title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾ÑÑÐµÐ¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ (p-1)-Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° ÐÐ¾Ð»Ð»Ð°ÑÐ´Ð° Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ°ÑÐ¿Ð°ÑÐ°Ð»Ð»ÐµÐ»Ð¸Ð²Ð°Ð½Ð¸Ñ Ð½Ð° GPU
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° ÑÐ¿ÐµÑÐ¸Ð°Ð»Ð¸ÑÑÐ°
titleType: Subtitle
title: Investigation of the possibilities of optimizing the Pollard factorization algorithm P-1 based on parallelization on the GPU
Creators: name: ÐÑÐ²Ð¾Ð², ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2024
Subjects: ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ ÑÐ¸ÑÐµÐ» Ð½Ð° Ð¼Ð½Ð¾Ð¶Ð¸ÑÐµÐ»Ð¸
Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼ ÐÐ¾Ð»Ð»Ð°ÑÐ´Ð°
Ð°Ð¿Ð¿Ð°ÑÐ°ÑÐ½Ð¾Ðµ ÑÑÐºÐ¾ÑÐµÐ½Ð¸Ðµ
factorization
pollard's algorithm
hardware acceleration

description: Ð¦ÐµÐ»ÑÑ ÑÐ°Ð±Ð¾ÑÑ ÑÐ²Ð»ÑÐµÑÑÑ Ð¿Ð¾Ð¸ÑÐº ÑÑÑÐ°ÑÐµÐ³Ð¸Ð¸ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¸ÑÐµÑÐºÐ¾Ð¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸Â  (P-1)-Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° ÐÐ¾Ð»Ð»Ð°ÑÐ´Ð° Ð² ÑÑÐµÐ´Ðµ Ð¼Ð½Ð¾Ð³Ð¾Ð¿Ð¾ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð±Ð¾Ð»ÐµÐµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾Ð³Ð¾ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ñ ÑÐ¸ÑÐµÐ» Ð½Ð° Ð¼Ð½Ð¾Ð¶Ð¸ÑÐµÐ»Ð¸. ÐÐ°Ð´Ð°ÑÐ¸, ÑÐµÑÐ°ÐµÐ¼ÑÐµ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ¸ Ð´Ð»Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð´Ð»Ð¸Ð½Ð½Ð¾Ð¹ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÐ¸ Ð² ÑÑÐµÐ´Ðµ Ð¿Ð°ÑÐ°Ð»Ð»ÐµÐ»ÑÐ½ÑÑ Ð²ÑÑÐ¸ÑÐ»ÐµÐ½Ð¸Ð¹ Ð½Ð° Ð³ÑÐ°ÑÐ¸ÑÐµÑÐºÐ¾Ð¼ Ð¿ÑÐ¾ÑÐµÑÑÐ¾ÑÐµ. 2. Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° ÑÐ¿Ð¾ÑÐ¾Ð±Ð° Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸ Ð¾ÑÐ¸Ð³Ð¸Ð½Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° P-1 Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸ÐµÐ¼ Ð°Ð¿Ð¿Ð°ÑÐ°ÑÐ½Ð¾Ð³Ð¾ ÑÑÐºÐ¾ÑÐµÐ½Ð¸Ñ Ð¸ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½Ð¾Ð¹ Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ¸. 3. ÐÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð°Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½Ð¾Ð³Ð¾ ÑÐ¿Ð¾ÑÐ¾Ð±Ð°. 4. ÐÑÐµÐ½ÐºÐ° ÑÐºÐ¾ÑÐ¾ÑÑÐ¸ Ð¸ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ ÑÐ°Ð±Ð¾ÑÑ ÑÐ¾Ð·Ð´Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÑÐµÐ´ÑÑÐ²Ð°. Ð ÑÐ¾Ð´Ðµ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð° ÑÐ¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð° Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ° Ð´Ð»Ð¸Ð½Ð½Ð¾Ð¹ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÐ¸ ÑÑÐ°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ°Ð·ÑÑÐ´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð² ÑÑÐµÐ´Ðµ Ð¼Ð½Ð¾Ð³Ð¾Ð¿Ð¾ÑÐ¾ÑÐ½Ð¾Ð³Ð¾ Ð¸ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð¸Ñ. ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð°Ð½Ð°Ð»Ð¸Ð·Ð¸ÑÐ¾Ð²Ð°Ð½Ñ ÑÐ¾Ð²ÑÐµÐ¼ÐµÐ½Ð½ÑÐµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð² Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° P-1 Ð½Ð° GPU. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð¾ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð¾ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¾ Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ¾Ð² Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¸ Ð¼ÐµÑÐ¾Ð´Ð° P-1 Ð½Ð° Ð³ÑÐ°ÑÐ¸ÑÐµÑÐºÐ¾Ð¼ Ð¿ÑÐ¾ÑÐµÑÑÐ¾ÑÐµ, Ð±ÑÐ»Ð¸ Ð¿ÑÐ¾Ð´ÐµÐ¼Ð¾Ð½ÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ñ Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾ÑÑÐ¸ ÑÐ¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÑ ÑÐ¿Ð¾ÑÐ¾Ð±Ð¾Ð² Ð¿Ð¾ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ñ ÑÐ¸ÑÐµÐ» ÑÐ°Ð·Ð»Ð¸ÑÐ½Ð¾Ð¹ ÑÐ°Ð·ÑÑÐ´Ð½Ð¾ÑÑÐ¸. ÐÑÐ»Ð¸ ÑÐ´ÐµÐ»Ð°Ð½Ñ Ð²ÑÐ²Ð¾Ð´Ñ Ð¾ Ð¿ÑÐ¸Ð¼ÐµÐ½Ð¸Ð¼Ð¾ÑÑÐ¸ ÐºÐ°Ð¶Ð´Ð¾Ð³Ð¾ Ð¸Ð· ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½ÑÑ ÑÐ¿Ð¾ÑÐ¾Ð±Ð¾Ð² Ð½Ð° Ð¿ÑÐ°ÐºÑÐ¸ÐºÐµ. ÐÐ¾Ð»ÑÑÐµÐ½Ð½ÑÐµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¼Ð¾Ð³ÑÑ Ð±ÑÑÑ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð² ÐºÐ°ÑÐµÑÑÐ²Ðµ Ð¾ÑÐ½Ð¾Ð²Ñ Ð´Ð»Ñ Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð±Ð¾Ð»ÐµÐµ ÑÐ¾Ð²ÐµÑÑÐµÐ½Ð½ÑÑ ÑÐ¿Ð¾ÑÐ¾Ð±Ð¾Ð² ÑÐ°ÐºÑÐ¾ÑÐ¸Ð·Ð°ÑÐ¸Ð¸ ÑÐ¸ÑÐµÐ» Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° P-1.
description: The purpose of the study is to find a strategy for algorithmic optimization of the Pollard P-1 algorithm in a multi-threading environment for more efficient numbers factorization. The research set the following goals: 1. Development of a library for implementing long arithmetic in a parallel computing environment on a graphics processor. 2. Development of a method for modifying the original P-1 algorithm using hardware acceleration and the developed library. 3. Software implementation of the developed method. 4. Scanning speed assessment and efficiency of the developed software implementation. During the work, a library of long static-bit arithmetic was designed for use in a multi-threaded execution environment. Modern studies on the application of the P-1 method on GPUs were analyzed. The work resulted in the development of several variants of an algorithmic modification of the P-1 method on a graphics processor. Capabilities of the designed methods for decomposing numbers of various bit depths were demonstrated. Conclusions were drawn about the applicability of each of the developed methods in practice. The results obtained can be used as a basis for designing more advanced methods for factoring numbers based on the P-1 algorithm.