On sharp transitions in making squares

In the fastest-performing integer factoring algorithms, one creates a sequence of integers (in a pseudo-random way) and wishes to rapidly determine a subsequence whose product is a square. In 1994 Pomerance stated the following problem which encapsulates all of the key issues: Select integers a1;a2;:::; at random from the interval [1;x], until some (nonempty) subsequence has product equal to a square. Find a good estimate for the expected stopping time of this process. A good solution should allow one to determine the optimal choice of parameters in many factoring algorithms. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1 o(1) the rst subsequence whose product equals a square

[1]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[2]  Prasad Tetali,et al.  Running Time Predictions for Factoring Algorithms , 2008, ANTS.

[3]  Carl Pomerance,et al.  Smooth numbers and the quadratic sieve , 2008 .

[4]  P. Leroux.  Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals , 2004, Electron. J. Comb..

[5]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[6]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[7]  Andrew Granville,et al.  Large character sums , 1999, math/9903196.

[8]  Neil J. Calkin.  Dependent Sets of Constant Weight Binary Vectors , 1997, Comb. Probab. Comput..

[9]  G. Tenenbaum.  Introduction to Analytic and Probabilistic Number Theory , 1995 .

[10]  Carl Pomerance,et al.  The Role of Smooth Numbers in Number Theoretic Algorithms , 1995 .

[11]  Carl Pomerance,et al.  The Development of the Number Field Sieve , 1994 .

[12]  H. Lenstra,et al.  Factoring integers with the number field sieve , 1993 .

[13]  R. Durrett.  Probability: Theory and Examples , 1993 .

[14]  Robert D. Silverman.  The multiple polynomial quadratic sieve , 1987 .

[15]  Adolf Hildebrand,et al.  On integers free of large prime factors , 1986 .

[16]  Carl Pomerance,et al.  The Quadratic Sieve Factoring Algorithm , 1985, EUROCRYPT.

[17]  J. Dixon.  Asymptotically fast factorization of integers , 1981 .

[18]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[19]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[20]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .