Daniel Shanks’ Square Forms Factorization

Square form factorization

We present a detailed analysis of SQUFOF, Daniel Shanks’ Square Form Factorization algorithm. We give the average time and space requirements for SQUFOF. We analyze the effect of multipliers, either used for a single factorization or when racing the algorithm in parallel.

Categorization and Adaptive Behavior: the Role of Associative Processes in Symbolic Concept Learning1

A new tendency is emerging in cognitive science that pays attention to the possible commonalities between elementary processes of learning (such as classical or instrumental conditioning) and more complex inductive processes (like categorization or causality judgements) (see, e.g., Estes, 1985; Gluck & Bower, 1988; Holland et al., 1986; Pearce, 1988; Shanks & Dickinson, 1987; Schlimmer & Grange...

A Sieve Method for Factoring Numbers

A One Line Factoring Algorithm

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of fac...

On the Computation of Quadratic 2-class Groups

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D 0; 1 mod 4 and the factorization of D, computes the structure of the 2-Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log jDj.