An integer factorization algorithm which uses diffusion as a computational engine
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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)^2log (log N) log (loglog 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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