{"id":166314,"date":"2018-11-06T17:22:47","date_gmt":"2018-11-07T01:22:47","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/efficient-quantum-circuits-for-binary-elliptic-curve-arithmetic-reducing-t-gate-complexity\/"},"modified":"2018-11-06T17:22:47","modified_gmt":"2018-11-07T01:22:47","slug":"efficient-quantum-circuits-for-binary-elliptic-curve-arithmetic-reducing-t-gate-complexity","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/efficient-quantum-circuits-for-binary-elliptic-curve-arithmetic-reducing-t-gate-complexity\/","title":{"rendered":"Efficient quantum circuits for binary elliptic curve arithmetic: reducing T-gate complexity"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Elliptic curves over finite fields GF(2<sup>n<\/sup>) play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of T-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in GF(2<sup>n<\/sup>) in depth O(n log n) using a polynomial basis representation, which may be of independent interest.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Elliptic curves over finite fields GF(2n) play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of T-gates needed to implement [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Quant. Inform. & Comp.","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Quant. Inform. & Comp.","msr_number":"7&8","msr_organization":"","msr_pages_string":"631\u2013644","msr_page_range_start":"631","msr_page_range_end":"644","msr_series":"","msr_volume":"13","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"B. Amento, M. R\u00f6tteler, R. 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LIQUi|&gt; can be used to translate a quantum algorithm written in the form of a high-level program into the low-level machine instructions for a quantum device. LIQUi|&gt; is being developed by the Quantum Architectures and Computation Group (QuArC)\u00a0at Microsoft Research. 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