{"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,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[243138],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-166314","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-quantum","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Quant. 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