27,070 research outputs found
Factorization in the Cloud: Integer Factorization Using F# and Windows Azure
Implementations are presented of two common algorithms for integer factorization, Pollard’s “p – 1” method and the SQUFOF method. The algorithms are implemented in the F# language, a functional programming language developed by Microsoft and officially released for the first time in 2010. The algorithms are thoroughly tested on a set of large integers (up to 64 bits in size), running both on a physical machine and a Windows Azure machine instance.
Analysis of the relative performance between the two environments indicates comparable performance when taking into account the difference in computing power. Further analysis reveals that the relative performance of the Azure implementation tends to improve as the magnitudes of the integers increase, indicating that such an approach may be suitable for larger, more complex factorization tasks.
Finally, several questions are presented for future research, including the performance of F# and related languages for more efficient, parallelizable algorithms, and the relative cost and performance of factorization algorithms in various environments, including physical hardware and commercial cloud computing offerings from the various vendors in the industry
Shallow Depth Factoring Based on Quantum Feasibility Labeling and Variational Quantum Search
Large integer factorization is a prominent research challenge, particularly
in the context of quantum computing. This holds significant importance,
especially in information security that relies on public key cryptosystems. The
classical computation of prime factors for an integer has exponential time
complexity. Quantum computing offers the potential for significantly faster
computational processes compared to classical processors. In this paper, we
propose a new quantum algorithm, Shallow Depth Factoring (SDF), to factor a
biprime integer. SDF consists of three steps. First, it converts a factoring
problem to an optimization problem without an objective function. Then, it uses
a Quantum Feasibility Labeling (QFL) method to label every possible solution
according to whether it is feasible or infeasible for the optimization problem.
Finally, it employs the Variational Quantum Search (VQS) to find all feasible
solutions. The SDF utilizes shallow-depth quantum circuits for efficient
factorization, with the circuit depth scaling linearly as the integer to be
factorized increases. Through minimizing the number of gates in the circuit,
the algorithm enhances feasibility and reduces vulnerability to errors.Comment: 10 pages, 3 figure
Statistical Mechanical Formulation and Simulation of Prime Factorization of Integers
We propose a new formulation of the problem of prime factorization of
integers. With replica exchange Monte Carlo simulation, the behavior which is
seemed to indicate exponential computational hardness is observed. But this
formulation is expected to give a new insight into the computational complexity
of this problem from a statistical mechanical point of view.Comment: 5 pages, 5figures, Proceedings of 4th YSM-SPIP (Sendai, 14-16
December 2012
Shorter addition chain for smooth integers using decomposition method.
An efficient computation of scalar multiplication in elliptic curve cryptography can be achieved by reducing the original problem into a chain of additions and doublings. Finding the shortest addition chain is an NP-problem. To produce the nearest possible shortest chain, various methods were introduced and most of them depends on the representation of a positive integer n into a binary form. Our method works out the given n by twice decomposition, first into its prime powers and second, for each prime into a series of 2's from which a set of rules based on addition and doubling is defined. Since prime factorization is computationally a hard problem, this method is only suitable for smooth integers. As an alternative, the need to decompose n can be avoided by choosing n of the form p1 e1p2 e2⋯r er. This shall not compromise the security of ECC since its does not depend on prime factorization problem. The result shows a significant improvement over existing methods especially when n grows very large
- …