Related papers: Computing an Integer Prime Factoring in O(n^2.5)
This paper has been withdrawn by the authors due to a problem with *efficiently* predicting the large fourier coefficients. It is being reworked and will be resubmitted in the near future.
Ideally, the time that an incremental algorithm uses to process a change should be a function of the size of the change rather than, say, the size of the entire current input. Based on a formalization of ``the set of things changed'' by an…
We improve the "sieve" part of the number field sieve used in factoring integer and computing discrete logarithm. The runtime of our method is shorter than that of existing methods. Under some reasonable assumptions, we prove that it is…
The road to computing on quantum devices has been accelerated by the promises that come from using Shor's algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and…
We consider the error due to a single bit-flip in a floating point number. We assume IEEE 754 double precision arithmetic, which encodes binary floating point numbers in a 64-bit word. We assume that the bit-flip happens randomly so it has…
This paper has been withdrawn by the author due a few mistakes in the paper.
This paper has been withdrawn.
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…
This paper has been withdrawn by the author due to serious flaws in certain proofs. For instance, the method used to construct certain automorphic representations is flawed.
We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…
The paper has been withdrawn
This paper has been withdrawn by the authors. There was an erroneous estimate of the degree of a transformed polynomial, making the method appear more effective than it really is. We thank an anonymous referee for pointing out this error.
This paper has been withdrawn.
Factorization is the most fundamental way to determine if a number $n$ is prime or composite. Yet, this approach becomes impracticable when considering large values of $n$, a difficulty that is exploited by cryptographic protocols. We…
This paper has been withdrawn.
Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
This paper has been withdrawn because of serious errors.
This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional…
This paper has been withdrawn by the author, due to a crucial error in the proof of Thm.1