{"post_stream":{"posts":[{"id":13073,"name":"Igor Gulamov","username":"snjax","avatar_template":"/user_avatar/ethresear.ch/snjax/{size}/2945_2.png","created_at":"2018-12-01T07:23:50.601Z","cooked":"\u003ch1\u003e\u003ca name=\"p-13073-short-sntark-exclusion-proofs-for-plasma-1\" class=\"anchor\" href=\"#p-13073-short-sntark-exclusion-proofs-for-plasma-1\"\u003e\u003c/a\u003eShort S[NT]ARK exclusion proofs for Plasma\u003c/h1\u003e\n\u003cp\u003eReading \u003ca href=\"https://ethresear.ch/t/a-sketch-for-a-stark-based-accumulator/4382\"\u003eSTARK-based accumulators\u003c/a\u003e proposed by \u003ca class=\"mention\" href=\"/u/vbuterin\"\u003e@vbuterin\u003c/a\u003e, I summarize that current prooving schema looks so:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eArbitrary slices\u003c/li\u003e\n\u003cli\u003eAligned slices\u003c/li\u003e\n\u003cli\u003eSTARK-based accumulator\u003c/li\u003e\n\u003cli\u003eSTARK-based exclusion proof for history reduction\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eLet’s consider some reduction of these stages up to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eArbitrary slices\u003c/li\u003e\n\u003cli\u003eS[NT]ARK-based exclusion proof for history reduction\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eWe do not need to do anything with aligned slices (\u003cspan class=\"math\"\u003e\\div \\log N\u003c/span\u003e complexity) and do not need to do anything with inclusion/exclusion proof for the aligned slices (also \u003cspan class=\"math\"\u003e\\div \\log N\u003c/span\u003e complexity).\u003c/p\u003e\n\u003cp\u003eAs it turned out, zk-SNARKs are useful to make batch exclusion Merkle proof for Plasma Cashflow.\u003c/p\u003e\n\u003cp\u003eThe state of Plasma Cashflow is looking something like this:\u003c/p\u003e\n\u003ca href=\"https://raw.githubusercontent.com/snjax/drawio/master/plasma%20cashflow%20state.svg?sanitize=true\" target=\"_blank\" rel=\"noopener nofollow ugc\"\u003e\n\u003cp\u003e\u003cimg src=\"https://ethresear.ch/uploads/default/original/2X/e/e2c583d55559acdbbd07830ae7eb3bf32b817745.svg\" alt=\"plasma cashflow state\" data-base62-sha1=\"wm755YVwKwDuSxNPwINSCiBxQAR\" width=\"690\" height=\"254\"\u003e\u003c/p\u003e\n\u003c/a\u003e\u003cp\u003e\u003ca href=\"https://raw.githubusercontent.com/snjax/drawio/master/plasma%20cashflow%20state.svg?sanitize=true\" target=\"_blank\" rel=\"noopener nofollow ugc\"\u003eView with better resolution.\u003cbr\u003e\n\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe space of plasma at the picture equals to \u003ccode\u003e[0, 1000000)\u003c/code\u003e. So, each block contains this interval inside the root node. There are transactions included in the block and voids inside the leaves.\u003c/p\u003e\n\u003cp\u003eIt is enough to prove the existence of a NULL node at the current slice for any chunk of blocks to prove exclusion of the slice.\u003c/p\u003e\n\u003cp\u003eWe do not need to prove tx validity, signatures or something like this. Here is the example circuit written on pseudocode, it is a very simple construction with Merkle prooves inside only:\u003c/p\u003e\n\u003cpre\u003e\u003ccode class=\"lang-auto\"\u003egadget ExclusionProof(slice, blockSum:public, nullIntervals[N], sumMerkleProof[N] :private)\n  for i:= 1..N:\n    root[i]:=SumMerkleProof(nullInterval[i], NULL, sumMerkleProof[i])\n    nullInterval[i].x1 \u0026lt;= slice.x1 \n    slice.x2 \u0026lt;= nullInterval[i].x2\n  blockSum == hashsum(root)\n\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eBelow I represent computations for 10k tps plasma with one block per 5 minutes publishing:\u003c/p\u003e\n\u003cp\u003eThere are about \u003ccode\u003e3000000\u003c/code\u003e tx per block. The Merkle tree depth is \u003ccode\u003e22\u003c/code\u003e. If we use 160bit cryptography, there are about 30k constraints per block. If we use 10M constraint SNARK, it can prove 300 blocks with 300 bytes proof size.\u003c/p\u003e\n\u003cp\u003eRaw Merkle proof of 300 blocks weights 300 kilobytes. So, we got x1000 disk space reduction to store the history.\u003c/p\u003e\n\u003cp\u003eThere are 100k 5minute blocks per year, so the history of the coin without reduction weights about 100 Mb and history of the coin with reduction weights 100 kb.\u003c/p\u003e\n\u003cp\u003eThe S[NT]ARKs do not need to be checked onchain. That means that STARKs and recursive SNARKs may be used. One thing we need to implement onchain: plasma state and challenges using S[NT]ARK-friendly hash functions. We can do it not expensive through truebit or SNARKs+truebit.\u003c/p\u003e\n\u003ch1\u003e\u003ca name=\"p-13073-related-links-2\" class=\"anchor\" href=\"#p-13073-related-links-2\"\u003e\u003c/a\u003eRelated links\u003c/h1\u003e\n\u003cp\u003e@barryWhiteHat \u003ca href=\"https://ethresear.ch/t/roll-up-roll-back-snark-side-chain-17000-tps/3675\"\u003eRoll_up / roll_back snark side chain ~17000 tps\u003c/a\u003e\u003cbr\u003e\n@vbuterin \u003ca href=\"https://ethresear.ch/t/a-sketch-for-a-stark-based-accumulator/4382\"\u003eA sketch for a STARK-based accumulator\u003c/a\u003e\u003cbr\u003e\n\u003ca href=\"https://www.youtube.com/watch?v=0ApUUoWYt8U\" rel=\"noopener nofollow ugc\"\u003ePlasma call \u003cspan class=\"hashtag-raw\"\u003e#16\u003c/span\u003e\u003c/a\u003e\u003cbr\u003e\n\u003ca href=\"https://hackmd.io/DgzmJIRjSzCYvl4lUjZXNQ?view\" rel=\"noopener nofollow ugc\"\u003ePlasma cashflow spec\u003c/a\u003e\u003cbr\u003e\n@vbuterin, \u003ca href=\"https://ethresear.ch/t/rsa-accumulators-for-plasma-cash-history-reduction/3739\"\u003eRSA Accumulators for Plasma Cash history reduction\u003c/a\u003e\u003cbr\u003e\nAlessandro Chiesa, Lynn Chua, Matthew Weidner \u003ca href=\"https://arxiv.org/pdf/1803.02067.pdf\" rel=\"noopener nofollow ugc\"\u003eOn cycles of pairing-friendly elliptic curves\u003c/a\u003e\u003cbr\u003e\nEli Ben-Sasson, Alessandro Chiesa, Eran Tromer, Madars Virza \u003ca href=\"https://eprint.iacr.org/2014/595.pdf\" rel=\"noopener nofollow ugc\"\u003eScalable Zero Knowledge via Cycles of Elliptic Curves\u003c/a\u003e\u003c/p\u003e","post_number":1,"post_type":1,"posts_count":5,"updated_at":"2025-12-29T15:41:54.236Z","reply_count":0,"reply_to_post_number":null,"quote_count":0,"incoming_link_count":161,"reads":100,"readers_count":99,"score":825.0,"yours":false,"topic_id":4438,"topic_slug":"short-s-nt-ark-exclusion-proofs-for-plasma","display_username":"Igor Gulamov","primary_group_name":null,"flair_name":null,"flair_url":null,"flair_bg_color":null,"flair_color":null,"flair_group_id":null,"badges_granted":[],"version":8,"can_edit":false,"can_delete":false,"can_recover":false,"can_see_hidden_post":false,"can_wiki":false,"link_counts":[{"url":"https://ethresear.ch/t/a-sketch-for-a-stark-based-accumulator/4382","internal":true,"reflection":false,"title":"A sketch for a STARK-based accumulator","clicks":61},{"url":"https://raw.githubusercontent.com/snjax/drawio/master/plasma%20cashflow%20state.svg?sanitize=true","internal":false,"reflection":false,"clicks":13},{"url":"https://ethresear.ch/t/rsa-accumulators-for-plasma-cash-history-reduction/3739","internal":true,"reflection":false,"title":"RSA Accumulators for Plasma Cash history reduction","clicks":3},{"url":"https://arxiv.org/pdf/1803.02067.pdf","internal":false,"reflection":false,"clicks":2},{"url":"https://hackmd.io/DgzmJIRjSzCYvl4lUjZXNQ?view","internal":false,"reflection":false,"title":"🔮💸🌊 Plasma Cashflow - HackMD","clicks":1},{"url":"https://eprint.iacr.org/2014/595.pdf","internal":false,"reflection":false,"clicks":1},{"url":"https://ethresear.ch/t/roll-up-roll-back-snark-side-chain-17000-tps/3675","internal":true,"reflection":false,"title":"Roll_up / roll_back snark side chain ~17000 tps","clicks":0},{"url":"https://www.youtube.com/watch?v=0ApUUoWYt8U","internal":false,"reflection":false,"title":"YouTube","clicks":0},{"url":"https://ethresear.ch/t/plasma-prime-design-proposal/4222/9","internal":true,"reflection":true,"title":"Plasma Prime design proposal","clicks":1}],"read":true,"user_title":"","bookmarked":false,"actions_summary":[],"moderator":false,"admin":false,"staff":false,"user_id":2428,"hidden":false,"trust_level":2,"deleted_at":null,"user_deleted":false,"edit_reason":"downloaded local copies of images","can_view_edit_history":true,"wiki":false,"post_url":"/t/short-s-nt-ark-exclusion-proofs-for-plasma/4438/1","can_accept_answer":false,"can_unaccept_answer":false,"accepted_answer":false,"topic_accepted_answer":null,"can_vote":false},{"id":13092,"name":"Vbuterin","username":"vbuterin","avatar_template":"/user_avatar/ethresear.ch/vbuterin/{size}/91_2.png","created_at":"2018-12-01T17:26:35.253Z","cooked":"\u003cp\u003eThe reason to use hash-chain accumulators rather than Merkle proofs is that with a Merkle tree you need to check 22 hashes, so there’s a ~22x increase in the amount of proving you need to do per value. Also, the implementation complexity goes up considerably (eg. it risks getting to the point where you can’t hand-roll a STARK, you need to use an HLL, which adds overhead and bug surface area…)\u003c/p\u003e\n\u003cp\u003eMy understanding is that at present with reasonably optimized SNARKs and using MIMC you can prove a branch in ~2 seconds. With STARKs you can get that down to ~0.2, and replacing the Merkle branch with a hash chain ~0.01, and probably even less with improved hash functions (Jarvis?) so you get to the point where it actually is viable for a Plasma server.\u003c/p\u003e","post_number":2,"post_type":1,"posts_count":5,"updated_at":"2018-12-01T17:26:35.253Z","reply_count":1,"reply_to_post_number":null,"quote_count":0,"incoming_link_count":54,"reads":79,"readers_count":78,"score":290.8,"yours":false,"topic_id":4438,"topic_slug":"short-s-nt-ark-exclusion-proofs-for-plasma","display_username":"Vbuterin","primary_group_name":null,"flair_name":null,"flair_url":null,"flair_bg_color":null,"flair_color":null,"flair_group_id":null,"badges_granted":[],"version":1,"can_edit":false,"can_delete":false,"can_recover":false,"can_see_hidden_post":false,"can_wiki":false,"read":true,"user_title":null,"bookmarked":false,"actions_summary":[],"moderator":false,"admin":true,"staff":true,"user_id":102,"hidden":false,"trust_level":4,"deleted_at":null,"user_deleted":false,"edit_reason":null,"can_view_edit_history":true,"wiki":false,"post_url":"/t/short-s-nt-ark-exclusion-proofs-for-plasma/4438/2","can_accept_answer":false,"can_unaccept_answer":false,"accepted_answer":false,"topic_accepted_answer":null},{"id":13095,"name":"Igor Gulamov","username":"snjax","avatar_template":"/user_avatar/ethresear.ch/snjax/{size}/2945_2.png","created_at":"2018-12-01T20:46:14.805Z","cooked":"\u003cp\u003eThank you for your explanation! I consider the accumulators on zk-STARKS as promising construction.\u003c/p\u003e\n\u003cp\u003eI see one issue in the accumulators: we need to push \u003cspan class=\"math\"\u003e\\sim (\\log N)^2\u003c/span\u003e objects inside it to prove the inclusion of one arbitrary slice.\u003c/p\u003e\n\u003cp\u003eNow I am working at reducing the size of log-proofs. At the current stage, we can reduce inclusion proof size up to 5 times and increase exclusion proof size up to 2 times (or vise versa).\u003c/p\u003e\n\u003cp\u003eWe are going to check the accumulators with STARK-friendly hash functions in our plasma\u003c/p\u003e","post_number":3,"post_type":1,"posts_count":5,"updated_at":"2018-12-01T21:45:53.482Z","reply_count":1,"reply_to_post_number":2,"quote_count":0,"incoming_link_count":2,"reads":79,"readers_count":78,"score":30.8,"yours":false,"topic_id":4438,"topic_slug":"short-s-nt-ark-exclusion-proofs-for-plasma","display_username":"Igor Gulamov","primary_group_name":null,"flair_name":null,"flair_url":null,"flair_bg_color":null,"flair_color":null,"flair_group_id":null,"badges_granted":[],"version":2,"can_edit":false,"can_delete":false,"can_recover":false,"can_see_hidden_post":false,"can_wiki":false,"read":true,"user_title":"","reply_to_user":{"id":102,"username":"vbuterin","name":"Vbuterin","avatar_template":"/user_avatar/ethresear.ch/vbuterin/{size}/91_2.png"},"bookmarked":false,"actions_summary":[],"moderator":false,"admin":false,"staff":false,"user_id":2428,"hidden":false,"trust_level":2,"deleted_at":null,"user_deleted":false,"edit_reason":null,"can_view_edit_history":true,"wiki":false,"post_url":"/t/short-s-nt-ark-exclusion-proofs-for-plasma/4438/3","can_accept_answer":false,"can_unaccept_answer":false,"accepted_answer":false,"topic_accepted_answer":null},{"id":13109,"name":"Vbuterin","username":"vbuterin","avatar_template":"/user_avatar/ethresear.ch/vbuterin/{size}/91_2.png","created_at":"2018-12-02T12:29:52.846Z","cooked":"\u003cp\u003eSee \u003ca class=\"mention\" href=\"/u/karl\"\u003e@karl\u003c/a\u003e’s work in making plasma prime work by assigning primes to slices instead of fixed coin IDs. You can get the complexity down to O(1) by doing it that way.\u003c/p\u003e","post_number":4,"post_type":1,"posts_count":5,"updated_at":"2018-12-02T12:29:52.846Z","reply_count":1,"reply_to_post_number":3,"quote_count":0,"incoming_link_count":5,"reads":70,"readers_count":69,"score":44.0,"yours":false,"topic_id":4438,"topic_slug":"short-s-nt-ark-exclusion-proofs-for-plasma","display_username":"Vbuterin","primary_group_name":null,"flair_name":null,"flair_url":null,"flair_bg_color":null,"flair_color":null,"flair_group_id":null,"badges_granted":[],"version":1,"can_edit":false,"can_delete":false,"can_recover":false,"can_see_hidden_post":false,"can_wiki":false,"read":true,"user_title":null,"reply_to_user":{"id":2428,"username":"snjax","name":"Igor Gulamov","avatar_template":"/user_avatar/ethresear.ch/snjax/{size}/2945_2.png"},"bookmarked":false,"actions_summary":[],"moderator":false,"admin":true,"staff":true,"user_id":102,"hidden":false,"trust_level":4,"deleted_at":null,"user_deleted":false,"edit_reason":null,"can_view_edit_history":true,"wiki":false,"post_url":"/t/short-s-nt-ark-exclusion-proofs-for-plasma/4438/4","can_accept_answer":false,"can_unaccept_answer":false,"accepted_answer":false,"topic_accepted_answer":null},{"id":13173,"name":"Igor Gulamov","username":"snjax","avatar_template":"/user_avatar/ethresear.ch/snjax/{size}/2945_2.png","created_at":"2018-12-04T23:49:28.452Z","cooked":"\u003cp\u003eI have overviewed last \u003ca class=\"mention\" href=\"/u/karl\"\u003e@karl\u003c/a\u003e’s works on Plasma, but have not seen the solution. Perhaps, \u003ca class=\"mention\" href=\"/u/karl\"\u003e@karl\u003c/a\u003e is planning to commit the solution \u003ca href=\"https://github.com/karlfloersch/research\" rel=\"noopener nofollow ugc\"\u003ehere\u003c/a\u003e, but it is still tagged “top secret”.\u003c/p\u003e\n\u003cp\u003eAlso, we have discussed the O(1) solution in our workshop after the devcon4 and found history split issue: the plasma operator can forge queer chains and hide them from honest users. Users can easily prove the consistency and validity of the real chain offchain. But plasma operator can try to exit from a queer chain and it is not easy to expand history back onchain and find the defect.\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"https://ethresear.ch/uploads/default/original/2X/9/96f1a3ee2f8325f729cc26846c6685687da66c39.svg\" alt=\"chains\" data-base62-sha1=\"lxjkCEyLU4cMcbTvdh2tYAFAazD\" width=\"591\" height=\"176\"\u003e\u003c/p\u003e\n\u003cp\u003eI did not look at the problem intently, but now I see, that the complexity is about \u003cspan class=\"math\"\u003e\\log_2 M\u003c/span\u003e, where M is a number of transactions with the coin in the real chain from the deposit or checkpoint.\u003c/p\u003e\n\u003cp\u003eThere is not so a huge number of challenges for coin used once per day (lesser than 9 steps for a year) and more complex problem for coin used every 5 minutes (17 steps for a year). The steps can be batched (for example, \u003cspan class=\"math\"\u003e\\times 8\u003c/span\u003e for one request size and \u003cspan class=\"math\"\u003e\\div 3\u003c/span\u003e for a number of requests).\u003c/p\u003e\n\u003cp\u003eProbably I missed the best solution. 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