A Note on Integer Factorization Using Lattices

We revisit Schnorr's lattice-based integer factorization algorithm, now with an effective point of view. We present effective versions of Theorem 2 of Schnorr's "Factoring integers and computing discrete logarithms via diophantine approximation" paper, as well as new elementary properties of the Prime Number Lattice bases of Schnorr and Adleman

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

Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

Optimal ancilla-free Clifford+T approximation of z-rotations

We consider the problem of approximating arbitrary single-qubit z-rotations by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of T-count m + O(log(log(1/epsilon))), where m is the T-count of the second-to-optimal solution. In the typical case, this yields circuit approximations of T-count 3log_2(1/epsilon) + O(log(log(1/epsilon))). Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is O(polylog(1/epsilon)).Comment: 40 pages. New in v3: added a section on worst-case behavio