{"id":3027,"date":"2021-10-11T19:10:16","date_gmt":"2021-10-11T19:10:16","guid":{"rendered":"https:\/\/starkware.co\/?page_id=3027"},"modified":"2024-02-08T12:30:05","modified_gmt":"2024-02-08T12:30:05","slug":"stark-math-a-very-short-primer","status":"publish","type":"page","link":"https:\/\/starkware.co\/stark-math-a-very-short-primer\/","title":{"rendered":"STARK Math \u2013 A Very Short Primer"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"3027\" class=\"elementor elementor-3027\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1d2e497 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1d2e497\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ce2a86\" data-id=\"1ce2a86\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a385401 elementor-widget elementor-widget-spacer\" data-id=\"a385401\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.19.0 - 05-02-2024 *\/\n.elementor-column .elementor-spacer-inner{height:var(--spacer-size)}.e-con{--container-widget-width:100%}.e-con-inner>.elementor-widget-spacer,.e-con>.elementor-widget-spacer{width:var(--container-widget-width,var(--spacer-size));--align-self:var(--container-widget-align-self,initial);--flex-shrink:0}.e-con-inner>.elementor-widget-spacer>.elementor-widget-container,.e-con>.elementor-widget-spacer>.elementor-widget-container{height:100%;width:100%}.e-con-inner>.elementor-widget-spacer>.elementor-widget-container>.elementor-spacer,.e-con>.elementor-widget-spacer>.elementor-widget-container>.elementor-spacer{height:100%}.e-con-inner>.elementor-widget-spacer>.elementor-widget-container>.elementor-spacer>.elementor-spacer-inner,.e-con>.elementor-widget-spacer>.elementor-widget-container>.elementor-spacer>.elementor-spacer-inner{height:var(--container-widget-height,var(--spacer-size))}.e-con-inner>.elementor-widget-spacer.elementor-widget-empty,.e-con>.elementor-widget-spacer.elementor-widget-empty{position:relative;min-height:22px;min-width:22px}.e-con-inner>.elementor-widget-spacer.elementor-widget-empty .elementor-widget-empty-icon,.e-con>.elementor-widget-spacer.elementor-widget-empty .elementor-widget-empty-icon{position:absolute;top:0;bottom:0;left:0;right:0;margin:auto;padding:0;width:22px;height:22px}<\/style>\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b439436 elementor-widget elementor-widget-heading\" data-id=\"b439436\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.19.0 - 05-02-2024 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h1 class=\"elementor-heading-title elementor-size-default\">STARK Math \u2013 A Very Short Primer<\/h1>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6dee4b2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6dee4b2\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1764fcf\" data-id=\"1764fcf\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-073bf42 elementor-widget elementor-widget-spacer\" data-id=\"073bf42\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a11c1db elementor-widget elementor-widget-text-editor\" data-id=\"a11c1db\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.19.0 - 05-02-2024 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<h2><b>Inclusive Accountability on Blockchains<\/b><\/h2><p><span style=\"font-weight: 400;\">The integrity of blockchains is premised on inclusive accountability: all citizens of the world are invited to verify the <a href=\"https:\/\/starkware.co\/resource\/extreme-integrity-in-decentralized-world\/\" target=\"_blank\" rel=\"noopener\">computational integrity<\/a> of the blockchain. None is excluded. This creates two problems:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Scalability: Scaling throughput to ever-growing demand would necessarily exclude some participants from verifying integrity, due to their limited computational resources.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Privacy: Without cryptographic blinding of information, all blockchain transactions are publicly exposed, which is unacceptable to businesses and individuals alike.<\/span><\/li><\/ul><h2><b>Theoretical Computational Complexity to the Rescue<\/b><\/h2><p><span style=\"font-weight: 400;\">Starting in the 1980\u2019s and through the 1990\u2019s a brilliant line of theoretical works on <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Interactive_proof_system\"><span style=\"font-weight: 400;\">Interactive Proofs<\/span><\/a><span style=\"font-weight: 400;\"> (IP), <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Zero-knowledge_proof\"><span style=\"font-weight: 400;\">Zero Knowledge Proofs<\/span><\/a><span style=\"font-weight: 400;\"> (ZKP) and <\/span><a href=\"https:\/\/en.wikipedia.org\/wiki\/Probabilistically_checkable_proof\"><span style=\"font-weight: 400;\">Probabilistically Checkable Proofs<\/span><\/a><span style=\"font-weight: 400;\"> (PCP) have shown how to completely resolve both scalability and privacy issues. But, as the famous saying goes, \u201cIn Theory, there is no difference between theory and practice. Practically speaking, there is\u201d. Early constructions were, practically, unusable.<\/span><\/p><h2><b>STARKs as Culmination of the \u201cTheory to Practice\u201d Journey<\/b><\/h2><p><span style=\"font-weight: 400;\">Several paths emerged in the new millennium to reach concrete efficiency, mostly to solve the privacy problem. The most prevalent ones are based on number theoretic cryptographic primitives like bilinear pairings over elliptic curves (SNARKs and Bulletproofs are two notable examples).<\/span><\/p><p><span style=\"font-weight: 400;\">The path that <a href=\"https:\/\/starkware.co\/stark\/\" target=\"_blank\" rel=\"noopener\">STARKs <\/a>pursue is older (or \u201cclassic\u201d) \u2013 relying on battle-tested hash functions as their sole cryptographic building block. To bridge the chasm to practicality, STARK relies on recent breakthroughs by StarkWare\u2019s team members \u2013 <\/span><a href=\"https:\/\/eccc.weizmann.ac.il\/report\/2004\/060\/\"><span style=\"font-weight: 400;\">quasi-linear PCPs<\/span><\/a><span style=\"font-weight: 400;\">, <\/span><a href=\"https:\/\/eprint.iacr.org\/2016\/116\"><span style=\"font-weight: 400;\">interactive oracle proofs<\/span><\/a><span style=\"font-weight: 400;\"> (IOPs) and fast algebraic local coding protocols like <\/span><a href=\"https:\/\/eccc.weizmann.ac.il\/report\/2017\/134\/\"><span style=\"font-weight: 400;\">FRI<\/span><\/a><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">As a result, at scale the STARK prover and verifier are fastest in class (outperforming all number-theoretic constructions) and reach vastly greater scale than competitors while relying on fewer and safer cryptographic assumptions. In particular, they are post-quantum secure and transparent, meaning that they do not use dangerous \u201ccryptographic toxic waste\u201d in their setup process and have no associated trapdoors that might compromise their security.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8b96607 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8b96607\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9a9c036\" data-id=\"9a9c036\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2bccdec elementor-widget elementor-widget-spacer\" data-id=\"2bccdec\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>STARK Math \u2013 A Very Short Primer Inclusive Accountability on Blockchains The integrity of blockchains is premised on inclusive accountability: all citizens of the world are invited to verify the computational integrity of the blockchain. None is excluded. This creates two problems: Scalability: Scaling throughput to ever-growing demand would necessarily exclude some participants from verifying<\/p>\n","protected":false},"author":4,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"footnotes":""},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>STARK Math \u2013 A Very Short Primer | StarkWare<\/title>\n<meta name=\"description\" content=\"The path that STARKs pursue is older or classic relying on battle-tested hash functions as their sole cryptographic building block. 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