{"id":{"repo_id":"nps","oai_identifier":"oai:calhoun.nps.edu:10945/2603"},"canonical_url":"https://search.dev.ndltd.org/etd/nps/oai:calhoun.nps.edu:10945/2603","repository":{"repo_id":"nps","name":"Naval Postgraduate School","base_url":"https://calhoun.nps.edu/server/oai/request"},"display":{"title":"Exploring fields with shift registers","abstract":"The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x).","abstract_html":"The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x).","abstract_has_math":false,"creators":["Radowicz, Jody L."],"institution":"Monterey, CA; Naval Postgraduate School","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Computer Science (CS)","school":null,"contributors":[],"advisors":["Dinolt, George","Fredricksen, Harold"],"committee_chairs":[],"committee_members":[],"year":2006,"date_issued":"2006-09","date_published":"2006-09","updated_at":"2026-07-27T20:26:38Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10945/2603","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Dinolt, George","Fredricksen, Harold"]},{"key":"dc:contributor.department","label":"Department","values":["Computer Science (CS)"]},{"key":"dc:creator","label":"Author","values":["Radowicz, Jody L."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2012-03-14T17:35:41Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2012-03-14T17:35:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2006-09"]},{"key":"dc:publisher","label":"Institution","values":["Monterey, CA; Naval Postgraduate School"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10945/2603"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x)."]},{"key":"dc:title","label":"Title","values":["Exploring fields with shift registers"]}]}],"canonical_facts":{"dc:contributor.advisor":["Dinolt, George","Fredricksen, Harold"],"dc:contributor.department":["Computer Science (CS)"],"dc:creator":["Radowicz, Jody L."],"dc:date.accessioned":["2012-03-14T17:35:41Z"],"dc:date.available":["2012-03-14T17:35:41Z"],"dc:date.issued":["2006-09"],"dc:description.abstract":["The S-Boxes used in the AES algorithm are generated by field extensions of the Galois field over two elements, called GF(2). Therefore, understanding the field extensions provides a method of analysis, potentially efficient implementation, and efficient attacks. Different polynomials can be used to generate the fields, and we explore the set of polynomials x^ 2 + x + a^J over GF(2^n) where a is a primitive element of GF(2^n). The results of this work are the first steps towards a full understanding of the field that AES computation occurs in-GF(2^8). The charts created with the data we gathered detail which power of the current primitive root is equal to previous primitive roots for fields up through GF(2^16) created by polynomials of the form x^2 + x + a^i for a primitive element a. Currently, a C++ program will also provide all the primitive polynomials of the form x^2 + x+ a^i for a primitive element a over the fields through GF(2^32). This work also led to a deeper understanding of certain elements of a field and their equivalent shift register state. In addition, given an irreducible polynomial 2 f(x) = x^2 + a^i x + a^j over GF(2^n), the period (and therefore the primitivity) can be determined by a new theorem without running the shift register generated by f(x)."],"dc:identifier.uri":["https://hdl.handle.net/10945/2603"],"dc:publisher":["Monterey, CA; Naval Postgraduate School"],"dc:title":["Exploring fields with shift registers"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T20:26:38Z"}