We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor’s algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible integer and modular arithmetic through windowing techniques and more adaptive placement of uncomputing steps, and improve over previous quantum circuits for modular inversion by reformulating the binary Euclidean algorithm. Overall, we obtain an affine Weierstrass point addition circuit that has lower depth and uses fewer T gates than previous circuits. While previous work mostly focuses on minimizing the total number of qubits, we present various trade-offs between different cost metrics including the number of qubits, circuit depth and T-gate count. Finally, we provide a full implementation of point addition in the Q# quantum programming language that allows unit tests and automatic quantum resource estimation for all components.<\/p>\n","protected":false},"excerpt":{"rendered":"
We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor’s algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible integer and modular arithmetic through windowing techniques and more adaptive placement of uncomputing steps, and improve over previous quantum circuits for modular inversion […]<\/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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13558],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-645252","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2020-4-15","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":"","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":1,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":"","msr_related_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/eprint.iacr.org\/2020\/077","label_id":"243112","label":0}],"msr_attachments":[],"msr-author-ordering":[{"type":"text","value":"Thomas H\u00e4ner","user_id":0,"rest_url":false},{"type":"text","value":"Samuel Jaques","user_id":0,"rest_url":false},{"type":"user_nicename","value":"Michael Naehrig","user_id":32976,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Michael Naehrig"},{"type":"user_nicename","value":"Martin Roetteler","user_id":32823,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Martin Roetteler"},{"type":"text","value":"Mathias Soeken","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[199565],"msr_event":[],"msr_group":[],"msr_project":[428250],"publication":[],"video":[],"download":[],"msr_publication_type":"inproceedings","related_content":{"projects":[{"ID":428250,"post_title":"Post-quantum Cryptography","post_name":"post-quantum-cryptography","post_type":"msr-project","post_date":"2018-04-30 12:33:53","post_modified":"2024-09-30 21:14:11","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/post-quantum-cryptography\/","post_excerpt":"Cryptography in the era of quantum computers The private communication of individuals and organizations is protected online by cryptography. Cryptography protects our information as it travels over and is stored on the internet\u2014whether making a purchase from an online store, uploading data to the cloud, or accessing work email remotely. Our research and engineering work has focused on protecting private information and communication from the possible threat of future quantum computers. 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