From dfff3baa4871eaa484e2a3709e014a0cc434d045 Mon Sep 17 00:00:00 2001 From: TAKAMI Torao Date: Tue, 13 Jul 2021 22:05:22 +0900 Subject: [PATCH 1/5] doc: add ostracon VRF+BLS feature document --- Makefile | 14 +- docs/.vuepress/config.js | 5 + docs/.vuepress/public/.gitkeep | 1 + docs/drafts/features/bft_round.png | Bin 0 -> 159811 bytes .../features/bls_signature_aggregation.png | Bin 0 -> 97081 bytes docs/drafts/features/consensus_ja.md | 115 + docs/drafts/features/index.md | 5 + docs/drafts/features/index_ja.md | 65 + docs/drafts/features/layered_structure.png | Bin 0 -> 111632 bytes docs/drafts/features/math_prove.png | Bin 0 -> 2196 bytes docs/drafts/features/math_verify.png | Bin 0 -> 1428 bytes .../features/signature_aggregation_ja.md | 43 + docs/drafts/features/vrf-based_round.png | Bin 0 -> 65479 bytes docs/drafts/features/vrf_block_generation.png | Bin 0 -> 50060 bytes docs/drafts/features/vrf_block_validation.png | Bin 0 -> 90714 bytes docs/drafts/features/vrf_election.png | Bin 0 -> 45175 bytes docs/drafts/features_ja.md | 1 + docs/drafts/figures.drawio | 1 + docs/package-lock.json | 6797 ++++++++--------- docs/versions | 2 +- 20 files changed, 3283 insertions(+), 3766 deletions(-) create mode 100644 docs/.vuepress/public/.gitkeep create mode 100644 docs/drafts/features/bft_round.png create mode 100644 docs/drafts/features/bls_signature_aggregation.png create mode 100644 docs/drafts/features/consensus_ja.md create mode 100644 docs/drafts/features/index.md create mode 100644 docs/drafts/features/index_ja.md create mode 100644 docs/drafts/features/layered_structure.png create mode 100644 docs/drafts/features/math_prove.png create mode 100644 docs/drafts/features/math_verify.png create mode 100644 docs/drafts/features/signature_aggregation_ja.md create mode 100644 docs/drafts/features/vrf-based_round.png create mode 100644 docs/drafts/features/vrf_block_generation.png create mode 100644 docs/drafts/features/vrf_block_validation.png create mode 100644 docs/drafts/features/vrf_election.png create mode 100644 docs/drafts/features_ja.md create mode 100644 docs/drafts/figures.drawio diff --git a/Makefile b/Makefile index 87609575e..dbd059bb0 100644 --- a/Makefile +++ b/Makefile @@ -233,14 +233,16 @@ DESTINATION = ./index.html.md ### Documentation ### ############################################################################### +BRANCH := $(shell git branch --show-current) +BRANCH_URI := $(shell git branch --show-current | sed 's/[\#]/%23/g') build-docs: cd docs && \ - while read p; do \ - (git checkout $${p} . && npm install && VUEPRESS_BASE="/$${p}/" npm run build) ; \ - mkdir -p ~/output/$${p} ; \ - cp -r .vuepress/dist/* ~/output/$${p}/ ; \ - cp ~/output/$${p}/index.html ~/output ; \ - done < versions ; + npm install && \ + VUEPRESS_BASE="/$(BRANCH_URI)/" npm run build && \ + mkdir -p ~/output/$(BRANCH) && \ + cp -r .vuepress/dist/* ~/output/$(BRANCH)/ && \ + for f in `find . -name '*.png' | grep -v '/node_modules/'`; do if [ ! -e `dirname $$f` ]; then mkdir `dirname $$f`; fi; cp $$f ~/output/$(BRANCH)/`dirname $$f`; done && \ + echo '' > ~/output/index.html .PHONY: build-docs sync-docs: diff --git a/docs/.vuepress/config.js b/docs/.vuepress/config.js index e7c078fbf..5c7742f33 100644 --- a/docs/.vuepress/config.js +++ b/docs/.vuepress/config.js @@ -1,6 +1,11 @@ module.exports = { title: 'Ostracon', base: process.env.VUEPRESS_BASE, + head: [ + ['script', { src: 'https://polyfill.io/v3/polyfill.min.js?features=es6' }], + ['script', { id: 'MathJax-script', src: 'https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js', async: "async"}], + ['script', { }, 'window.MathJax = { tex: { inlineMath: [[\'$\',\'$\'], [\'\\\\(\',\'\\\\)\']] } };'] + ], themeConfig: { repo: 'line/ostracon', docsRepo: 'line/ostracon', diff --git a/docs/.vuepress/public/.gitkeep b/docs/.vuepress/public/.gitkeep new file mode 100644 index 000000000..088cf7f83 --- /dev/null +++ b/docs/.vuepress/public/.gitkeep @@ -0,0 +1 @@ +🏺 diff --git 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000000000..f9d9b891e --- /dev/null +++ b/docs/drafts/features/index_ja.md @@ -0,0 +1,65 @@ +# Ostracon: A Fast, Secure Consensus Layer for The Blockchain of New Token Economy + +Version 1.0 :: [English](index.md) + +## Ostracon Overview + +Ostracon は LINE Blockchain エコシステムにビザンチン障害耐性を持つ分散合意メカニズムを提供するコアコンポーネントです。アプリケーションが +実行するトランザクションの順序を確定し、トランザクションのコンテナであるブロックの生成、検証を行います。 + +LINE Blockchain はインターネット上の電子サービスのみならず、金融や産業にも適用可能な合意メカニズムをもたらすために技術選定について達成 +すべきいくつかの方針を設定しています。 + +1. **セキュリティ**: 暗号理論に基づいた実用に十分な完全性と健全性を持つ。 +2. **整合性**: 強い整合性 (ファイナリティ) の合意アルゴリズムを持つ。 +3. **障害耐性**: ビザンチン障害を含むシステム障害に対して Safety と Liveness を持つ。 +4. **パフォーマンスとスケーラビリティ**: 1000TPS+ の速度性能を持つ。 +5. **チェーン間接続**: LINE Blockchain 以外のブロックチェーンとの相互接続性を持つ。 + +ファイナリティとパフォーマンスの観点から Bitcoin のような Proof of Work よりも BFT (Byzantine Fault-Tolerance) に基づく P2P +合意アルゴリズムの方が適しています。中でもブロックチェーンに最適化された近代的な設計が行われている Tendermint-BFT は我々の方針に最も +近い実装でした (さらに良いことに Cosmos Hub とも接続できます)。 + +我々は我々のブロックチェーンをさらに改善するために Tendermint-BFT に二つの新しい暗号技術を導入しています。その一つである**検証可能な +疑似乱数** (VRF) は、ブロックを生成する Proposer ノードの選出にランダム性をもたせて未来の選出を予測困難にすることを目的に導入されました。 +このランダム性の導入により、悪意を持つ攻撃者に攻撃の猶予を与えたり、将来のある時点を狙って参加者どうしで共謀することを困難にする効果が期待 +できます。 + +もう一つの機能は **BLS 署名**です。双線形写像に基づく BLS 署名は複数の電子署名をたった一つの電子署名に集約できる特徴を持ちます。多くの +ブロックチェーンプロトコルでは、ブロックを承認するために多数の署名を保存しなければなりませんが、BLS 署名集約を有効にすることでそのフット +プリントを削減し、通信速度やストレージ消費量を大きく改善する効果が期待できます。 + +## Layered Structure + +LINE Blockchain ノードを構成する Application, Consensus および Networking の 3 つのレイヤーのうち、Ostracon には Consensus と +Networking レイヤーが含まれています。 + +![Layered Structure](layered_structure.png) + +まだブロックに取り込まれていないトランザクションは mempool と呼ばれる Network レイヤーのアンチエントロピー機構 (ゴシッピング) によって +各ノード間で共有されます。ここで、Network および Consensus レイヤーではトランザクションを単純なバイナリとして扱い、そのデータの内容には +関与しません。 + +## Specifications and Technology Stack + +| Specifications | Policy / Algorithms | Methods / Implementations | +|:----------------------|:------------------------------|:------------------------------------------------| +| Participation | Permissioned | Consortium or Private | +| Election | Proof of Stake | VRF-based Weighted Sampling without Replacement + SplitMix64 | +| Agreement | Strong Consistency w/Finality | Tendermint-BFT | +| Signature | Elliptic Curve Cryptography | Ed25519, *BLS12-381**1 | +| Hash | SHA2 | SHA-256, SHA-512 | +| HSM | *N/A* | *No support for VRF or signature aggregation* | +| Key Auth Protocol | Station-to-Station | | +| Tx Sharing Protocol | Gossiping | mempool | +| Application Protocol | ABCI | | +| Interchain Protocol | IBC (Cosmos Hub) | | +| Storage | Embedded KVS | LevelDB | +| Message Recovery | WAL | | + +*1 experimental implementation. + +## Ostracon Features + +* [Extending Tendermint-BFT with VRF-based Election](consensus_ja.md) +* [BLS Signature Aggregation](signature_aggregation_ja.md) diff --git a/docs/drafts/features/layered_structure.png b/docs/drafts/features/layered_structure.png new file mode 100644 index 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+非中央集権的な合意メカニズムを持つブロックチェーンではブロックを生成するたびに十分な数の投票 (署名) を収集する必要があります。合意の参加者が +多くなれば安全性も向上しますが、一方で署名の数によってブロックサイズが肥大化し検証にも時間がかかるためパフォーマンスの劣化が問題となります。 +この問題を解決するために Bitcoin ([BIP340](https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki)) や Ethereum 2.0 +では署名集約を取り入れてパフォーマンスを改善しようと取り組んでいます。 + +BLS 署名は双線形写像を使用した署名集約が可能な署名アルゴリズムの一種です。BLS 署名に関する最初の論文は非常に小さなサイズで実現できる電子署名 +として出版されましたが、ペアリング暗号とよばれるその技術からは他にもしきい値署名やブラインド署名のようないくつかの興味深い特徴が考案されました。 +Ostracon でも BLS 署名によって収集した署名を単一の署名に集約することで 1) ブロックサイズの削減と 2) 検証回数の削減でパフォーマンスの向上を +図っています。 + +![BLS Signature Aggregation](bls_signature_aggregation.png) + +## Public Key Abstraction + +BLS 署名の導入に伴い、Ostracon ではノードごとにスキームの異なる署名鍵を同一のブロックチェーンインスタンス上で利用できるように設計を抽象化 +しました。つまり Ostracon ノードの運営者はノードを構築するときに、高速で実績のある Ed25519 署名と署名集約が可能な BLS 署名を選択できる +ようになります。この柔軟性によって、将来にもっと良い署名アルゴリズムを試験/採用する場合や、特定の公開鍵アルゴリズムに脆弱性が発見された場合に +柔軟に対処することができます。 + +## Why is this an experimental status? + +BLS の導入にあたり、残念ながら BLS 署名集約は Ostracon のフォーク元である Tendermint の設計といくつかの点で競合していることが分かっています。 +典型的な例は軽量ノードの Light Validation と呼ばれる簡易検証です。合意に参加したすべてのノードの公開鍵を持っていないクライアントであっても、 +BFT 仮定に基づいて全 Voter 数の 2/3+1 個の投票の検証が成功すれば正しいブロックブロックとみなすことができます。しかし BLS 署名では合意に +参加した公開鍵が一つでも欠けていれば集約されたすべての署名を検証できなくなります。 + +性能の観点では、単一の署名生成/検証においては BLS 署名よりも Ed25519 署名の方が高速という事実があります。その遅さをブロックサイズ削減や +検証回数削減による改善が上回る分水嶺がどこなのかを注意深く調査しています。 + +| アルゴリズム | 秘密鍵 | 公開鍵 | 署名 | 署名作成 | 署名検証 | +|:------------------|------:|------:|-----:|--------:|--------:| +| ECDSA (secp256k1) | 96B | 64B | 64B | 92μs | 124μs | +| Ed25519 | 64B | 32B | 64B | 49μs | 130μs | +| BLS12-381 | 32B | 96B | 48B | 233μs | 1,149μs | + +Table: 署名アルゴリズムの空間効率と処理時間効率。署名作成/検証のメッセージ長は 1024 バイト。 + +我々は将来的に BLS 署名をサポートする予定ですが、このような問題に対して下位互換性のない修正を行う予定があることから、現在では experimental +ステータスとしています。 diff --git a/docs/drafts/features/vrf-based_round.png b/docs/drafts/features/vrf-based_round.png new file mode 100644 index 0000000000000000000000000000000000000000..32ed5aae742e75a430e4d3ecf8dcfb0ca54453c2 GIT binary patch literal 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"sha1-zg6+ql94yxiSXqfYENe1mwEP1RU=" - }, - "string-width": { - "version": "3.1.0", - "resolved": "https://registry.npmjs.org/string-width/-/string-width-3.1.0.tgz", - "integrity": "sha512-vafcv6KjVZKSgz06oM/H6GDBrAtz8vdhQakGjFIvNrHA6y3HCF1CInLy+QLq8dTJPQ1b+KDUqDFctkdRW44e1w==", - "requires": { - "emoji-regex": "^7.0.1", - "is-fullwidth-code-point": "^2.0.0", - "strip-ansi": "^5.1.0" - } - }, - "strip-ansi": { - "version": "5.2.0", - "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-5.2.0.tgz", - "integrity": "sha512-DuRs1gKbBqsMKIZlrffwlug8MHkcnpjs5VPmL1PAh+mA30U0DTotfDZ0d2UUsXpPmPmMMJ6W773MaA3J+lbiWA==", - "requires": { - "ansi-regex": "^4.1.0" - } } } }, diff --git a/docs/versions b/docs/versions index 061ca5668..ba2906d06 100644 --- a/docs/versions +++ b/docs/versions @@ -1 +1 @@ -main main +main From f0963d30c141e647cd2e1ddb5cd57322977644d0 Mon Sep 17 00:00:00 2001 From: TAKAMI Torao Date: Mon, 2 Aug 2021 19:13:27 +0900 Subject: [PATCH 2/5] fix a typo --- docs/drafts/features/consensus_ja.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/drafts/features/consensus_ja.md b/docs/drafts/features/consensus_ja.md index f8429906b..eb9050b8b 100644 --- a/docs/drafts/features/consensus_ja.md +++ b/docs/drafts/features/consensus_ja.md @@ -6,7 +6,7 @@ Tendermint-BFT に基づく Ostracon のブロック生成メカニズムは以 ここで、ブロックの世代を*高さ*、この 3 つのブロック承認プロセスを*ラウンド*と呼んでいます。 **選出フェーズ**. 候補ノードの中から １ つの Proposer と複数の Voter を選出します。これは一般的な分散システムにおけるリーダー選挙と同じ -ですが、ブロックチェーンではアイク的な妨害によってシステム全体の性能を低下させないために作為的な選出ができないように設計する必要があります。 +ですが、ブロックチェーンでは悪意を持った妨害によってシステム全体の性能を低下させないために作為的な選出ができないように設計する必要があります。 また公平性を保証するために Ostracon の選挙には中央集権的な機関が介在していないことにも注意してください。選挙結果はすべてのノードで決定論的に 算出できるため、各ノードは「自分が Proposer または Voter に当選しているか」を自律的に判断することができます。 From ff97defb489a3354043cb217d30885c943ce6c85 Mon Sep 17 00:00:00 2001 From: TAKAMI Torao Date: Tue, 3 Aug 2021 21:51:20 +0900 Subject: [PATCH 3/5] fix wording of VRF-validation --- docs/drafts/features/consensus_ja.md | 28 +++++++++++++----- docs/drafts/features/math_expression.png | Bin 0 -> 3334 bytes docs/drafts/features/math_prove.png | Bin 2196 -> 2327 bytes docs/drafts/features/math_verify.png | Bin 1428 -> 1504 bytes ...rf-based_round.png => vrf_based_round.png} | Bin 5 files changed, 21 insertions(+), 7 deletions(-) create mode 100644 docs/drafts/features/math_expression.png rename docs/drafts/features/{vrf-based_round.png => vrf_based_round.png} (100%) diff --git a/docs/drafts/features/consensus_ja.md b/docs/drafts/features/consensus_ja.md index eb9050b8b..482465288 100644 --- a/docs/drafts/features/consensus_ja.md +++ b/docs/drafts/features/consensus_ja.md @@ -19,14 +19,28 @@ Tendermint-BFT に基づく Ostracon のブロック生成メカニズムは以 正式に*承認*されます。反対に、定足数の賛成票が集まらなければ提案されたブロックは拒否され新しいラウンドで選挙または投票からやり直しとなります (Tendermint-BFT には拒否の理由によってショートカットする経路がいくつかあります)。 -![VRF-based Block Generation Round](vrf-based_round.png) +![VRF-based Block Generation Round](vrf_based_round.png) ## VRF-based Consensus Group Election VRF は暗号論的疑似乱数として使用できるハッシュ値 $t$ を生成するアルゴリズムです。VRF が一般的なハッシュ関数や疑似乱数生成器と異なる点は、 -公開鍵を使用して 「本当にその乱数がメッセージ $m$ に基づいて生成されたものか」を第三者が検証できる点です。乱数の生成者はメッセージ $m$ と -秘密鍵 $S_k$ を使って証明 $\pi$ (VRF Proof; ハッシュ値 $t$ の元となる値) を生成します。他方、検証者は入手した証明 $\pi$ が偽造された -ものでないことを $S_k$ に対応する公開鍵 $P_k$ とメッセージ $m$ を使って同じハッシュ値 $t$ が生成されるかを検証します。 +公開鍵を使用して 「本当にその乱数がメッセージ $m$ に基づいて生成されたものか」を第三者が検証できる点です。 + +乱数の生成者 $k$ は自身の秘密鍵 $S_k$ を使ってメッセージ $m$ から証明 (VRF Proof) $\pi$ を生成します:式(1)。ここでハッシュ値 $t$ +は証明 $pi$ から生成することができます:式(2)。一方、検証者は入手した証明 $\pi$ が本当にその生成者によってメッセージ $m$ に対して +生成されたことを検証するために、$S_k$ に対応する公開鍵 $P_k$ と $m$、$pi$ を使って同一のハッシュ値 $t$ が生成されるかを確認します:式(3)。 + +![VRF Expression](math_expression.png) + +```math +\begin{eqnarray} +\pi & = & {\rm vrf\_prove}(S_k, m) \\ +t & = & {\rm vrf\_proof\_to\_hash}(\pi) +\end{eqnarray} +\begin{equation} +{\rm vrf\_proof\_to\_hash}(\pi) \overset{\text{?}}{=} {\rm vrf\_verify}(P_k, m, \pi) +\end{equation} +``` Ostracon では、あるブロックを作成した Proposer による*無作為で検証可能な乱数*によって次の Proposer と Voter を決定します。そして ブロックにはそのための VRF Proof フィールド $\pi$ が追加されています。 @@ -46,8 +60,8 @@ $r$ に基づいて算出した新しい VRF Proof $\pi'$ をブロックに設 ```math \begin{eqnarray*} m_h & = & {\rm SHA256}(h \,\|\, r \,\|\, t_{h-1}) \\ -\pi_h & = & {\rm prove}(S_i, m_h) \\ -t_h & = & {\rm prove\_to\_hash}(\pi_h) +\pi_h & = & {\rm vrf\_prove}(S_i, m_h) \\ +t_h & = & {\rm vrf\_proof\_to\_hash}(\pi_h) \end{eqnarray*} ``` @@ -71,7 +85,7 @@ prevote, precommit, commit を経て複製され、定足数以上の有効票 ![VRF Verify](math_verify.png) ```math -{\rm verify}(P_i, m_h, \pi_h) \overset{\text{?}}{=} {\rm proof\_to\_hash}(\pi_h) +{\rm vrf\_verify}(P_i, m_h, \pi_h) \overset{\text{?}}{=} {\rm vrf\_proof\_to\_hash}(\pi_h) ``` この一連のラウンドを繰り返すことによって無作為なランダムサンプリングをすべてのブロック生成に渡って連鎖させることができます。 diff --git a/docs/drafts/features/math_expression.png b/docs/drafts/features/math_expression.png new file mode 100644 index 0000000000000000000000000000000000000000..cf1f553abce3f4e59935ed5bb9d857d0f5973412 GIT binary patch literal 3334 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b/docs/drafts/features/vrf_based_round.png similarity index 100% rename from docs/drafts/features/vrf-based_round.png rename to docs/drafts/features/vrf_based_round.png From 24d7cd16ca56369c462ec0b58153a9eee9481460 Mon Sep 17 00:00:00 2001 From: TAKAMI Torao Date: Thu, 5 Aug 2021 18:31:07 +0900 Subject: [PATCH 4/5] squash --- docs/drafts/features/consensus_ja.md | 22 +++++----------- docs/drafts/features/index_ja.md | 26 ++++++++++++++++++- .../features/signature_aggregation_ja.md | 10 +++---- 3 files changed, 37 insertions(+), 21 deletions(-) diff --git a/docs/drafts/features/consensus_ja.md b/docs/drafts/features/consensus_ja.md index 482465288..39607a291 100644 --- a/docs/drafts/features/consensus_ja.md +++ b/docs/drafts/features/consensus_ja.md @@ -23,12 +23,13 @@ Tendermint-BFT に基づく Ostracon のブロック生成メカニズムは以 ## VRF-based Consensus Group Election -VRF は暗号論的疑似乱数として使用できるハッシュ値 $t$ を生成するアルゴリズムです。VRF が一般的なハッシュ関数や疑似乱数生成器と異なる点は、 -公開鍵を使用して 「本当にその乱数がメッセージ $m$ に基づいて生成されたものか」を第三者が検証できる点です。 +VRF は暗号論的疑似乱数として使用できるハッシュ値 $t$ を生成するアルゴリズムです。VRF が一般的なハッシュ関数や疑似乱数生成器と異なるのは、 +秘密鍵の所有者のみがハッシュ値 $t$ を算出でき、対応する公開鍵を持つ人であれば誰でもそのハッシュ値の正当性を検証できる点です。 -乱数の生成者 $k$ は自身の秘密鍵 $S_k$ を使ってメッセージ $m$ から証明 (VRF Proof) $\pi$ を生成します:式(1)。ここでハッシュ値 $t$ -は証明 $pi$ から生成することができます:式(2)。一方、検証者は入手した証明 $\pi$ が本当にその生成者によってメッセージ $m$ に対して -生成されたことを検証するために、$S_k$ に対応する公開鍵 $P_k$ と $m$、$pi$ を使って同一のハッシュ値 $t$ が生成されるかを確認します:式(3)。 +乱数の生成者 $k$ は式 (1) のように自身の秘密鍵 $S_k$ を使ってメッセージ $m$ から証明 (VRF Proof) $\pi$ を生成します。ここでハッシュ値 +$t$ は式 (2) を使って証明 $pi$ から生成することができます。一方、検証者は証明 $\pi$ が秘密鍵 $S_k$ の所有者によってメッセージ $m$ に +対して生成されたものであることを検証するために、$S_k$ に対する公開鍵 $P_k$, $m$, $\pi$ を式 (3) に適用して同一のハッシュ値 $t$ が +生成されることを確認します。 ![VRF Expression](math_expression.png) @@ -97,7 +98,7 @@ prevote, precommit, commit を経て複製され、定足数以上の有効票 ある行動を行ったノードに対して懲罰を与えることができます。一方で、それらが明らかになるのは必要最小限の期間のみであり、1 ブロックより先の Proposer や Voter を予測することは依然として困難です。 -VRF は現在のところ ed25519 鍵を使用して実装されています。BLS 署名を選択したとしても VRF の算出を行うために Ed25519 鍵がセットで生成 +VRF は現在のところ Ed25519 鍵を使用して実装されています。BLS 署名を選択したとしても VRF の算出を行うために Ed25519 鍵がセットで生成 されます。 ## Voters @@ -118,12 +119,3 @@ Ostracon の合意スキームは少数のノードが故障していても正 コンセンサスグループに選ばれないことが理想的です。とりわけ一般的な非同期メッセージングの問題が原因ではないケース、つまり意図的に行ったと 思われる不正な行為に対しては (悪意の有無に関わらず) その挙動の evidence が共有されて Stake の没収によって選出候補から排除する措置が 取られます。 - - diff --git a/docs/drafts/features/index_ja.md b/docs/drafts/features/index_ja.md index f9d9b891e..8228b7009 100644 --- a/docs/drafts/features/index_ja.md +++ b/docs/drafts/features/index_ja.md @@ -13,7 +13,7 @@ LINE Blockchain はインターネット上の電子サービスのみならず 1. **セキュリティ**: 暗号理論に基づいた実用に十分な完全性と健全性を持つ。 2. **整合性**: 強い整合性 (ファイナリティ) の合意アルゴリズムを持つ。 3. **障害耐性**: ビザンチン障害を含むシステム障害に対して Safety と Liveness を持つ。 -4. **パフォーマンスとスケーラビリティ**: 1000TPS+ の速度性能を持つ。 +4. **パフォーマンスとスケーラビリティ**: 2 秒に 1 つのブロックを生成し、1000TPS+ の速度性能を持つ。 5. **チェーン間接続**: LINE Blockchain 以外のブロックチェーンとの相互接続性を持つ。 ファイナリティとパフォーマンスの観点から Bitcoin のような Proof of Work よりも BFT (Byzantine Fault-Tolerance) に基づく P2P @@ -56,6 +56,7 @@ Networking レイヤーが含まれています。 | Interchain Protocol | IBC (Cosmos Hub) | | | Storage | Embedded KVS | LevelDB | | Message Recovery | WAL | | +| Block Generation Time | 2 seconds | | *1 experimental implementation. @@ -63,3 +64,26 @@ Networking レイヤーが含まれています。 * [Extending Tendermint-BFT with VRF-based Election](consensus_ja.md) * [BLS Signature Aggregation](signature_aggregation_ja.md) + +## Consideration with Other Consensus Schemes + +他のブロックチェーンではどのようなコンセンサス機構を採用しているのでしょうか? Ostracon の方向性を決定するために多くの比較と検討を行いました。 + +**Bitcoin** や **Ethereum** で採用している PoW は最も有名なブロックチェーン向けコンセンサス機構です。これらはパブリックチェーンとして +運用している実績がありますが、十分な時間が経過しないと結果が覆る可能性があるという機能的な制約を持ちます。これは、短期には lost update 問題を +引き起こし、長期には必要なパフォーマンスが確保できないという問題が顕著に現れることから、PoW は検討初期の段階で選択肢から外れました。 + +**Tendermint** が合意アルゴリズムに採用している Tendermint-BFT はブロックチェーン向けによく考慮された設計です。短時間でファイナリティを +保証できる点も我々の方針に適していました。一方で、選出アルゴリズムに採用している加重ラウンドロビンは決定論的に動作するため、誰でも将来の +Proposer を知り得ることから標的を見つけて攻撃を準備しやすい点があります。このため Ostracon では攻撃の可能性を軽減する目的で VRF を使って +予測不可能なアルゴリズムに置き換えています。 + +**Algorand** は我々とは大きく異なる方法で VRF を使用しています。Algorand では選挙が始まるとそれぞれのノードが VRF 乱数を生成して次の +Validator に当選しているかをノード自身が判断します (すべてのノードが一斉にコイントスするのと似ています)。これは PoW のハッシュ計算で +当選を引き当てる方法と比較して、大量の計算時間と電力消費を省略しつつ暗号論的な安全性を保証している優れた方法です。一方で、選出される +Validator 数が決定的ではなく二項分布に従うランダムな振る舞い含むことや、当選ノード間の相互認識でプロトコルが複雑性が上がること、当選した +にもかかわらず役割をサボタージュしたノードを見つけることができないこと、といったいくつかの理由により適用は難しいと判断しました。 + +他にいくつものコンセンサス機構について考慮しましたが、役割選出と合意アルゴリズムに関しては現在の選択が現実的に最良に近い選択と考えています。 +しかし、Ostracon は特定の研究理論に対しての実験的証明や実証実験をゴールとしていないため、将来的により良いアルゴリズムの提案があればそれを +採用する準備があります。 diff --git a/docs/drafts/features/signature_aggregation_ja.md b/docs/drafts/features/signature_aggregation_ja.md index 6d8dccdc9..74c9772b2 100644 --- a/docs/drafts/features/signature_aggregation_ja.md +++ b/docs/drafts/features/signature_aggregation_ja.md @@ -8,7 +8,7 @@ では署名集約を取り入れてパフォーマンスを改善しようと取り組んでいます。 BLS 署名は双線形写像を使用した署名集約が可能な署名アルゴリズムの一種です。BLS 署名に関する最初の論文は非常に小さなサイズで実現できる電子署名 -として出版されましたが、ペアリング暗号とよばれるその技術からは他にもしきい値署名やブラインド署名のようないくつかの興味深い特徴が考案されました。 +として出版されましたが、ペアリングとよばれるその技術からは他にもしきい値署名やブラインド署名のようないくつかの興味深い特徴が考案されました。 Ostracon でも BLS 署名によって収集した署名を単一の署名に集約することで 1) ブロックサイズの削減と 2) 検証回数の削減でパフォーマンスの向上を 図っています。 @@ -16,10 +16,10 @@ Ostracon でも BLS 署名によって収集した署名を単一の署名に集 ## Public Key Abstraction -BLS 署名の導入に伴い、Ostracon ではノードごとにスキームの異なる署名鍵を同一のブロックチェーンインスタンス上で利用できるように設計を抽象化 -しました。つまり Ostracon ノードの運営者はノードを構築するときに、高速で実績のある Ed25519 署名と署名集約が可能な BLS 署名を選択できる -ようになります。この柔軟性によって、将来にもっと良い署名アルゴリズムを試験/採用する場合や、特定の公開鍵アルゴリズムに脆弱性が発見された場合に -柔軟に対処することができます。 +BLS 署名の導入に伴い、Ostracon ではノードごとにスキームの異なる署名鍵を同一のブロックチェーンインスタンス上で利用できるように設計を見直し +ました。つまり Ostracon ノードの運営者はノードを構築するときに、高速で実績のある Ed25519 署名と署名集約が可能な BLS 署名を選択できる +ようになります。この柔軟性によって、将来にもっと良い署名アルゴリズムを試験/採用する場合や、実装に脆弱性が発見された場合に柔軟に対処することが +できます。 ## Why is this an experimental status? From 2c5159eaaaf7e1974da2f3a059225385cca358d7 Mon Sep 17 00:00:00 2001 From: TAKAMI Torao Date: Tue, 17 Aug 2021 15:59:08 +0900 Subject: [PATCH 5/5] fix some pointeds --- docs/drafts/features/consensus_ja.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/drafts/features/consensus_ja.md b/docs/drafts/features/consensus_ja.md index 39607a291..f13aebf79 100644 --- a/docs/drafts/features/consensus_ja.md +++ b/docs/drafts/features/consensus_ja.md @@ -27,8 +27,8 @@ VRF は暗号論的疑似乱数として使用できるハッシュ値 $t$ を 秘密鍵の所有者のみがハッシュ値 $t$ を算出でき、対応する公開鍵を持つ人であれば誰でもそのハッシュ値の正当性を検証できる点です。 乱数の生成者 $k$ は式 (1) のように自身の秘密鍵 $S_k$ を使ってメッセージ $m$ から証明 (VRF Proof) $\pi$ を生成します。ここでハッシュ値 -$t$ は式 (2) を使って証明 $pi$ から生成することができます。一方、検証者は証明 $\pi$ が秘密鍵 $S_k$ の所有者によってメッセージ $m$ に -対して生成されたものであることを検証するために、$S_k$ に対する公開鍵 $P_k$, $m$, $\pi$ を式 (3) に適用して同一のハッシュ値 $t$ が +$t$ は式 (2) を使って証明 $pi$ から生成することができます。一方、検証者はハッシュ値 $t$ が秘密鍵 $S_k$ の所有者によってメッセージ $m$ に +基づいて生成されたものであることを検証するために、$S_k$ に対する公開鍵 $P_k$ と $m$, $\pi$ を式 (3) に適用して同一のハッシュ値 $t$ が 生成されることを確認します。 ![VRF Expression](math_expression.png)