Verkle tree construction | Ethereum Basis Weblog

A Verkle tree is a dedication scheme that works much like a Merkle tree, however has a lot smaller witnesses. It really works by changing the hashes in a Merkle tree with a vector dedication, which makes wider branching components extra environment friendly.

Because of Kevaundray Wedderburn for suggestions on the submit.

Overview

For particulars on how verkle bushes work, see:

The goal of this submit is to clarify the concrete format of the draft verkle tree EIP. It’s aimed toward consumer builders who wish to implement verkle bushes and are searching for an introduction earlier than delving deeper into the EIP.

Verkle bushes introduce quite a few adjustments to the tree construction. Essentially the most vital adjustments are:

  • a change from 20 byte keys to 32 byte keys (to not be confused with 32 byte addresses, which is a separate change);
  • the merge of the account and storage tries; and eventually
  • The introduction of the verkle trie itself, which makes use of vector commitments as an alternative of hashes.

Because the vector dedication scheme for the verkle tree, we use Pedersen commitments. Pedersen commitments are primarily based on elliptic curves. For an introduction to Pedersen commitments and how you can use them as polynomial or vector commitments utilizing Inside Product Argumentss, see right here.

The curve we’re utilizing is Bandersnatch. This curve was chosen as a result of it’s performant, and in addition as a result of it’ll enable environment friendly SNARKs in BLS12_381 to purpose in regards to the verkle tree sooner or later. This may be helpful for rollups in addition to permitting an improve the place all witnesses will be compressed into one SNARK as soon as that turns into sensible, without having an extra dedication replace.

The curve order/scalar area dimension of bandersnatch is p = 13108968793781547619861935127046491459309155893440570251786403306729687672801, which is a 253 bit prime. On account of this, we will solely safely decide to bit strings of at most 252 bits, in any other case the sphere overflows. We selected a branching issue (width) of 256 for the verkle tree, which implies every dedication can decide to as much as 256 values of 252 bits every (or to be exact, integers as much as p – 1). We write this as Commit(v₀, v₁, …, v₂₅₅) to decide to the record v of size 256.

Format of the verkle tree

One of many design objectives with the verkle tree EIP is to make accesses to neighbouring positions (e.g. storage with virtually the identical handle or neighbouring code chunks) low cost to entry. So as to do that, a key consists of a stem of 31 bytes and a suffix of 1 byte for a complete of 32 bytes. The important thing scheme is designed in order that “shut” storage areas are mapped to the identical stem and a unique suffix. For particulars please have a look at the EIP draft.

The verkle tree itself is then composed of two sorts of nodes:

  • Extension nodes, that symbolize 256 values with the identical stem however totally different suffixes
  • Inside nodes, which have as much as 256 kids, which will be both different internal nodes or extension nodes.

The dedication to an extension node is a dedication to a 4 component vector; the remaining positions will likely be 0. It’s:

C₁ and C₂ are two additional commitments that decide to all of the values with stem equal to stem. The explanation we have to commitments is that values have 32 bytes, however we will solely retailer 252 bits per area component. A single dedication would thus not be sufficient to retailer 256 values. So as an alternative C₁ shops the values for suffix 0 to 127, and C₂ shops 128 to 255, the place the values are cut up in two to be able to match into the sphere dimension (we’ll come to that later.)

The extension along with the commitments C₁ and C₂ are known as “extension-and-suffix tree” (EaS for brief).

Determine 1 Illustration of a stroll via a verkle tree for the important thing 0xfe0002abcd..ff04: the trail goes via 3 inside nodes with 256 kids every (254, 0, 2), one extension node representing abcd..ff and the 2 suffix tree commitments, together with the worth for 04, v₄. Word that stem is definitely the primary 31 bytes of the important thing, together with the trail via the inner nodes.

Dedication to the values leaf nodes

Every extension and suffix tree node comprise 256 values. As a result of a worth is 256 bits vast, and we will solely retailer 252 bits safely in a single area component, 4 bits could be misplaced if we merely tried so retailer one worth in a single area component.

To bypass this downside, we selected to partition the group of 256 values into two teams of 128 values every. Every 32-byte worth in a gaggle is cut up into two 16-byte values. So a worth vᵢ∈ 𝔹₃₂ is became v⁽ˡᵒʷᵉʳ⁾ᵢ ∈ 𝔹₁₆ and v⁽ᵘᵖᵖᵉʳ⁾ᵢ∈ 𝔹₁₆ such that v⁽ˡᵒʷᵉʳ⁾ᵢ ++ v⁽ᵘᵖᵖᵉʳ⁾ᵢ= vᵢ.

A “leaf marker” is added to the v⁽ˡᵒʷᵉʳ⁾ᵢ, to distinguish between a leaf that has by no means been accessed and a leaf that has been overwritten with 0s. No worth ever will get deleted from a verkle tree. That is wanted for upcoming state expiry schemes. That marker is ready on the 129th bit, i.e. v⁽ˡᵒʷᵉʳ ᵐᵒᵈⁱᶠⁱᵉᵈ⁾ᵢ = v⁽ˡᵒʷᵉʳ⁾ᵢ + 2¹²⁸ if vᵢ has been accessed earlier than, and v⁽ˡᵒʷᵉʳ ᵐᵒᵈⁱᶠⁱᵉᵈ⁾ᵢ = 0 if vᵢ has by no means been accessed.

The 2 commitments C₁ and C₂ are then outlined as

Dedication of extension nodes

The dedication to an extension node consists of an “extension marker”, which is simply the #1, the 2 subtree commitments C₁ and C₂, and the stem of the important thing resulting in this extension node.

In contrast to extension nodes within the Merkle-Patricia tree, which solely comprise the part of the important thing that bridges the mother or father inside node to the kid inside node, the stem covers the entire key as much as that time. It is because verkle bushes are designed with stateless proofs in thoughts: if a brand new key’s inserted that “splits” the extension in two, the older sibling needn’t be up to date, which permits for a smaller proof.

Dedication of Inner nodes

Inner nodes have the easier calculation methodology for his or her commitments: the node is seen as a vector of 256 values, which might be the (area illustration of the) root dedication of every of their 256 subtrees. The dedication for an empty subtree is 0. If the subtree will not be empty, then the dedication for the inner node is

the place the Cᵢ are the kids of the inner node, and 0 if a toddler is empty.

Insertion into the tree

Determine 2 is an illustration of the method of inserting a brand new worth into the tree, which will get attention-grabbing when the stems collide on a number of preliminary bytes.

Determine 2 Worth v₁₉₂ is inserted at location 0000010000...0000 in a verkle tree containing solely worth v₁₂₇ at location 0000000000...0000. As a result of the stems differ on the third byte, two inside nodes are added till the differing byte. Then one other “extension-and-suffix” tree is inserted, with a full 31-byte stem. The preliminary node is untouched, and C²₀ has the identical worth as C⁰₀ earlier than the insertion.

Shallower bushes, smaller proofs

The verkle tree construction makes for shallower bushes, which reduces the quantity of saved information. Its actual energy, nonetheless, comes from the flexibility to supply smaller proofs, i.e. witnesses. This will likely be defined within the subsequent article.

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