Differencing methods for Korobov-type exponential sums

Vandehey, Joseph

doi:10.1007/s11854-019-0038-2

Cited by 1 publication

(2 citation statements)

References 17 publications

(27 reference statements)

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“…By using the bounds for the Korobov-type exponential sums as proved in [KM12], we will deduce an upper bound for the second largest eigenvalue λ X r,S * of the half-lazy random walk on X r,S . We note here that results related to those from [KM12] are given in [LLW98], [Mo09] and [Va09].…”

Section: Proof Of Theoremsupporting

confidence: 57%

“…From [Ba79] and [Lo75], we can express the eigenvalues of the associated Laplacian in terms of certain exponential sums. As it turns out, optimal bounds for these exponential bounds are known; see [KM12] as well as [Va09]. When combining these results, we show that after O((log N ) 2 ) diffusion steps one determines the cardinality of V , thus the order of b modulo N .…”

Section: Introductionmentioning

confidence: 59%

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An integer factorization algorithm which uses diffusion as a computational engine

Cadavid¹,

Paulina²,

Jorgenson³

et al. 2021

Preprint

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In this article we develop an algorithm which computes a divisor of an integer N , which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If N has m distinct prime factors, then the probability that our algorithm runs successfully is at least p(m) = 1 − (m + 1)/2 m . We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most O((log N ) 2 ) deterministic steps and at most O((log N ) 2 ) diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most O((log N ) 2 log(log N ) log(log log N )) quantum steps on a quantum computer.As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of N = 33 and N = 1363.

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Section: Proof Of Theoremsupporting

confidence: 57%