var Component=(()=>{var E=Object.create;var s=Object.defineProperty;var p=Object.getOwnPropertyDescriptor;var F=Object.getOwnPropertyNames;var y=Object.getPrototypeOf,f=Object.prototype.hasOwnProperty;var D=(n,l)=>()=>(l||n((l={exports:{}}).exports,l),l.exports),C=(n,l)=>{for(var r in l)s(n,r,{get:l[r],enumerable:!0})},a=(n,l,r,t)=>{if(l&&typeof l=="object"||typeof l=="function")for(let o of F(l))!f.call(n,o)&&o!==r&&s(n,o,{get:()=>l[o],enumerable:!(t=p(l,o))||t.enumerable});return n};var u=(n,l,r)=>(r=n!=null?E(y(n)):{},a(l||!n||!n.__esModule?s(r,"default",{value:n,enumerable:!0}):r,n)),A=n=>a(s({},"__esModule",{value:!0}),n);var i=D((b,c)=>{c.exports=_jsx_runtime});var m={};C(m,{default:()=>h,frontmatter:()=>g});var e=u(i()),g={title:"A Hackers Guide to Layer 2: Zero Merkle Trees from Scratch",date:"2023-06-15",lastmod:"2023-06-15",tags:["blockchain","blockchain-development","zero-knowledge-proofs","merkle-tree","javascript"],draft:!1,author_id:"carter",excerpt:"This is the second weekly installment of A Hackers Guide to Layer 2: Building a zkVM from Scratch.",og_image:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/post-banner.png",belongs:["learns"]};function d(n){let l={a:"a",blockquote:"blockquote",code:"code",em:"em",figure:"figure",h3:"h3",h4:"h4",li:"li",ol:"ol",p:"p",pre:"pre",span:"span",strong:"strong",ul:"ul",...n.components};return(0,e.jsxs)(e.Fragment,{children:[(0,e.jsxs)(l.blockquote,{children:[` `,(0,e.jsx)(l.p,{children:(0,e.jsxs)(l.em,{children:["Follow ",(0,e.jsx)(l.a,{href:"https://twitter.com/cmpeq",children:(0,e.jsx)(l.strong,{children:"@cmpeq"})})," on twitter to stay up to date with future installments."]})}),` `]}),` `,(0,e.jsxs)(l.blockquote,{children:[` `,(0,e.jsx)(l.p,{children:(0,e.jsxs)(l.em,{children:["Tutorial code snippets: ",(0,e.jsx)(l.a,{href:"https://github.com/cf/zero-merkle-tree-tutorial",children:(0,e.jsx)(l.strong,{children:"github.com/cf/zero-merkle-tree-tutorial"})})]})}),` `]}),` `,(0,e.jsxs)(l.h4,{id:"in-this-tutorial-we-will",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#in-this-tutorial-we-will",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"In this tutorial, we will:"]}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsxs)(l.li,{children:["Learn how to build an efficient ",(0,e.jsx)(l.strong,{children:"zero merkle tree"})," implementation capable of processing changes in trees with ",(0,e.jsx)(l.strong,{children:"quadrillions of leaves"})," , and implement it in TypeScript from scratch."]}),` `,(0,e.jsxs)(l.li,{children:["Learn how to build an ",(0,e.jsx)(l.strong,{children:"append-only merkle tree"})," which only requires ",(0,e.jsx)(l.strong,{children:"O(log(n) ) storage"})," , and implement it in TypeScript from scratch."]}),` `,(0,e.jsxs)(l.li,{children:["Learn about ",(0,e.jsx)(l.strong,{children:"Spiderman proofs"})," and how to efficiently generate/verify ",(0,e.jsx)(l.strong,{children:"batch updates"})," to a merkle tree with a ",(0,e.jsx)(l.strong,{children:"single, succinct proof"})," ."]}),` `]}),` `,(0,e.jsxs)(l.h4,{id:"note",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#note",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Note"]}),` `,(0,e.jsxs)(l.p,{children:["Before reading this tutorial, it is ",(0,e.jsx)(l.strong,{children:(0,e.jsx)(l.em,{children:"highly recommended"})})," to first work through the ",(0,e.jsx)(l.a,{href:"a-hackers-guide-to-layer-2-merkle-trees-from-scratch-f682fc31bced",children:(0,e.jsx)(l.strong,{children:"first installment"})})," where we cover merkle tree fundamentals, and we will be frequently referencing content from what we learned in the previous guide."]}),` `,(0,e.jsxs)(l.blockquote,{children:[` `,(0,e.jsx)(l.p,{children:(0,e.jsx)(l.em,{children:"What I cannot create, I do not understand."})}),` `]}),` `,(0,e.jsxs)(l.blockquote,{children:[` `,(0,e.jsx)(l.p,{children:(0,e.jsx)(l.em,{children:"Richard Feynman"})}),` `]}),` `,(0,e.jsxs)(l.h3,{id:"introduction",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#introduction",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Introduction"]}),` `,(0,e.jsx)(l.p,{children:"In the previous installment of A Hackers Guide to Layer 2, we covered the basics of merkle trees and built a simple merkle tree database implementation."}),` `,(0,e.jsxs)(l.p,{children:["While what we built was simple and usable for small trees, our previous merkle tree implementation will be ",(0,e.jsx)(l.strong,{children:"very slow"})," when working with large trees:"]}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsxs)(l.li,{children:["Every time we called getRoot (to compute the tree\u2019s merkle root), we had to ",(0,e.jsx)(l.strong,{children:"rehash the whole merkle tree"})," ( ",(0,e.jsx)(l.strong,{children:"O(2^height)"})," time complexity)"]}),` `,(0,e.jsxs)(l.li,{children:["Creating a new merkle tree instance required us to pass the tree\u2019s height and ",(0,e.jsx)(l.strong,{children:"an array of leaves"})," with size 2^height to the constructor (memory is also ",(0,e.jsx)(l.strong,{children:"O(2^n)"})," )"]}),` `]}),` `,(0,e.jsxs)(l.p,{children:["Unfortunately, our ",(0,e.jsx)(l.strong,{children:"zkVM"})," will need to work with merkle trees that are both ",(0,e.jsx)(l.strong,{children:"large"})," (height > 32) and ",(0,e.jsx)(l.strong,{children:"sparse"})," (most of the leaves are zero) ."]}),` `,(0,e.jsxs)(l.p,{children:["We will explain why this is the case in future installments, but it is intuitively related to the fact that our zkVM will use merkle trees to ",(0,e.jsx)(l.strong,{children:"verifiably store and organize persistent data"})," (think of merkle trees as a verifiable hard drive + file system for our zkVM) ."]}),` `,(0,e.jsxs)(l.p,{children:["That being said, we are not out of luck, as we can take advantage of merkle tree\u2019s ",(0,e.jsx)(l.strong,{children:"self-similarity"})," to build a much more efficient database for our use case."]}),` `,(0,e.jsxs)(l.h3,{id:"the-zero-hashes",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#the-zero-hashes",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"The Zero Hashes"]}),` `,(0,e.jsx)(l.p,{children:"To begin tackling the problem of building a merkle tree better suited for our large/sparse use case, we can gain some insight by first examining an empty merkle tree and looking for patterns that we can take advantage of:"}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/empty-merkle-tree.png",alt:"An Empty Merkle Tree (TreeHeight = 3)","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:"An Empty Merkle Tree (TreeHeight = 3)"}),` `,(0,e.jsx)(l.p,{children:"If we recall that a node\u2019s value is equal to the hash of its children, we can draw rectangles around the nodes which have the same value:"}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/same-value-nodes.png",alt:"Since the nodes of the bottom level all have the same value, the nodes on the level above will also have the same value, since the nodes above have a value depedant only on the hash of the nodes below","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:"Since the nodes of the bottom level all have the same value, the nodes on the level above will also have the same value, since the nodes above have a value depedant only on the hash of the nodes below"}),` `,(0,e.jsxs)(l.p,{children:["Aha! When a merkle tree is empty, it looks like the ",(0,e.jsx)(l.strong,{children:"all the nodes on each level have the same value"})," . The value of the nodes on each level of an empty merkle tree are known as the ",(0,e.jsx)(l.strong,{children:"Zero Hashes"})," and are often denoted as ",(0,e.jsx)(l.strong,{children:(0,e.jsx)(l.em,{children:"Z\u2099"})})," where ",(0,e.jsx)(l.strong,{children:"n"})," = ",(0,e.jsx)(l.strong,{children:"TreeHeight"})," - ",(0,e.jsx)(l.strong,{children:"Level"})," ."]}),` `,(0,e.jsxs)(l.p,{children:["Since we know that the bottom nodes have a value ",(0,e.jsx)(l.em,{children:"Z"})," \u2080 = 0, lets write an expression for ",(0,e.jsx)(l.strong,{children:(0,e.jsx)(l.em,{children:"Z\u2099."})})]}),` `,(0,e.jsx)(l.p,{children:"Recalling the formula for computing the value of node in a merkle tree that we learned in the previous installment:"}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/node-value-formula.png",alt:"The formula for computing the value of a merkle tree node","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:"The formula for computing the value of a merkle tree node"}),` `,(0,e.jsxs)(l.p,{children:["We can just plug-in ",(0,e.jsx)(l.strong,{children:"zero"})," for the leaves and use the ",(0,e.jsx)(l.strong,{children:"level-wise"})," ",(0,e.jsx)(l.strong,{children:"self-similarity"})," that we observed in the tree to write an expression for ",(0,e.jsx)(l.strong,{children:(0,e.jsx)(l.em,{children:"Z\u2099"})})," :"]}),` `,(0,e.jsxs)(l.p,{children:["Hence we can say that, for an empty merkle tree with height = ",(0,e.jsx)(l.strong,{children:"TreeHeight"})," :"]}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/zero-hash-formula.png",alt:"Zero hash formula","data-align":"center"}),` `,(0,e.jsxs)(l.p,{children:["Great, now let\u2019s demonstrate our knowledge of the zero hashes by writing a TypeScript function that computes the array of zero hashes for a given ",(0,e.jsx)(l.code,{children:"treeHeight"})," :"]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"ts","data-theme":"Aura Theme",__rawString__:`import shajs from "sha.js"; type MerkleNodeValue = string; function hash(leftNode: MerkleNodeValue, rightNode: MerkleNodeValue) { return shajs("sha256") .update(leftNode + rightNode, "hex") .digest("hex"); } function computeZeroHashes(height: number): MerkleNodeValue[] { let currentZeroHash = "0000000000000000000000000000000000000000000000000000000000000000"; const zeroHashes: MerkleNodeValue[] = [currentZeroHash]; for (let i = 1; i <= height; i++) { currentZeroHash = hash(currentZeroHash, currentZeroHash); zeroHashes.push(currentZeroHash) } return zeroHashes; } function example1(){ console.log("[example1] the first 32 zero hashes are: "+JSON.stringify(computeZeroHashes(32), undefined, 2)); } example1(); `,children:(0,e.jsxs)(l.code,{"data-language":"ts","data-theme":"Aura 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`,(0,e.jsx)(l.span,{"data-line":"",children:" "}),` `,(0,e.jsx)(l.span,{"data-line":"",children:" "}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"function"}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" example1"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:"()"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"{"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" console."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"log"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:'"[example1] the first 32 zero hashes are: 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`,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"example1"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"();"})]})]})})}),` `,(0,e.jsx)(l.p,{children:"Running this code, we get the output:"}),` `,(0,e.jsx)(l.p,{children:"Great! Now that we have written a function to compute the zero hashes, we can write a function to obtain the value of any node in an empty merkle tree using our zero hashes:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`function getNodeValue(treeHeight: number, level: number): MerkleNodeValue{ const zeroHashes = computeZeroHashes(treeHeight); return zeroHashes[treeHeight-level]; } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"function"}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" getNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:"(treeHeight"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:" number"}),(0,e.jsx)(l.span,{style:{color:"#F694FF",fontStyle:"italic"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:" level"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:" number"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:")"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"{"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" zeroHashes "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" computeZeroHashes"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(treeHeight);"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" return"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" zeroHashes[treeHeight"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"-"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"level];"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})})]})})}),` `,(0,e.jsxs)(l.h3,{id:"implementing-a-zmt-database",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#implementing-a-zmt-database",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Implementing a ZMT Database"]}),` `,(0,e.jsx)(l.p,{children:"We now know how to compute the value any node in an empty merkle tree, let\u2019s use our knowledge to write our efficient merkle tree database."}),` `,(0,e.jsxs)(l.p,{children:["In the previous installment of our series, we already proved that, when you update a leaf in a merkle tree, ",(0,e.jsx)(l.strong,{children:"only the nodes alone the leaf\u2019s merkle path will change"})," :"]}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/zero-hashes-example.png",alt:"Zero hashes example","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:"Using this principle, we make a merkle tree database which only stores all the nodes in the tree that aren\u2019t zero hashes, and follows the procedure:"}),` `,(0,e.jsx)(l.p,{children:"To find the value of node N(level, index):"}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsxs)(l.li,{children:["Check if it exists in the database. ",(0,e.jsx)(l.strong,{children:"If it does, return the value"})," ."]}),` `,(0,e.jsx)(l.li,{children:(0,e.jsx)(l.strong,{children:"If it does not exist, return the corresponding zero hash for the node\u2019s level"})}),` `,(0,e.jsxs)(l.li,{children:["If you want to update a node, first set the leaf, compute the leaf\u2019s merkle path, and then ",(0,e.jsx)(l.strong,{children:"save all the new node values in the merkle path to the database"})]}),` `]}),` `,(0,e.jsx)(l.p,{children:"In this way, we will create a database which only needs to store the non-zero hash nodes, perfect for our large, sparse merkle trees!"}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s start by writing a simple data store which follows our rule for getting nodes from the datastore (return if it exists, else fallback to a zero hash):"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura 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Now we can write a ZeroMerkleTree class which uses the NodeStore:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`class ZeroMerkleTree { height: number; nodeStore: NodeStore; constructor(height: number){ this.height = height; this.nodeStore = new NodeStore(height); } } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"class"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:" ZeroMerkleTree"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" {"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" 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`,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})})]})})}),` `,(0,e.jsx)(l.p,{children:"Following what we learned in the previous installment, we know that we can compute the merkle path of a node by dividing the index by 2 each time we go up a level in the merkle path:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"javascript","data-theme":"Aura Theme",__rawString__:`function getMerklePathOfNode(level, index){ const merklePath = []; for(let currentLevel=level;currentLevel>=0;currentLevel--){ merklePath.push({ level: currentLevel, index: index, }); index = Math.floor(index/2); } return merklePath; } `,children:(0,e.jsxs)(l.code,{"data-language":"javascript","data-theme":"Aura 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level:"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" currentLevel"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" index:"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" });"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" Math."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"floor"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(index"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"/"}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"2"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:");"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" return"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" merklePath;"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})})]})})}),` `,(0,e.jsxs)(l.p,{children:["Applying this knowledge, lets write a ",(0,e.jsx)(l.code,{children:"setLeaf"})," function which sets a leaf in the tree and updates its merkle path according to the same logic as our getMerklePathOfNode function and the even-odd hashing rule for siblings:"]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`class ZeroMerkleTree { height: number; nodeStore: NodeStore; constructor(height: number){ this.height = height; this.nodeStore = new NodeStore(height); } setLeaf(index: number, value: MerkleNodeValue){ // start traversing the leaf's merkle path at the leaf node let currentIndex = index; let currentValue = value; // don't set the root (level = 0) in the loop, as it has no sibling for(let level = this.height; level > 0; level--){ // set the current node in the tree this.nodeStore.set(level, currentIndex, currentValue); if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); } // set current index to the index of the parent node currentIndex = Math.floor(currentIndex/2); } // set the root node (level = 0, index = 0) to current value this.nodeStore.set(0, 0, currentValue); } } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"class"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:" ZeroMerkleTree"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" {"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" height"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:" 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this"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"nodeStore"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"set"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"0"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:" 0"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" currentValue);"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" getRoot"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:"()"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" {"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" return"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" this"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"nodeStore"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"get"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"0"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:" 0"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:");"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:" "}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"function"}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" example2"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:"()"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"{"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" leavesToSet "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" ["})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000001"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 1"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000003"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 3"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000003"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 3"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000007"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 7"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000004"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 4"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000002"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 2"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000000"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 0"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000006"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 6"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" ];"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" tree "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" new"}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" ZeroMerkleTree"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"3"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:");"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" leavesToSet."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"forEach"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:"(leaf"}),(0,e.jsx)(l.span,{style:{color:"#F694FF",fontStyle:"italic"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE",fontStyle:"italic"},children:" index)"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"=>"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"{"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" tree."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"setLeaf"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(index"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" leaf);"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" });"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" console."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"log"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:'"[example2] the root is: "'}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"+"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"tree."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"getRoot"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"());"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"example2"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"();"})]})]})})}),` `,(0,e.jsx)(l.p,{children:"Running this function yields the result:"}),` `,(0,e.jsx)(l.pre,{children:(0,e.jsx)(l.code,{children:`[example2] the root is: 7e286a6721a66675ea033a4dcdec5abbdc7d3c81580e2d6ded7433ed113b7737 `})}),` `,(0,e.jsxs)(l.p,{children:["We can see if we got everything right by comparing it with the merkle root we calculated from a tree with the same leaves in our ",(0,e.jsx)(l.a,{href:"https://medium.com/@carterfeldman/a-hackers-guide-to-layer-2-merkle-trees-from-scratch-f682fc31bced#c21e",children:"previous installment"})," :"]}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/root-matches.png",alt:"The root matches!","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:(0,e.jsx)(l.strong,{children:"The root matches!"})}),` `,(0,e.jsxs)(l.p,{children:["A perfect match \u{1F389}, now we can make it even more useful by returning a ",(0,e.jsx)(l.strong,{children:"delta merkle proof"})," whenever we call ",(0,e.jsx)(l.code,{children:"setLeaf"})," ."]}),` `,(0,e.jsx)(l.p,{children:"Recalling our previous definition of delta merkle proofs we will need the following data:"}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsx)(l.li,{children:"index of the leaf (index)"}),` `,(0,e.jsx)(l.li,{children:"the siblings of the leaf\u2019s merkle path (siblings)"}),` `,(0,e.jsx)(l.li,{children:"the root before the leaf was changed (oldRoot)"}),` `,(0,e.jsx)(l.li,{children:"the value of the leaf before it was changed (oldValue)"}),` `,(0,e.jsx)(l.li,{children:"the root after the leaf is changed (newRoot)"}),` `,(0,e.jsx)(l.li,{children:"the new value of the leaf (newValue)"}),` `]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`type MerkleNodeValue = string; interface IDeltaMerkleProof { index: number; siblings: MerkleNodeValue[]; oldRoot: MerkleNodeValue; oldValue: MerkleNodeValue; newRoot: MerkleNodeValue; newValue: MerkleNodeValue; } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"type"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" ="}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" string"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"interface"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" IDeltaMerkleProof"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" {"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" index"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:" number"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" siblings"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#F694FF",fontStyle:"italic"},children:"[]"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" oldRoot"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" oldValue"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" newRoot"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:" newValue"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})})]})})}),` `,(0,e.jsx)(l.p,{children:"To do this, we can modify the start of our setLeaf function to record the value of the oldRoot and oldValue before we make our update + add a siblings array to keep track of the siblings of the leaf\u2019s merkle path:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`... setLeaf(index: number, value: MerkleNodeValue): IDeltaMerkleProof{ // get the old root and old value for the delta merkle proof const oldRoot = this.nodeStore.get(0, 0); const oldValue = this.nodeStore.get(this.height, index); // siblings array for delta merkle proof const siblings: MerkleNodeValue[] = []; ... `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"..."})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:" setLeaf"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(index: number"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" value: MerkleNodeValue): IDeltaMerkleProof{"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // get the old root and old value for the delta merkle proof"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" oldRoot "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" this"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"nodeStore"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"get"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"0"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:" 0"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:");"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" oldValue "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" this"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"nodeStore"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"get"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"this"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"height,"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index);"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // siblings array for delta merkle proof"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" siblings"}),(0,e.jsx)(l.span,{style:{color:"#A277FF",fontStyle:"italic"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF",fontStyle:"italic"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#F694FF",fontStyle:"italic"},children:"[] "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" [];"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"..."})})]})})}),` `,(0,e.jsx)(l.p,{children:"Then, we can modify the body of the loop where we hash the node on the merkle path with its sibling to also push the siblings value to our siblings array:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`... setLeaf(index: number, value: MerkleNodeValue): IDeltaMerkleProof{ ... for(let level = this.height; level > 0; level--){ // set the current node in the tree this.nodeStore.set(level, currentIndex, currentValue); if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); // add the right sibling to the siblings array siblings.push(rightSibling); // <---- HERE }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); // add the left sibling to the siblings array siblings.push(leftSibling); // <---- HERE } // set current index to the index of the parent node currentIndex = Math.floor(currentIndex/2); } ... `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura 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hash"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(currentValue"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" rightSibling);"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // add the right sibling to the siblings array"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" siblings."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"push"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(rightSibling); "}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:"// <---- HERE"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"else"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"{"})]}),` 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siblings."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"push"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(leftSibling); "}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:"// <---- HERE"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // set current index to the index of the parent node"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" currentIndex "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" 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the delta merkle proof const oldRoot = this.nodeStore.get(0, 0); const oldValue = this.nodeStore.get(this.height, index); // siblings array for delta merkle proof const siblings: MerkleNodeValue[] = []; // start traversing the leaf's merkle path at the leaf node let currentIndex = index; let currentValue = value; // don't set the root (level = 0) in the loop, as it has no sibling for(let level = this.height; level > 0; level--){ // set the current node in the tree this.nodeStore.set(level, currentIndex, currentValue); if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); // add the right sibling to the siblings array siblings.push(rightSibling); }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); // add the left sibling to the siblings array siblings.push(leftSibling); } // set current index to the index of the parent node currentIndex = Math.floor(currentIndex/2); } // set the root node (level = 0, index = 0) to current value this.nodeStore.set(0, 0, currentValue); return { // <--- HERE index, siblings, oldRoot, oldValue, newValue: value, newRoot: currentValue, }; } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"setLeaf"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(index: number"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" value: MerkleNodeValue): IDeltaMerkleProof{"})]}),` 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number; value: MerkleNodeValue; } interface IDeltaMerkleProof { index: number; siblings: MerkleNodeValue[]; oldRoot: MerkleNodeValue; oldValue: MerkleNodeValue; newRoot: MerkleNodeValue; newValue: MerkleNodeValue; } function hash(leftNode: MerkleNodeValue, rightNode: MerkleNodeValue) { return shajs("sha256") .update(leftNode + rightNode, "hex") .digest("hex"); } function computeZeroHashes(height: number): MerkleNodeValue[] { let currentZeroHash = "0000000000000000000000000000000000000000000000000000000000000000"; const zeroHashes: MerkleNodeValue[] = [currentZeroHash]; for (let i = 1; i <= height; i++) { currentZeroHash = hash(currentZeroHash, currentZeroHash); zeroHashes.push(currentZeroHash) } return zeroHashes; } class NodeStore { nodes: {[id: string]: MerkleNodeValue}; height: number; zeroHashes: MerkleNodeValue[]; constructor(height: number){ this.nodes = {}; this.height = height; this.zeroHashes = computeZeroHashes(height); } contains(level: number, index: number): boolean { // check if the node exists in the data store return Object.hasOwnProperty.call(this.nodes, level+"_"+index); } set(level: number, index: number, value: MerkleNodeValue){ // set the value of the node in the data store this.nodes[level+"_"+index] = value; } get(level: number, index: number): MerkleNodeValue { if(this.contains(level, index)){ // if the node is in the datastore, return it return this.nodes[level+"_"+index]; }else{ // if the node is NOT in the data store, return the correct zero hash for the node's level return this.zeroHashes[this.height-level]; } } } class ZeroMerkleTree { height: number; nodeStore: NodeStore; constructor(height: number){ this.height = height; this.nodeStore = new NodeStore(height); } setLeaf(index: number, value: MerkleNodeValue): IDeltaMerkleProof{ // get the old root and old value for the delta merkle proof const oldRoot = this.nodeStore.get(0, 0); const oldValue = this.nodeStore.get(this.height, index); // siblings array for delta merkle proof const siblings: MerkleNodeValue[] = []; // start traversing the leaf's merkle path at the leaf node let currentIndex = index; let currentValue = value; // don't set the root (level = 0) in the loop, as it has no sibling for(let level = this.height; level > 0; level--){ // set the current node in the tree this.nodeStore.set(level, currentIndex, currentValue); if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); // add the right sibling to the siblings array siblings.push(rightSibling); }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); // add the left sibling to the siblings array siblings.push(leftSibling); } // set current index to 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delta merkle proof for index 6 is valid [example3] delta merkle proof for index 7 is valid [example3] the delta merkle proofs are: [ { "index": 0, "siblings": [ "0000000000000000000000000000000000000000000000000000000000000000", "f5a5fd42d16a20302798ef6ed309979b43003d2320d9f0e8ea9831a92759fb4b", "db56114e00fdd4c1f85c892bf35ac9a89289aaecb1ebd0a96cde606a748b5d71" ], "oldRoot": "c78009fdf07fc56a11f122370658a353aaa542ed63e44c4bc15ff4cd105ab33c", "oldValue": "0000000000000000000000000000000000000000000000000000000000000000", "newValue": "0000000000000000000000000000000000000000000000000000000000000001", "newRoot": "f06e424318b067ae608de0ef0035e9f48a2658cc59e7f94f9f94600b2a36eac6" }, ... /* omitted for brevity, check codepen for full result */ ] `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example3] delta merkle proof for 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All of the delta merkle proofs are valid!"}),` `,(0,e.jsxs)(l.h3,{id:"chain-delta-merkle-proofs",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#chain-delta-merkle-proofs",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Chain Delta Merkle Proofs"]}),` `,(0,e.jsxs)(l.p,{children:["Along with verifying delta merkle proofs, another important check that our zkVM will have to perform is known as a ",(0,e.jsx)(l.strong,{children:"chain delta merkle proof"})," verification."]}),` `,(0,e.jsxs)(l.p,{children:[(0,e.jsx)(l.strong,{children:"A chain delta merkle proof verification ensures that a series of delta merkle proofs are sequential"})," , meaning that you have provided ",(0,e.jsx)(l.strong,{children:"all"})," the proofs for the ",(0,e.jsx)(l.strong,{children:"state transition"})," from the first delta merkle proof\u2019s oldRoot to the last delta merkle proof in the series\u2019s newRoot."]}),` `,(0,e.jsx)(l.p,{children:"We can perform this verification by asserting that, for each delta merkle proof in the series, the oldRoot of the current proof equals the newRoot of the previous proof:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`... for(let i=0;i0 && deltaProof.oldRoot !== deltaMerkleProofs[i-1].newRoot){ // <------ [HERE] // the previous proof's new root should be the same as this proof's old root console.error( "[example4] ERROR: delta merkle proof for index "+deltaProof.index + " has a different old root than the previous delta merkle proof's new root" ); throw new Error("delta merkle proof root sequence mismatch"); }else{ console.log("[example4] delta merkle proof for index "+deltaProof.index+" is valid"); } } ... `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura 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`,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"..."})})]})})}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s update our code to include a chain delta merkle proof verification and check to make sure our code is working properly:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"ts","data-theme":"Aura Theme",__rawString__:`import shajs from "sha.js"; type MerkleNodeValue = string; interface IMerkleProof { root: MerkleNodeValue; siblings: MerkleNodeValue[]; index: number; value: MerkleNodeValue; } interface IDeltaMerkleProof { index: number; siblings: MerkleNodeValue[]; oldRoot: MerkleNodeValue; oldValue: MerkleNodeValue; newRoot: MerkleNodeValue; newValue: MerkleNodeValue; } function hash(leftNode: MerkleNodeValue, rightNode: MerkleNodeValue) { return shajs("sha256") .update(leftNode + rightNode, "hex") .digest("hex"); } function computeZeroHashes(height: number): MerkleNodeValue[] { let currentZeroHash = "0000000000000000000000000000000000000000000000000000000000000000"; const zeroHashes: MerkleNodeValue[] = [currentZeroHash]; for (let i = 1; i <= height; i++) { currentZeroHash = hash(currentZeroHash, currentZeroHash); zeroHashes.push(currentZeroHash) } return zeroHashes; } class NodeStore { nodes: {[id: string]: MerkleNodeValue}; height: number; zeroHashes: MerkleNodeValue[]; constructor(height: number){ this.nodes = {}; this.height = height; this.zeroHashes = computeZeroHashes(height); } contains(level: number, index: number): boolean { // check if the node exists in the data store return Object.hasOwnProperty.call(this.nodes, level+"_"+index); } set(level: number, index: number, value: MerkleNodeValue){ // set the value of the node in the data store this.nodes[level+"_"+index] = value; } get(level: number, index: number): MerkleNodeValue { if(this.contains(level, index)){ // if the node is in the datastore, return it return this.nodes[level+"_"+index]; }else{ // if the node is NOT in the data store, return the correct zero hash for the node's level return this.zeroHashes[this.height-level]; } } } class ZeroMerkleTree { height: number; nodeStore: NodeStore; constructor(height: number){ this.height = height; this.nodeStore = new NodeStore(height); } setLeaf(index: number, value: MerkleNodeValue): IDeltaMerkleProof{ // get the old root and old value for the delta merkle proof const oldRoot = this.nodeStore.get(0, 0); const oldValue = this.nodeStore.get(this.height, index); // siblings array for delta merkle proof const siblings: MerkleNodeValue[] = []; // start traversing the leaf's merkle path at the leaf node let currentIndex = index; let currentValue = value; // don't set the root (level = 0) in the loop, as it has no sibling for(let level = this.height; level > 0; level--){ // set the current node in the tree this.nodeStore.set(level, currentIndex, currentValue); if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); // add the right sibling to the siblings array siblings.push(rightSibling); }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); // add the left sibling to the siblings array siblings.push(leftSibling); } // set current index to the index of the parent node currentIndex = Math.floor(currentIndex/2); } // set the root node (level = 0, index = 0) to current value this.nodeStore.set(0, 0, currentValue); return { index, siblings, oldRoot, oldValue, newValue: value, newRoot: currentValue, }; } getRoot(): MerkleNodeValue { return this.nodeStore.get(0, 0); } } function computeMerkleRootFromProof(siblings: MerkleNodeValue[], index: number, value: MerkleNodeValue): MerkleNodeValue{ // start our merkle node path at the leaf node let merklePathNodeValue = value; let merklePathNodeIndex = index; // we follow the leaf's merkle path up to the root, // computing the merkle path's nodes using the siblings provided as we go alone for(let i=0;itree.setLeaf(index, leaf)); // verify the delta merkle proofs for(let i=0;i0 && deltaProof.oldRoot !== deltaMerkleProofs[i-1].newRoot){ // the previous proof's new root should be the same as this proof's old root console.error( "[example4] ERROR: delta merkle proof for index "+deltaProof.index + " has a different old root than the previous delta merkle proof's new root" ); throw new Error("delta merkle proof root sequence mismatch"); }else{ console.log("[example4] delta merkle proof for index "+deltaProof.index+" is valid"); } } console.log("[example4] the delta merkle proofs are:\\n"+JSON.stringify(deltaMerkleProofs, null, 2)); } example4(); `,children:(0,e.jsxs)(l.code,{"data-language":"ts","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"import"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" shajs "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"from"}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "sha.js"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:";"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:" "}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"type"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:" MerkleNodeValue"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" ="}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" 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`,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" const"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" leavesToSet "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"="}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" ["})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000001"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 1"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "0000000000000000000000000000000000000000000000000000000000000003"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // 3"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' 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`,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // the previous proof's new root should be the same as this proof's old root"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" console."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"error"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"("})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "[example4] ERROR: delta merkle proof for index "'}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"+"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"deltaProof."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"index"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" +"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:` " has a different old root than the previous delta merkle proof's new root"`})}),` 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`,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"example4"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"();"})]})]})})}),` `,(0,e.jsx)(l.p,{children:"Running example4, we get the result:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"csharp","data-theme":"Aura Theme",__rawString__:`[example4] delta merkle proof for index 0 is valid [example4] delta merkle proof for index 1 is valid [example4] delta merkle proof for index 2 is valid [example4] delta merkle proof for index 3 is valid [example4] delta merkle proof for index 4 is valid [example4] delta merkle proof for index 5 is valid [example4] delta merkle proof for index 6 is valid [example4] delta merkle proof for index 7 is valid [example4] the delta merkle proofs are: [ ... /* omitted for brevity, check codepen for full result */ ] `,children:(0,e.jsxs)(l.code,{"data-language":"csharp","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"["}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:"example4"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"] delta merkle proof "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"for"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index 0 is valid"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 1 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 2 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 3 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 4 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 5 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 6 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] delta merkle proof for index 7 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example4] the delta merkle proofs are:"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"["})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"... /* omitted for brevity, check codepen for full result */"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"]"})})]})})}),` `,(0,e.jsx)(l.p,{children:"Great! No errors \u{1F389}, we have proves the changes we made are valid (delta merkle proof) and sequential (chain delta merkle proof) ."}),` `,(0,e.jsxs)(l.h3,{id:"finishing-up-our-zmt-database",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#finishing-up-our-zmt-database",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Finishing up our ZMT Database"]}),` `,(0,e.jsxs)(l.p,{children:["The final import function we will want to implement is a ",(0,e.jsx)(l.code,{children:"getLeaf"})," function which returns the value of a leaf and an accompanying merkle proof of its inclusion within the tree. To do this, we just need to grab the value of the leaf, and then walk its merkle path to obtain the siblings and root using the same strategy we used for computing the delta merkle proof in setLeaf:"]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`class ZeroMerkleTree { ... getLeaf(index: number): IMerkleProof { // siblings array for merkle proof const siblings: MerkleNodeValue[] = []; // get value for the merkle proof const value = this.nodeStore.get(this.height, index); // start traversing the leaf's merkle path at the leaf node let currentIndex = index; let currentValue = value; // don't set the root (level = 0) in the loop, as it has no sibling for(let level = this.height; level > 0; level--){ if(currentIndex % 2 === 0){ // if the current index is even, then it has a sibling on the right (same level, index = currentIndex+1) const rightSibling = this.nodeStore.get(level, currentIndex+1); currentValue = hash(currentValue, rightSibling); // add the right sibling to the siblings array siblings.push(rightSibling); }else{ // if the current index is odd, then it has a sibling on the left (same level, index = currentIndex-1) const leftSibling = this.nodeStore.get(level, currentIndex-1); currentValue = hash(leftSibling, currentValue); // add the left sibling to the siblings array siblings.push(leftSibling); } // set current index to the index of the parent node currentIndex = Math.floor(currentIndex/2); } // current value is the root const root = currentValue; return { root, siblings, index, value, }; } ... `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"class"}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:" 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`,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" }"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"..."})})]})})}),` `,(0,e.jsxs)(l.p,{children:["We can now put everything together to and use the verifyMerkleProof function we wrote in the ",(0,e.jsx)(l.a,{href:"https://medium.com/@carterfeldman/a-hackers-guide-to-layer-2-merkle-trees-from-scratch-f682fc31bced#ce75",children:"previous installment"})," to verify the merkle proofs generated by getLeaf ( ",(0,e.jsx)(l.a,{href:"https://gist.github.com/cf/5ae6a374a936a7034a1ac6c850732b84?file=example5.ts#file-example5-ts",children:"full code gist"})," ):"]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`/* see https://gist.github.com/cf/1b04459735c0b86dda56714205566c97 */ ... function example5(){ const leavesToSet = [ "0000000000000000000000000000000000000000000000000000000000000001", // 1 "0000000000000000000000000000000000000000000000000000000000000003", // 3 "0000000000000000000000000000000000000000000000000000000000000003", // 3 "0000000000000000000000000000000000000000000000000000000000000007", // 7 "0000000000000000000000000000000000000000000000000000000000000004", // 4 "0000000000000000000000000000000000000000000000000000000000000002", // 2 "0000000000000000000000000000000000000000000000000000000000000000", // 0 "0000000000000000000000000000000000000000000000000000000000000006", // 6 ]; const tree = new ZeroMerkleTree(3); const deltaMerkleProofs = leavesToSet.map((leaf, index)=>tree.setLeaf(index, leaf)); // verify the delta merkle proofs for(let i=0;i0 && deltaProof.oldRoot !== deltaMerkleProofs[i-1].newRoot){ // the previous proof's new root should be the same as this proof's old root console.error( "[example5] ERROR: delta merkle proof for index "+deltaProof.index + " has a different old root than the previous delta merkle proof's new root" ); throw new Error("delta merkle proof root sequence mismatch"); }else{ console.log("[example5] delta merkle proof for index "+deltaProof.index+" is valid"); } } // don't print the delta merkle proofs to avoid clutter //console.log("[example5] the delta merkle proofs are:\\n"+JSON.stringify(deltaMerkleProofs, null, 2)); // verify each leaf's merkle proof for(let i=0;i"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"tree."}),(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"setLeaf"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(index"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" leaf));"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" // verify the delta merkle proofs"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" 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`,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"example5"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"();"})]})]})})}),` `,(0,e.jsx)(l.p,{children:"Running the code we get the result:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"csharp","data-theme":"Aura Theme",__rawString__:`[example5] delta merkle proof for index 0 is valid [example5] delta merkle proof for index 1 is valid [example5] delta merkle proof for index 2 is valid [example5] delta merkle proof for index 3 is valid [example5] delta merkle proof for index 4 is valid [example5] delta merkle proof for index 5 is valid [example5] delta merkle proof for index 6 is valid [example5] delta merkle proof for index 7 is valid [example5] merkle proof for index 0 is valid merkle proof for index 0: { "root": "7e286a6721a66675ea033a4dcdec5abbdc7d3c81580e2d6ded7433ed113b7737", "siblings": [ "0000000000000000000000000000000000000000000000000000000000000003", "6b0e4bcd4368ba74e6a99ee69334c2593bcae1170d77048854d228664218c56b", "81b1e323f0e91a785dfd155817e09949a7d66fe8fdc4f31f39530845e88ab63c" ], "index": 0, "value": "0000000000000000000000000000000000000000000000000000000000000001" } [example5] merkle proof for index 1 is valid ... /* excluded for brevity, check the codepen above for the full result */ `,children:(0,e.jsxs)(l.code,{"data-language":"csharp","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"["}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:"example5"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"] delta merkle proof "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"for"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index 0 is valid"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 1 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 2 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 3 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 4 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 5 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 6 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] delta merkle proof for index 7 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"[example5] merkle proof for index 0 is valid"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"merkle proof for index 0: {"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "root": "7e286a6721a66675ea033a4dcdec5abbdc7d3c81580e2d6ded7433ed113b7737",'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "siblings": ['})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "0000000000000000000000000000000000000000000000000000000000000003",'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "6b0e4bcd4368ba74e6a99ee69334c2593bcae1170d77048854d228664218c56b",'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "81b1e323f0e91a785dfd155817e09949a7d66fe8fdc4f31f39530845e88ab63c"'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" ],"})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "index": 0,'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:' "value": "0000000000000000000000000000000000000000000000000000000000000001"'})}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"["}),(0,e.jsx)(l.span,{style:{color:"#82E2FF"},children:"example5"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"] merkle proof "}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:"for"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" index 1 is valid"})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"... /* excluded for brevity, check the codepen above for the full result */"})})]})})}),` `,(0,e.jsx)(l.p,{children:"Fantastic!"}),` `,(0,e.jsxs)(l.p,{children:[(0,e.jsx)(l.strong,{children:"We have now written a merkle tree database which we can use with our zkVM"})," \u{1F389}\u{1F389}\u{1F389}"]}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/spiderman-proof-animation.gif",alt:"Spiderman proof animation","data-align":"center"}),` `,(0,e.jsxs)(l.p,{children:["To show off our new super-powered merkle tree, lets create a merkle tree of height 50, which has 1,125,899,906,842,624 leaves, and set values for the 1337th and 999,999,999,999th node ( ",(0,e.jsx)(l.a,{href:"https://gist.github.com/cf/5ae6a374a936a7034a1ac6c850732b84?file=example6.ts#file-example6-ts",children:"link to code"})," )"]}),` `,(0,e.jsxs)(l.p,{children:["At 64 bytes of storage per leaf, our previous implementation would need ",(0,e.jsx)(l.strong,{children:"64 petabytes of RAM"})," just to store the tree (not even modern super computers are capable of this), yet with our new ZeroMerkleTree implementation we can do this in the blink of an eye on laptop and only consume 6400 bytes:"]}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`// for the whole code check the gist at https://gist.github.com/cf/9df9b18cd5877fb4fc5c8b31a3acf77d ... function example6(){ const tree = new ZeroMerkleTree(50); const deltaA = tree.setLeaf(999999999999,"0000000000000000000000000000000000000000000000000000000000000008"); const deltaB = tree.setLeaf(1337,"0000000000000000000000000000000000000000000000000000000000000007"); const proofA = tree.getLeaf(999999999999); const proofB = tree.getLeaf(1337); console.log("verifyDeltaMerkleProof(deltaA): "+verifyDeltaMerkleProof(deltaA)); console.log("verifyDeltaMerkleProof(deltaB): "+verifyDeltaMerkleProof(deltaB)); console.log("deltaA.newRoot === deltaB.oldRoot: "+(deltaA.newRoot === deltaB.oldRoot)); console.log("verifyMerkleProof(proofA): "+verifyMerkleProof(proofA)); console.log("verifyMerkleProof(proofB): "+verifyMerkleProof(proofB)); console.log("proofA: "+JSON.stringify(proofA, null, 2)); console.log("proofB: "+JSON.stringify(proofB, null, 2)); } example6(); `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:"// for the whole code check the gist at https://gist.github.com/cf/9df9b18cd5877fb4fc5c8b31a3acf77d"})}),` 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`,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"example6"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"();"})]})]})})}),` `,(0,e.jsx)(l.p,{children:"Which prints the result:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`verifyDeltaMerkleProof(deltaA): true verifyDeltaMerkleProof(deltaB): true deltaA.newRoot === deltaB.oldRoot: true verifyMerkleProof(proofA): true verifyMerkleProof(proofB): true proofA: { "root": "40db8b6edad868d911c8b9aea2692ee80b2e87ac407b8d1a5efe30419e843991", "siblings": ... /* omitted for brevity, check the code pen for the full result */ ], "index": 999999999999, "value": "0000000000000000000000000000000000000000000000000000000000000008" } proofB: { "root": "40db8b6edad868d911c8b9aea2692ee80b2e87ac407b8d1a5efe30419e843991", "siblings": [ ... /* omitted for brevity, check the code pen for the full result */ ], "index": 1337, "value": "0000000000000000000000000000000000000000000000000000000000000007" } `,children:(0,e.jsxs)(l.code,{"data-language":"typescript","data-theme":"Aura Theme",style:{display:"grid"},children:[(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"verifyDeltaMerkleProof"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(deltaA): "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"true"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"verifyDeltaMerkleProof"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"(deltaB): "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"true"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"deltaA."}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:"newRoot"}),(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" 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`,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "index"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"999999999999"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "value"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:'"0000000000000000000000000000000000000000000000000000000000000008"'})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#FFCA85"},children:"proofB"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:":"}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" {"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "root"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:'"40db8b6edad868d911c8b9aea2692ee80b2e87ac407b8d1a5efe30419e843991"'}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "siblings"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": ["})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#A277FF"},children:" ..."}),(0,e.jsx)(l.span,{style:{color:"#6D6D6D"},children:" /* omitted for brevity, check the code pen for the full result */"})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:" ]"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "index"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:"1337"}),(0,e.jsx)(l.span,{style:{color:"#F694FF"},children:","})]}),` `,(0,e.jsxs)(l.span,{"data-line":"",children:[(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:' "value"'}),(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:": "}),(0,e.jsx)(l.span,{style:{color:"#61FFCA"},children:'"0000000000000000000000000000000000000000000000000000000000000007"'})]}),` `,(0,e.jsx)(l.span,{"data-line":"",children:(0,e.jsx)(l.span,{style:{color:"#EDECEE"},children:"}"})})]})})}),` `,(0,e.jsx)(l.p,{children:"We have climbed the impossible mountain, and built a merkle tree ready for our zkVM, all thanks to ZeroHashes and our ZMT \u{1F4AA}"}),` `,(0,e.jsxs)(l.h3,{id:"append-only-merkle-trees",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#append-only-merkle-trees",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"Append Only Merkle Trees"]}),` `,(0,e.jsxs)(l.p,{children:["Our ZMT implementation is perfect for our zkVM use case, but when we are building on legacy blockchains (EVM/zkEVM), storage can be expensive to the point where even our ZMT will prove too costly to make sense for use cases like ",(0,e.jsx)(l.strong,{children:"verifiable event logs"})," ."]}),` `,(0,e.jsx)(l.p,{children:"We will touch on verifiable event logs in more detail in a future installment where we explore trustless cross chain bridges, but for now it is sufficient to know that these write-only merkle trees are critical for trustlessly bridging data & assets from layer 1 to layer 2."}),` `,(0,e.jsxs)(l.h3,{id:"the-verifiable-event-log",children:[(0,e.jsx)(l.a,{"aria-label":"Link to section",className:"subheading-anchor",href:"#the-verifiable-event-log",children:(0,e.jsx)(l.span,{className:"icon icon-link"})}),"The Verifiable Event Log"]}),` `,(0,e.jsx)(l.p,{children:"The basic theory behind a verifiable event log is that we want to allow a smart contract to record events in the leaves of a merkle tree. Whenever a new event occurs, we append a leaf to the merkle tree (set the next non-zero leaf in the merkle tree to the hash of the event) and calculate the updated root. You can then process these events off-chain using zero knowledge proofs or a layer 2, and submit proofs which prove you have sync\u2019d all the events up to the latest event log root, thus providing trustless data transfer from layer 1 to layer 2."}),` `,(0,e.jsx)(l.p,{children:"The most important property of use case is that the smart contract only needs to write leaves to the merkle tree and never has to read back the previous event leaves (we only really care about reading the latest root) ."}),` `,(0,e.jsxs)(l.p,{children:["With our understanding of the ",(0,e.jsx)(l.strong,{children:"zero hashes"})," and some cleverness, it turns out that ",(0,e.jsx)(l.strong,{children:"we will be able to build an append-only merkle tree of height N which only needs to store O(log(N) ) hashes on chain"})," !"]}),` `,(0,e.jsx)(l.p,{children:"As we did before, let\u2019s get started by visualizing the changes in the merkle tree when we append leaves and see if we can find any patterns."}),` `,(0,e.jsx)(l.p,{children:"In the animation below, we visualize the delta merkle proof results from appending a leaf where:"}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsxs)(l.li,{children:["the ",(0,e.jsx)(l.strong,{children:"purple"})," node represents the ",(0,e.jsx)(l.strong,{children:"leaf"})," being appending"]}),` `,(0,e.jsxs)(l.li,{children:["the ",(0,e.jsx)(l.strong,{children:"blue"})," nodes represent the leaf\u2019s ",(0,e.jsx)(l.strong,{children:"merkle path"})]}),` `,(0,e.jsxs)(l.li,{children:["the ",(0,e.jsx)(l.strong,{children:"red"})," nodes represent the ",(0,e.jsx)(l.strong,{children:"siblings"})]}),` `]}),` `,(0,e.jsxs)(l.p,{children:["Recall that our goal is to be able to some how compute the new merkle root while storing the minimum amount of data possible. We also know from our previous installment that the information required to compute a merkle tree\u2019s root is known as a ",(0,e.jsx)(l.strong,{children:"merkle proof"})," and contains the following information"]}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsx)(l.li,{children:"index (index of the leaf)"}),` `,(0,e.jsx)(l.li,{children:"value (value of the leaf)"}),` `,(0,e.jsx)(l.li,{children:"siblings (siblings of the leaf\u2019s merkle path)"}),` `]}),` `,(0,e.jsx)(l.p,{children:"In the case of our append only merkle tree, the index and value will be passed to the function, so the only thing we would be missing for our calculation is a method for calculating the siblings for the next merkle proof using the data stored in our contract. If we used the ZMT implementation from the previous section, we would have to store all the nodes in the tree on-chain, which is undesirable."}),` `,(0,e.jsx)(l.p,{children:"One way we might go about tackling the problem is storing the previous node\u2019s merkle proof, and somehow writing a function which computes the new siblings from the previous nodes merkle proof."}),` `,(0,e.jsx)(l.p,{children:"If we could do that, we could accomplish our goal of O(log(n) ) storage by repeating the procedure below each time we append a leaf to the tree:"}),` `,(0,e.jsxs)(l.ol,{children:[` `,(0,e.jsx)(l.li,{children:"Read the previous merkle proof from the smart contract storage, and increment the index by 1"}),` `,(0,e.jsx)(l.li,{children:"Somehow calculate the new siblings for the leaf we are appending using the zero hashes and the previous merkle proof"}),` `,(0,e.jsx)(l.li,{children:"Compute the new merkle root using the new siblings and the value we are appending, and then overwrite the old merkle proof with the newly appended leaf\u2019s merkle proof to the smart contract storage"}),` `]}),` `,(0,e.jsx)(l.p,{children:"This of course begs the question: is it possible to compute the siblings of the next leaf in our append only merkle tree using only the zero hashes and previous merkle proof?"}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s examine the animation once again, but this time let\u2019s pay attention to how the siblings change as each leaf is appended to the tree. We will still color all the sibling nodes red, but this time we will make the sibling\u2019s text yellow if it was not a sibling for the previous node\u2019s merkle proof:"}),` `,(0,e.jsx)(l.p,{children:"Starting to see the pattern? The key to cracking this puzzle is remembering that the siblings are just the adjacent nodes to the leaf\u2019s merkle path, so when transitioning from leaf N to leaf N+1:"}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsx)(l.li,{children:"On the levels where the merkle paths of leaf N and leaf N+1 are the same, you can reuse the sibling from leaf N\u2019s merkle proof in leaf N+1\u2019s merkle proof"}),` `,(0,e.jsx)(l.li,{children:"On the levels where the merkle paths of leaf N and leaf N+1 are the different, you need to somehow compute what new sibling is."}),` `]}),` `,(0,e.jsx)(l.p,{children:"Great! So for the levels where the merkle path doesn\u2019t change, we know to just reuse the sibling from the previous leaf\u2019s proof, but what about the second case?"}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s examine some more append transition to get some intuition:"}),` `,(0,e.jsx)(l.p,{children:"Notice that, because it is an append-only merkle tree, our leaf index is always increasing, and therefore our path is always moving to the right."}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s write down what we know about the relationship between Leaf[n] and Leaf[n+1]\u2019s merkle paths/siblings on a tree with height H and then try to deduce a formula from the facts we know:"}),` `,(0,e.jsx)("img",{title:"",src:"/blog-posts/a-hackers-guide-to-layer-2-zero-merkle-trees-from-scratch/latex-formula.png",alt:"Sorry for the image embed, medium doesn't support LaTeX so I can't inline this","data-align":"center"}),` `,(0,e.jsx)(l.p,{children:(0,e.jsx)(l.em,{children:"Sorry for the image embed, medium doesn\u2019t support LaTeX so I can\u2019t inline this"})}),` `,(0,e.jsx)(l.p,{children:"Aha! So the only information we will need to generate a merkle proof for Leaf[n+1] is:"}),` `,(0,e.jsxs)(l.ul,{children:[` `,(0,e.jsx)(l.li,{children:"the zero hashes for the tree"}),` `,(0,e.jsx)(l.li,{children:"Leaf[n]\u2019s siblings/value"}),` `,(0,e.jsx)(l.li,{children:"Leaf[n]\u2019s merkle path (we can compute this using the siblings and value! )"}),` `,(0,e.jsx)(l.li,{children:"Leaf[n+1]\u2019s value"}),` `]}),` `,(0,e.jsx)(l.p,{children:"This is very cool, it looks like all we need to store when building our append only merkle tree is latest leaf\u2019s merkle proof!"}),` `,(0,e.jsx)(l.p,{children:"Let\u2019s write our implementation based on the facts we wrote down above:"}),` `,(0,e.jsx)(l.figure,{"data-rehype-pretty-code-figure":"",children:(0,e.jsx)(l.pre,{className:"Aura Theme",tabIndex:"0","data-language":"typescript","data-theme":"Aura Theme",__rawString__:`class AppendOnlyMerkleTree { height: number; lastProof: IMerkleProof; zeroHashes: MerkleNodeValue[]; constructor(height: number){ this.height = height; this.zeroHashes = computeZeroHashes(height); // create a dummy proof of all zero hashes for initialization (before we append any leaves, we know all the siblings will be zero hashes because it is an empty tree) this.lastProof = { root: this.zeroHashes[this.height], siblings: this.zeroHashes.slice(0, this.height), index: -1, value: this.zeroHashes[this.height], }; } appendLeaf(leafValue: string): IDeltaMerkleProof { const oldMerklePath = computeMerklePathFromProof(this.lastProof.siblings, this.lastProof.index, this.lastProof.value); // get the old root and old value for the delta merkle proof const oldRoot = this.lastProof.root; // the old value will always be empty, thats why its an append only tree :P const oldValue = "0000000000000000000000000000000000000000000000000000000000000000"; const prevIndex = this.lastProof.index; // append only tree = new index is always the previous index + 1 const newIndex = prevIndex+1; // keep track of the old siblings so we can use them for our delta merkle proof const oldSiblings =this.lastProof.siblings; const siblings: MerkleNodeValue[] = []; let multiplier = 1; for(let level=0;level