[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127827-en":3,"doc-seo-127827-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},127827,1099523885336,"Violet","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Analyzing the Potential of a statistically-based Integer Factorization Method for use in Machine Learning by means of Prime Characteristics, Statistics and Geometry - Dissertation","RSAの安全性依赖于整数因子分解的困难性，传统方法通常由硬编码规则限制搜索过程；因此，机器学习可通过利用质数统计信息与目标因子特性实现自适应搜索策略，从而为破解提供新思路。本研究旨在建立一种可融合“质数模式统计”和“质数特征”的智能分解算法研究基础。首先对真实与生成的质数进行20TB统计分析（最高512位），发现多种部分未被记载的结构性模式，并获得生成质数的上下界。据此将搜索目标从以√n为起点的区间进行统计距离平移，在最优情况下对目标因子p、q达到最高约99.08%的近似。进一步设计两个质数统计驱动的整数分解原型，对128位测试数完成分解，耗时范围从1小时到96小时，用以证明可行性并为后续机器学习实现奠定基础。","Fakultät für  \nMathematik und Informatik  \nAykan Inan  \nAnalyzing the Potential of a statistically-based Integer Factorization Method for use in Machine Learning by means of Prime Characteristics, Statistics and Geometry  \nDissertation  \nAnalyzing the Potential of a statistically-based Integer Factorization Method for use in Machine Learning by means of Prime Characteristics, Statistics and Geometry  \nDissertation for obtaining the academic degree of Dr. rer. nat. at the Faculty of Mathematics and Computer Science at the FernUniversität in Hagen  \nsubmitted by  \nAykan D. Inan, M.Sc.  \nHagen, 21.01.2023  \nAbstract  \nCurrently, the RSA crypto scheme is considered hard to be compromised since all integer factorization approaches are limited by their hard-coded algorithms. Machine learning might be a possible solution to this problem since smart algorithms can dynamically adjust their search scheme based on statistical information about primes and the searched prime factors.  \nThis work, therefore, aims to create the research basis for an intelligent factorization algorithm that can integrate dynamic rules from prime pattern statistics and prime characteristics. This ✜rst required a statistical analysis of 20TB of real and generated primes (up to 512 bits) . The analysis found numerous ✕in part undocumented✕ patterns including inter alia block patterns among primes as well as an upper and lower bound of generated primes. These results allowed e.g. statistical (distance) shifts into the requisite number range of the searched prime factors p and q with n = p · q in relation to √n as starting value. This shift provided up to 99 .08% approximation to the searched factors. On this basis, two prime statistic-based integer factorization prototypes were designed.  \nThese prototypes each integrated up to four statistical ✜ndings, factorizing the test numbers (128 bits) between one hour and 96 hours. Compared to the currently existing hard-coded methods, the evaluation of this low-performance speed is secondary at this stage, since these prototypes were designed to serve as proof of concept that a statisticallybased factorization algorithm is feasible but not to integrate them into actual machine learning, yet. This work, therefore, forms the promising foundation for a ✜rst machine learning setup.  \nZusammenfassung  \nDerzeit gilt das RSA-Verschlüsselungsverfahren als nahezu unangreifbar, da die Primfaktorzerlegung durch ihre festkodierten Algorithmen starren, rechen intensiven Regeln unterliegt. Maschinelles Lernen könnte hier die Lösung sein, da intelligente Algorithmenimstande sind ihr Suchschema dynamisch auf der Grundlage statistischer Informationen über Primzahlen und den gesuchten Primfaktoren anzupassen.  \nZiel dieser Arbeit ist es daher, die Forschungsgrundlage für einen intelligenten Faktorisierungsalgorithmus zu scha✛en, der dynamische Regeln aus Primmusterstatistiken und statistisch relevanter Primmerkmale integrieren kann. Dies erforderte zunächst eine statistische Analyse von 20 TB realer und generierter Primzahlen (bis zu 512 Bit) . Diese Analyse ergab zahlreiche ✕zum Teil bislang undokumentierter✕ Muster, darunter auch Blockmuster. Zudem wurde eine obere und untere Grenze für generierten Primzahlen gefunden. Diese Ergebnisse ermöglichten wiederum eine statistische (Distanz)-Verschiebungen in den Zahlenbereich der gesuchten Primfaktoren p und q mit n = p · q und √n als Startwert. Diese Verschiebungen machten eine bis zu 99,08 prozentige Annäherung an die gesuchten Faktoren möglich.  \nAuf Grundlage dieser Ergebnisse wurden zwei Prototypen entwickelt. Diese integrieren jeweils bis zu vier statistische Ergebnisse und faktorisieren die Testzahlen (128 Bit) in einem Zeitraum zwischen einer Stunde und 96 Stunden. Diese Prototypen demonstrierendaher die Machbarkeit eines statistisch basierten Faktorisierungsalgorithmus. Allerdingskönnen diese Prototypen aktuell noch nicht mit den hardkodierten Verfahren verglichen werden, da sie aktu","cbCaikxyXpXzf16b","https://ap.wps.com/l/cbCaikxyXpXzf16b","pdf",4363582,3,1,239,"English","en",105,"# Abstract\n## Forschungsziele und Motivation\n## Statistische Analyse von Primzahlen\n## Prototypen und Ergebnisse","[{\"question\":\"研究要解决的核心问题是什么？\",\"answer\":\"在传统整数因子分解方法受硬编码规则限制的情况下，探索如何利用机器学习与质数统计信息来构建更灵活的分解搜索策略。\"},{\"question\":\"研究首先做了什么关键工作？\",\"answer\":\"对真实与生成的质数进行20TB规模统计分析，覆盖最高512位，并提取出质数的结构性模式及生成质数的上下界。\"},{\"question\":\"两个因子分解原型的设计思路与性能如何？\",\"answer\":\"原型基于质数统计发现进行集成，每个原型最多使用四类统计结果，用于对128位测试数完成分解，耗时从约1小时到96小时，用于证明统计驱动方法的可行性。\"}]","Analyzing the Potential of a statistically-based Integer Factorization Method for use in Machine Learning by means of Prime Characteristics, Statistics and Geometry - Dissertation | PDF",1785942174,602,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"analyzing-the-potential-of-a-statistically-based-integer-factorization-method-for-use-in-machine-learning-by-means-of-prime-characteristics-statistics-and-geometry-dissertation","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":20},"https://docshare.wps.com/document/research-report/",{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/analyzing-the-potential-of-a-statistically-based-integer-factorization-method-for-use-in-machine-learning-by-means-of-prime-characteristics-statistics-and-geometry-dissertation/127827/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-24","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"研究要解决的核心问题是什么？","Question",{"text":76,"@type":77},"在传统整数因子分解方法受硬编码规则限制的情况下，探索如何利用机器学习与质数统计信息来构建更灵活的分解搜索策略。","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"研究首先做了什么关键工作？",{"text":81,"@type":77},"对真实与生成的质数进行20TB规模统计分析，覆盖最高512位，并提取出质数的结构性模式及生成质数的上下界。",{"name":83,"@type":74,"acceptedAnswer":84},"两个因子分解原型的设计思路与性能如何？",{"text":85,"@type":77},"原型基于质数统计发现进行集成，每个原型最多使用四类统计结果，用于对128位测试数完成分解，耗时从约1小时到96小时，用于证明统计驱动方法的可行性。","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]