a method by R.L. Graham and P.M. Winkler [“On Isometric Embeddings of Graphs,” Transactions of the American Mathematics Society, Vol. 288 (1985), pp. 527–533] of embedding a connected graph isometrically into Cartesian products opened another approach to this problem. In both approaches an equivalence relation σ that determines the prime factorization is constructed. The methods differ by the starting relations used. We show that σ can be obtained as the convex hull of the starting relation used by Sabidussi. Our result also holds for the relation determining the prime decomposition of infinite connected graphs with respect to the weak Cartesian product. Moreover, we show that this relation is the transitive closure of the union of the starting relations of Sabidussi and Winkler [“Factoring a Graph in Polynomial Time,” European Journal of Combinatorics, Vol. 8 (1987), pp. 209–212], thereby generalizing the result of T. Feder [“Product Graph Representations,” Journal of Graph Theory, Vol 16 (1993), pp. 467–488] from finite to infinite graphs.     

Factoring by Cyclic and Simulated Subsets

Let G be a finite abelian group whose p-component is an elementary p-group for each prime divisor p of ¦G¦. Assume that B, A₁,..., A_n are subsets of G such that each ¦A_i¦ is a prime, ¦B¦ is a product of two primes, and each A_i is either cyclic or simulated. If the product BA₁ ... A_n is direct and is equal to G, then at least one of the factors B, A₁, ..., A_n is periodic.     

Factoring Polynomials and the Knapsack Problem

For several decades the standard algorithm for factoring polynomials f with rational coefficients has been the Berlekamp-Zassenhaus algorithm. The complexity of this algorithm depends exponentially on n, where n is the number of modular factors of f. This exponential time complexity is due to a combinatorial problem: the problem of choosing the right subsets of these n factors. In this paper, this combinatorial problem is reduced to a type of Knapsack problem that can be solved with lattice reduction algorithms. The result is a practical algorithm that can factor polynomials that are far out of reach for previous algorithms. The presented solution to the combinatorial problem is different from previous lattice-based factorizers; these algorithms avoided the combinatorial problem by solving the entire factorization problem with lattice reduction. This led to lattices of large dimension and coefficients, and thus poor performance. This is why lattice-based algorithms, despite their polynomial time complexity, did not replace Berlekamp-Zassenhaus as the standard method. That is now changing; new versions of computer algebra systems such as Maple, Magma, NTL and Pari have already switched to the algorithm presented here.     

Factoring Effectiveness Factors!

It has previously been suggested that the effectiveness of strategic planning systems may be characterized in terms of a multi-attribute framework. This paper explores the possibility of reducing the postulated set of attributes to an ‘irreducible core’ by means of a factor analysis. In addition, the data set used in this analysis was explored to determine whether, in terms of their planning effectiveness, the organizations in the sample formed natural subgroups. Some interesting questions are generated by the analysis.     

Factoring weakly compact operators

The main result is that every weakly compact operator between Banach spaces factors through a reflexive Banach space. Applications of the result and technique of proof include new results (e.g., separable conjugate spaces embed isomorphically in spaces with boundedly complete bases; convex weakly compact sets are affinely homeomorphic to sets in a reflexive space) and simple proofs of known results (e.g., there is a reflexive space failing the Banach-Saks property; if X is separable, then $\text{X =}\text{Z}{}^7\text{Z}$ for some Z; there is a separable space which does not contain l₁ whose dual is nonseparable).     

Factoring by simulated subsets II

In earlier papers it has been shown that certain different types of conditions on the factors in a factorization of a finite abelian group by its subsets lead to the conclusion that one factor must be a subgroup. In this paper the common generalization is proved that this result still holds even if different factors satisfy different types of condition. It is also shown that one condition may be weakened without effecting the conclusion.     

Factoring operators through Hilbert space

We give an example of a pair of Banach spacesE andF so that neitherE′ norF has cotype 2, but every bounded linear operator fromE intoF factors through Hilbert space.     

Factoring polynomials modulo special primes

We consider the problem of factoring polynomials overGF(p) for those prime numbersp for which all prime factors ofp– 1 are small. We show that if we have a primitivet-th root of unity for every primet dividingp– 1 then factoring polynomials overGF(p) can be done in deterministic polynomial time.Research partially supported by Hungarian National Foundation for Scientific Research, Grant 1812.