--- title: A quick look at zero-knowledge proofs | daily.dev description: An explanation of zero-knowledge proofs framed purely through graph theory rather than cryptocurrency, walking through the classic 3-coloring interactive proof... canonical: https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: A quick look at zero-knowledge proofs | daily.dev og:description: An explanation of zero-knowledge proofs framed purely through graph theory rather than cryptocurrency, walking through the classic 3-coloring interactive proof... og:url: https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf og:image: https://api.daily.dev/og/posts/U8PVjyHpF.png og:image:alt: A quick look at zero-knowledge proofs og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # A quick look at zero-knowledge proofs **[Programming Digest](https://daily.dev/sources/programmingdigest)** · 12 min read · 0 upvotes · 0 comments ## Summary An explanation of zero-knowledge proofs framed purely through graph theory rather than cryptocurrency, walking through the classic 3-coloring interactive proof protocol from Goldreich, Micali, and Widgerson's paper. The post builds a Python implementation step by step: permuting colors, hashing them with nonces to simulate 'locked boxes', and having a verifier probabilistically check edges over many rounds. It also discusses probability of cheating detection, links to a live prover/verifier demo, and extends the idea to Sudoku and general NP-complete problems via polynomial reductions. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### How does a zero-knowledge proof for graph 3-coloring actually work without revealing the coloring? The prover permutes a valid 3-coloring, hashes each node's color with a random nonce to create 'locked boxes,' and sends these hashes to the verifier. The verifier picks a random edge and asks the prover to reveal the colors and nonces for its two endpoints. The verifier checks the hashes match and the colors differ, gaining tiny confidence per round; repeating this many times (based on Goldreich, Micali, and Widgerson's original protocol) makes cheating statistically implausible. _Developers exploring cryptographic protocols can find implementation walkthroughs like this one on daily.dev._ ### How many rounds of the 3-coloring zero-knowledge proof are needed to make cheating unlikely? For a graph with 1000 edges, about 4600 rounds bring the probability of successful cheating down to 1%, and 10,000 rounds bring it down to roughly 0.0045%. The bound comes from the formula (1 - m^-1)^rounds, where m is the number of edges, as stated in the original interactive proof paper by Goldreich, Micali, and Widgerson. _Anyone reasoning about protocol soundness trade-offs can track deep dives like this on daily.dev._ ### Why do you need to add a nonce when hashing colors in a zero-knowledge coloring proof? Without a nonce, two nodes sharing the same color would produce identical hashes, leaking the coloring's structure to the verifier even though the actual color names are hidden. Adding random per-node data (a nonce, ideally from a secure source like Python's secrets.token_hex()) before hashing makes each node's hash unique regardless of shared colors, preserving zero knowledge. _Developers implementing cryptographic hiding schemes can browse hands-on breakdowns like this via daily.dev._ ## Similar posts on daily.dev - [Zero-Knowledge Proofs for Verifiable MCP Tool Execution](https://daily.dev/posts/zero-knowledge-proofs-for-verifiable-mcp-tool-execution-c0bqqix43) · Security Boulevard · 0 upvotes · 0 comments - [Graph Coloring You Can See](https://daily.dev/posts/graph-coloring-you-can-see-iuk0aslhm) · Towards Data Science · 0 upvotes · 0 comments - [Zero knowlege proof of compositeness](https://daily.dev/posts/zero-knowlege-proof-of-compositeness-osesjzbo2) · Hacker News · 2 upvotes · 1 comments - [Zero-Knowledge Proofs for Privacy-Preserving Context Validation](https://daily.dev/posts/zero-knowledge-proofs-for-privacy-preserving-context-validation-ow9k48ypx) · Security Boulevard · 0 upvotes · 0 comments --- Tags: [#python](https://daily.dev/tags/python), [#algorithms](https://daily.dev/tags/algorithms), [#cryptography](https://daily.dev/tags/cryptography), [#graph-theory](https://daily.dev/tags/graph-theory), [#zero-knowledge-proofs](https://daily.dev/tags/zero-knowledge-proofs) [View this post on daily.dev](https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"A quick look at zero-knowledge proofs","url":"https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf"},"datePublished":"2026-08-30T18:45:29.330Z","dateModified":"2026-09-14T06:08:23.448Z","description":"An explanation of zero-knowledge proofs framed purely through graph theory rather than cryptocurrency, walking through the classic 3-coloring interactive proof...","image":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","thumbnailUrl":"https://media.daily.dev/image/upload/s--ZrL_HSsR--/f_auto/v1722860399/public/Placeholder%2006","isAccessibleForFree":true,"articleSection":"Programming Digest","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Programming Digest","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/4637f2287b44420aa9a44f432116043d","url":"https://daily.dev/sources/programmingdigest"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"python,algorithms,cryptography,graph-theory,zero-knowledge-proofs","timeRequired":"PT12M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Programming Digest","item":"https://daily.dev/sources/programmingdigest"},{"@type":"ListItem","position":3,"name":"A quick look at zero-knowledge proofs"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/a-quick-look-at-zero-knowledge-proofs-u8pvjyhpf#faq","mainEntity":[{"@type":"Question","name":"How does a zero-knowledge proof for graph 3-coloring actually work without revealing the coloring?","acceptedAnswer":{"@type":"Answer","text":"The prover permutes a valid 3-coloring, hashes each node's color with a random nonce to create 'locked boxes,' and sends these hashes to the verifier. 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