{"id":158375,"date":"1997-07-01T00:00:00","date_gmt":"1997-07-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/fast-software-exponentiation-in-gf2k\/"},"modified":"2021-03-18T12:58:23","modified_gmt":"2021-03-18T19:58:23","slug":"fast-software-exponentiation-in-gf2k","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/fast-software-exponentiation-in-gf2k\/","title":{"rendered":"Fast software exponentiation in GF(2k)"},"content":{"rendered":"

The authors present a new algorithm for computing $a^e$ where $a \\in GF(2^k<\/em>)$ and $e$ is a positive integer. The proposed algorithm is more suitable for implementation in software, and relies on the Montgomery multiplication in $GF(2^k<\/em>)$. The speed of the exponentiation algorithm largely depends on the availability of a fast method for multiplying two polynomials of length $w$ defined over GF(2). The theoretical analysis and experiments indicate that the proposed exponentiation method is at least 6 times faster than the exponentiation method using the standard multiplication when $w=8$. Furthermore, the availability of a 32-bit GF(2) polynomial multiplication instruction on the underlying processor would make the new exponentiation algorithm up to 37 times faster.<\/p>\n","protected":false},"excerpt":{"rendered":"

The authors present a new algorithm for computing $a^e$ where $a \\in GF(2^k)$ and $e$ is a positive integer. The proposed algorithm is more suitable for implementation in software, and relies on the Montgomery multiplication in $GF(2^k)$. The speed of the exponentiation algorithm largely depends on the availability of a fast method for multiplying two […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13546,13558],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[253996,249139,253990,254002,254011,253984,254005,253999,251878,254008],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-158375","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-research-area-security-privacy-cryptography","msr-locale-en_us","msr-field-of-study-arithmetic","msr-field-of-study-discrete-mathematics","msr-field-of-study-exponentiation","msr-field-of-study-exponentiation-by-squaring","msr-field-of-study-factorization-of-polynomials-over-finite-fields","msr-field-of-study-gf2","msr-field-of-study-knuths-up-arrow-notation","msr-field-of-study-modular-exponentiation","msr-field-of-study-multiplication","msr-field-of-study-polynomial-arithmetic"],"msr_publishername":"IEEE","msr_edition":"","msr_affiliation":"","msr_published_date":"1997-3-5","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"IEEE Computer Society Press","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":0,"msr_main_download":"224398","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/1997\/07\/c14expgf.pdf","label_id":"243132","label":0},{"type":"doi","viewUrl":"false","id":"false","title":"10.1109\/ARITH.1997.614899","label_id":"243106","label":0}],"msr_related_uploader":"","msr_attachments":[{"id":224398,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/1997\/07\/c14expgf.pdf"}],"msr-author-ordering":[{"type":"text","value":"C.K. 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