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diving into @ordepdev Merkle Trees

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@ordepdev @ordepdev

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@ordepdev

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@ordepdev

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https://people.eecs.berkeley.edu/~raluca/cs261-f15/readings/merkle.pdf 1987

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“[…] can sign an unlimited number of messages, and the signature size increases logarithmically.”

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“The general idea in this new system is to use an infinite tree of one-time signatures.”

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https://people.eecs.berkeley.edu/~raluca/cs261-f15/readings/merkle.pdf https://patents.google.com/patent/US4309569 http://www.merkle.com/papers/Thesis1979.pdf 1979

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1989 http://merkle.com/papers/Certified1979.pdf

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“[…] is a data structure used for efficiently summarizing and verifying the integrity of large sets of data.”

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“[…] it was used for the purpose of one-time signatures and authenticated public key distribution.”

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Summarize and Verify

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One-way Functions

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“A one-way function f is a function that is easy to compute but difficult to invert.”

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() () () 4a23ca438f3 065b90acc7 ccca99192c

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Why do we need them?

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“[…] being able to store and identify data with a fixed length output can create vast storage savings.”

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“The person who computes y=F(x) is the only person who knows x.”

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One-time Signatures

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“One time signatures are practical between a single pair of users who are willing to exchange the large amount of data necessary […]”

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“If 1000 messages are to be signed before new public authentication data is needed, over 20,000,000 bits or 2.5 megabytes must be stored as public information.”

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Improving one- time signatures

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https://people.eecs.berkeley.edu/~raluca/cs261-f15/readings/merkle.pdf http://www.merkle.com/papers/Thesis1979.pdf

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How to compute a Merkle Root?

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9ec4 L1 L2

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h(L1)

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9ec4 L1 L2

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h(L2)

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a871 L1 L2 7e6a 9ec4

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h(h(L1)||h(L2))

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aea9 a871 7e6a L1 L2 9ec4

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aea9 a871 7e6a L1 L2 9ec4 MerkleTree.Leaf

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defmodule MerkleTree.Leaf do defstruct [:hash, :value] @type hash :: String.t @type value :: String.t @type t :: %MerkleTree.Leaf{ hash: hash, value: value } end

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defmodule MerkleTree.Leaf do defstruct [:hash, :value] @type hash :: String.t @type value :: String.t @type t :: %MerkleTree.Leaf{ hash: hash, value: value } end

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aea9 a871 7e6a L1 L2 9ec4 MerkleTree.Node

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defmodule MerkleTree.Node do defstruct [:hash, :left, :right] @type hash :: String.t @type left :: MerkleTree.Node.t | MerkleTree.Leaf.t @type right :: MerkleTree.Node.t | MerkleTree.Leaf.t @type t :: %MerkleTree.Node{ hash: hash, left: left, right: right } end

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defmodule MerkleTree.Node do defstruct [:hash, :left, :right] @type hash :: String.t @type left :: MerkleTree.Node.t | MerkleTree.Leaf.t @type right :: MerkleTree.Node.t | MerkleTree.Leaf.t @type t :: %MerkleTree.Node{ hash: hash, left: left, right: right } end

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aea9 a871 7e6a L1 L2 9ec4 MerkleTree.Leaf MerkleTree.Node MerkleTree[:root]

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defmodule MerkleTree do defstruct [:root] @type root :: MerkleTree.Node.t @type t :: %MerkleTree{ root: root } end

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defmodule MerkleTree do defstruct [:root] @type root :: MerkleTree.Node.t @type t :: %MerkleTree{ root: root } end

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Hashing the data blocks

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defmodule MerkleTree.Crypto do def hash(input, type \\ :sha256) do type |> :crypto.hash("#{input}") |> Base.encode16 end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) end end

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defmodule MerkleTree.Leaf do def build(value, hash) do %Leaf{value: value, hash: hash} end end

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[“L1”, “L2”] |> MerkleTree.new

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[ %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } ]

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[ %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } ]

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[ %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } ]

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Hashing the nodes

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defmodule MerkleTree.Node do alias MerkleTree.Crypto def new(nodes) do nodes |> Enum.map_join(&(&1.hash)) |> Crypto.hash |> build(nodes) end end

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defmodule MerkleTree.Node do alias MerkleTree.Crypto def new(nodes) do nodes |> Enum.map_join(&(&1.hash)) |> Crypto.hash |> build(nodes) end end

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defmodule MerkleTree.Node do alias MerkleTree.Crypto def new(nodes) do nodes |> Enum.map_join(&(&1.hash)) |> Crypto.hash |> build(nodes) end end

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defmodule MerkleTree.Node do alias MerkleTree.Crypto def new(nodes) do nodes |> Enum.map_join(&(&1.hash)) |> Crypto.hash |> build(nodes) end end

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defmodule MerkleTree.Node do alias MerkleTree.Crypto def new(nodes) do nodes |> Enum.map_join(&(&1.hash)) |> Crypto.hash |> build(nodes) end end

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defmodule MerkleTree.Node do def build(hash, [left, right]) do %Node{hash: hash, left: left, right: right} end end

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MerkleTree.Node.new([ %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } ])

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%MerkleTree.Node{ hash: “F5FC2A0E6D4568040ED7B8D2C59A16B73B8C9EC9...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } }

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%MerkleTree.Node{ hash: “F5FC2A0E6D4568040ED7B8D2C59A16B73B8C9EC9...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } }

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%MerkleTree.Node{ hash: “F5FC2A0E6D4568040ED7B8D2C59A16B73B8C9EC9...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } }

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%MerkleTree.Node{ hash: “F5FC2A0E6D4568040ED7B8D2C59A16B73B8C9EC9...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “6B86B273FF34FCE19D6B804EFF5A3F5747ADA4EA...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D4735E3A265E16EEE03F59718B9B5D03019C07D8...” } }

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Building a bigger tree

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9ec4 L1 L2 L3 L4

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9ec4 L1 L2 L3 L4

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9ec4 L1 L2 L3 L4 7e6a

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9ec4 L1 L2 L3 L4 7e6a 842c

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9ec4 L1 L2 L3 L4 7e6a 842c 4aa0

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9ec4 842c 4aa0 L1 L2 L3 L4 7e6a aea9

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9ec4 4aa0 L1 L2 L3 L4 7e6a aea9 842c d886

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d886 9ec4 4aa0 L1 L2 L3 L4 7e6a aea9 842c 4b25

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All the way up

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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[“L1”, “L2”,“L3”, “L4”] |> Enum.chunk_every(2)

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[[“L1”, “L2”],[“L3”, “L4”]]

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do defp build([root]) do %MerkleTree{root: root} end defp build(nodes) do nodes |> Enum.chunk_every(2) |> Enum.map(&Node.new(&1)) |> build end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) |> build end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) |> build end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) |> build end end

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defmodule MerkleTree do alias MerkleTree.Leaf alias MerkleTree.Crypto def new(blocks) do blocks |> Enum.map(&%Leaf.build(&1, Crypto.hash(&1))) |> build end end

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[“L1”, “L2”] |> MerkleTree.new

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%MerkleTree{ root: %MerkleTree.Node{ hash: “8C569660D98A20D59DE10E134D81A8CE55...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “DFFE8596427FC50E8F64654A609AF135...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D76354D8457898445BB69E0DC0DC95FB...” } } }

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%MerkleTree{ root: %MerkleTree.Node{ hash: “8C569660D98A20D59DE10E134D81A8CE55...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “DFFE8596427FC50E8F64654A609AF135...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D76354D8457898445BB69E0DC0DC95FB...” } } }

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%MerkleTree{ root: %MerkleTree.Node{ hash: “8C569660D98A20D59DE10E134D81A8CE55...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “DFFE8596427FC50E8F64654A609AF135...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D76354D8457898445BB69E0DC0DC95FB...” } } }

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%MerkleTree{ root: %MerkleTree.Node{ hash: “8C569660D98A20D59DE10E134D81A8CE55...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “DFFE8596427FC50E8F64654A609AF135...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D76354D8457898445BB69E0DC0DC95FB...” } } }

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%MerkleTree{ root: %MerkleTree.Node{ hash: “8C569660D98A20D59DE10E134D81A8CE55...”, left: %MerkleTree.Leaf{ value: “L1”, hash: “DFFE8596427FC50E8F64654A609AF135...” }, right: %MerkleTree.Leaf{ value: “L2”, hash: “D76354D8457898445BB69E0DC0DC95FB...” } } }

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Audit Proof

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P2 9ec4 H3 L1 L2 L3 L4 H2 P1 H3 R

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P2 9ec4 h(L4) L1 L2 L3 L4 h(L2) P1 h(L3) R H3 H2 H3

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P2 9ec4 h(L4) L1 L2 L3 L4 H2 P1 h(L3) R H3 H3

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P2 9ec4 h(L4) L1 L2 L3 L4 h(L2) P1 h(L3) R H2 H3 H3

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P2 9ec4 h(L4) L1 L2 L3 L4 h(L2) P1 h(L3) R H2 H3 H3

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P2 9ec4 h(L4) L1 L2 L3 L4 h(L2) P1 h(L3) R H2 H3 H3

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How they are useful?

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Detect Inconsistencies

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L1 L4 Replica 1 L1 L3 L1 L4 L1 L2 L1 L4 L1 L2 Replica 2 Replica 3 Repair Coordinator

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L1 L4 Replica 1 H1 H3 L1 L3 L1 L4 H1 H2 L1 L2 L1 L4 H1 H2 L1 L2 Replica 2 Replica 3 Repair Coordinator

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L1 L4 Replica 1 B H1 H3 L1 L3 L1 L4 A H1 H2 L1 L2 L1 L4 A H1 H2 L1 L2 Replica 2 Replica 3 Repair Coordinator

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L1 L4 Replica 1 B H1 H3 L1 L3 L1 L4 A H1 H2 L1 L2 L1 L4 A H1 H2 L1 L2 Replica 2 Replica 3 Repair Coordinator

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B H1 H2 H1 H3 L1 L2 L1 L3 A Replica 2 Replica 3

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B H1 H2 H1 H3 L1 L2 L1 L3 A Replica 2 Replica 3

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B H1 H2 H1 H3 L1 L2 L1 L3 A Replica 2 Replica 3

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B H1 H2 H1 H3 L1 L2 L1 L2 A Replica 2 Replica 3

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B H1 H2 H1 H2 L1 L2 L1 L2 A Replica 2 Replica 3

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A H1 H2 H1 H2 L1 L2 L1 L2 A Replica 2 Replica 3

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Dynamo, Riak, and Cassandra use this to repair bad replicas!

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Peer-to-peer file sharing

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9cee

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9cee L3

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9cee b2d0 L3

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9cee b2d0 L3 8f14 b2d0

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9cee b2d0 L3 8f14 165f b2d0

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165f 8f14 L3 e831 b2d0 9cee b2d0

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165f 8f14 L3 e831 b2d0 9cee ✅ ✅ b2d0

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Copy-on-write

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Copy-on-write data structures are also called persistent data structures.

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 8f14

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 8f14 L4

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 8f14 L4 5e42

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 8f14 L4 5e42 3d0b

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a871 L1 L2 L3 L4 e4da e831 b2d0 L4 4a88 3d0b 5e42

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Instead of taking a full copy, we can share the same tree between both the copy and the original tree.

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 5e42 3d0b 4a88 8f14 L4

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165f a871 L1 L2 L3 e4da e831 b2d0 9cee L4 8f14 Version 1

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a871 L1 L2 L3 L4 e4da e831 b2d0 L4 4a88 3d0b 5e42 Version 2

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Wrapping-up

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Merkle trees are binary trees containing an infinite number of cryptographic hashes.

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Leaves contains hashes of data blocks.

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Nodes contains hashes of their children.

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Produce a root hash that summarizes an entire data set and it’s publicly distributed.

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Easily prove that a given data block exists in the tree.

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You can find them in…

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And lots of papers…

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@ordepdev Thank you.

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diving into @ordepdev Merkle Trees