{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/36618"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/36618","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"An Introduction to the General Number Field Sieve","abstract":"With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a \"large\" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a \"reasonable\" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined.","abstract_html":"With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a "large" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a "reasonable" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined.","abstract_has_math":false,"creators":["Briggs, Matthew Edward"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Brown, Ezra A."],"committee_members":["Parry, Charles J.","Day, Martin V."],"year":1998,"date_issued":"1998-04-17","date_published":"1998-04-17","updated_at":"2026-07-22T22:19:04Z","subjects":["Number Field Sieve","Factoring","Cryptography","Algebraic Number Theory"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-32298-93111"],"render_values":[{"text":"etd-32298-93111","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/36618","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Brown, Ezra A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Parry, Charles J.","Day, Martin V."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Briggs, Matthew Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:51:17Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:51:17Z","1998-04-23"]},{"key":"dc:date.issued","label":"Date","values":["1998-04-17"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Number Field Sieve","Factoring","Cryptography","Algebraic Number Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-32298-93111"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/36618"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a \"large\" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a \"reasonable\" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["An Introduction to the General Number Field Sieve"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Brown, Ezra A."],"dc:contributor.committeemember":["Parry, Charles J.","Day, Martin V."],"dc:contributor.department":["Mathematics"],"dc:creator":["Briggs, Matthew Edward"],"dc:date.accessioned":["2014-03-14T20:51:17Z"],"dc:date.available":["2014-03-14T20:51:17Z","1998-04-23"],"dc:date.issued":["1998-04-17"],"dc:description.abstract":["With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a \"large\" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a \"reasonable\" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined."],"dc:description.degree":["Master of Science"],"dc:identifier.other":["etd-32298-93111"],"dc:identifier.uri":["http://hdl.handle.net/10919/36618"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Number Field Sieve","Factoring","Cryptography","Algebraic Number Theory"],"dc:title":["An Introduction to the General Number Field Sieve"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:04Z"}