Sharp Transitions in Making Squares

In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a square product). In his lecture at the 1994 International Congress of Mathematicians, Pomerance observed that the following problem encapsulates all of the key issues: Select integers a_1, a_2, >... at random from the interval [1,x], until some (non-empty) subsequence has product equal to a square. Find good estimate for the expected stopping time of this process. A good solution to this problem should help one to determine the optimal choice of parameters for one's factoring algorithm, and therefore this is a central question. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1-o(1) the first subsequence whose product equals a square occurs after at least J_0^{1-o(1)} integers have been selected, but no more than J_0, for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this expected stopping time up to a constant factor, tightening Pomerance's interval to $$[ (\pi/4)(e^{-\gamma} - o(1))J_0, (e^{-\gamma} + o(1)) J_0],$$ where $\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also confirm the well established belief that, typically, none of the integers in the square product have large prime factors. We believe the upper of the two bounds to be asymptotically sharp

Factorization

The purpose of this thesis will focus on the two most efficient algorithms which are quadratic sieve and number field sieve. Background information such as definitions and theorems are given to help understand the concepts behind each method

Block Sieving Algorithms

Quite similiar to the Sieve of Erastosthenes, the best-known general algorithms for factoring large numbers today are memory-bounded processes. We develop three variations of the sieving phase and discuss them in detail. The fastest modification is tailored to RISC processors and therefore especially suited for modern workstations and massively parallel supercomputers. For a 116 decimal digit composite number we achieved a speedup greater than two on an IBM RS/6000 250 workstation

Quantum and Classical Combinatorial Optimizations Applied to Lattice-Based Factorization

The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby integers whose prime factors are all small. This paper demonstrates that the hope of factoring numbers of commercial significance using these methods is unfounded. Mathematically, this is because the density of smooth numbers (numbers all of whose prime factors are small) decays exponentially as n grows. Our experimental reproductions and analysis show that lattice-based factoring does not scale successfully to larger numbers, that the proposed quantum enhancements do not alter this conclusion, and that other simpler classical optimization heuristics perform much better for lattice-based factoring. However, many topics in this area have interesting applications and mathematical challenges, independently of factoring itself. We consider particular cases of the CVP, and opportunities for applying quantum techniques to other parts of the factorization pipeline, including the solution of linear equations modulo 2. Though the goal of factoring 1000-bit numbers is still out-of-reach, the combinatoric landscape is promising, and warrants further research with more circumspect objectives

Factoring integers defined by second and third order recurrence relations

Factoring the first two-hundred and fifty Fibonacci numbers using just trial division would take an unreasonable amount of time. Instead the problem must be attacked using modern factorization algorithms. We look not only at the Fibonacci numbers, but also at factoring integers defined by other second and third order recurrence relations. Specifically we include the Fibonacci, Tribonacci and Lucas numbers. We have verified the known factorizations of first 382 Fibonacci numbers and the first 185 Lucas numbers, we also completely factored the first 311 Tribonacci numbers