{"id":152745,"date":"2004-07-01T00:00:00","date_gmt":"2004-07-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/a-new-related-message-attack-on-rsa\/"},"modified":"2018-10-16T20:13:37","modified_gmt":"2018-10-17T03:13:37","slug":"a-new-related-message-attack-on-rsa","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/a-new-related-message-attack-on-rsa\/","title":{"rendered":"A New Related Message Attack on RSA"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Coppersmith, Franklin, Patarin, and Reiter show that given two RSA cryptograms x<sup>e<\/sup> mod N and (ax+b)<sup>e<\/sup> mod N for known constants a,b in Z<sub>N<\/sub>, one can compute x in O(elog\u00b2e) Z<sub>N<\/sub> -operations with some positive error probability. We show that given e cryptograms ci =(ai x+bi)<sup>emodN<\/sup>, i=0,1,&#8230;e-1, for any known constants ai ,bi in Z<sub>N<\/sub> one can deterministically compute x in O(e) Z<sub>N<\/sub> -operations that depend on the cryptograms, after a pre-processing that depends only on the constants. The complexity of the pre-processing is O(elog\u00b2e) Z<sub>N<\/sub> -operations, and can be amortized over many instances. We also consider a special case where the overall cost of the attack is O(e) Z<sub>N<\/sub> -operations. Our tools are borrowed from numerical-analysis and adapted to handle formal polynomials over finite-rings. To the best of our knowledge their use in cryptanalysis is novel.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Coppersmith, Franklin, Patarin, and Reiter show that given two RSA cryptograms xe mod N and (ax+b)e mod N for known constants a,b in ZN, one can compute x in O(elog\u00b2e) ZN -operations with some positive error probability. We show that given e cryptograms ci =(ai x+bi)emodN, i=0,1,&#8230;e-1, for any known constants ai ,bi in ZN [&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":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2004-65","msr_organization":"","msr_pages_string":"7","msr_page_range_start":"7","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"Microsoft 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