## Exploring fields with shift registers

 dc.contributor.advisor Dinolt, George dc.contributor.advisor Fredricksen, Harold dc.contributor.author Radowicz, Jody L. dc.contributor.corporate Naval Postgraduate School (U.S.) dc.contributor.department Computer Science (CS) 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). en_US dc.description.distributionstatement Approved for public release; distribution is unlimited. dc.description.uri http://archive.org/details/exploringfieldsw109452603 dc.format.extent xiv, 83 p. : ill. ; en_US dc.identifier.oclc 73174933 dc.identifier.uri https://hdl.handle.net/10945/2603 dc.publisher Monterey, CA; Naval Postgraduate School en_US dc.subject.lcsh Computer science en_US dc.subject.lcsh Polynomials en_US dc.subject.lcsh Exponential functions en_US dc.subject.lcsh Algorithms en_US dc.title Exploring fields with shift registers en_US dc.type Thesis en_US dspace.entity.type Publication etd.thesisdegree.discipline Computer Science en_US etd.thesisdegree.grantor Naval Postgraduate School en_US etd.thesisdegree.level Masters en_US etd.thesisdegree.name M.S. en_US etd.verified no en_US relation.isDepartmentOfPublication 67864e54-711d-4c0a-a6d4-439a011f2bd1 relation.isDepartmentOfPublication.latestForDiscovery 67864e54-711d-4c0a-a6d4-439a011f2bd1
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