An interactive-PCP (say, for the membership x \u2208 L) is a proof that can be verified by reading only one of its bits, with the help of a very short interactive-proof. We show that for membership in some languages L, there are interactive-PCPs that are significantly shorter than the known (non-interactive) PCPs for these languages. Our main result is that the satisfiability of a constant depth Boolean formula \u03a6(z1, . . . , zk) of size n (over the gates \u2227, \u2228, \u2295, \u00ac) can be proved by an interactive-PCP of size poly(k), followed by a short interactive proof of communication complexity polylog(n). That is, we obtain interactivePCPs of size polynomial in the size of the witness. This compares to the known (non-interactive) PCPs that are of size polynomial in the size of the instance. By reductions, this result extends to many other central NP languages (e.g., SAT, k-clique, Vertex-Cover, etc.). More generally, we show that the satisfiability of Vn i=1[\u03a6i(z1, . . . , zk) = 0], where each \u03a6i(z1, . . . , zk) is an arithmetic formula of size n (say, over GF[2]) that computes a polynomial of degree d, can be proved by an interactive-PCP of size poly(k, d), followed by a short interactive proof of communication complexity poly(d, log n). We give many cryptographic applications and motivations for our results. In particular, we show the following: 1. The satisfiability of a constant depth formula \u03a6(z1, . . . , zk) of size n (as above) has an interactive zero-knowledge proof of communication complexity poly(k) (rather than poly(n))1 . As before, this result extends to many other central NP languages. This zero-knowledge proof has some additional desired properties that will be elaborated on in the body of the paper. 2. Alice can commit to a Boolean formula \u039b of size m, by a message of size poly(m), and later on prove to Bob any N statements of the form \u039b(x1) = z1, . . . ,\u039b(xN ) = zN by a zero-knowledge proof of communication complexity poly(m, log N). Moreover, if \u039b is a constant depth Boolean formula then the zero-knowledge proof has communication complexity poly(log m, log N). We further motivate this application in the body of the paper.<\/p>\n","protected":false},"excerpt":{"rendered":"
An interactive-PCP (say, for the membership x \u2208 L) is a proof that can be verified by reading only one of its bits, with the help of a very short interactive-proof. We show that for membership in some languages L, there are interactive-PCPs that are significantly shorter than the known (non-interactive) PCPs for these languages. […]<\/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":"Automata, Languages and Programming; Lecture Notes in Computer Science, 2008","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"Automata, Languages and Programming; Lecture Notes in Computer Science, 2008","msr_doi":"10.1007\/978-3-540-70583-3_44","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":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2008-01-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13562,13546,13558],"msr-publication-type":[193716],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-344771","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computer-vision","msr-research-area-computational-sciences-mathematics","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Automata, Languages and Programming; Lecture Notes in Computer Science, 2008","msr_affiliation":"","msr_published_date":"2008-01-01","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":"344774","msr_publicationurl":"","msr_doi":"10.1007\/978-3-540-70583-3_44","msr_publication_uploader":[{"type":"file","title":"2008-interactive_pcp","viewUrl":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-content\/uploads\/2017\/01\/2008-Interactive_PCP.pdf","id":344774,"label_id":0},{"type":"doi","title":"10.1007\/978-3-540-70583-3_44","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"yael","user_id":34951,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=yael"},{"type":"text","value":"Ran Raz","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"inproceedings","related_content":[],"_links":{"self":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/344771","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":2,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/344771\/revisions"}],"predecessor-version":[{"id":539970,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/344771\/revisions\/539970"}],"wp:attachment":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media?parent=344771"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=344771"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=344771"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=344771"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=344771"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=344771"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=344771"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=344771"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=344771"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=344771"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=344771"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=344771"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=344771"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}