NEMで透明性ある抽選機能をつくる

NEMブロックチェーンをうまく使って、改ざんされてないと証明できる抽選機能を作れないかということで検証してみました。

抽選機能をどう作るかというと、乱数と呼ばれるランダムな数字をつくりだし、その数字の範囲に対応するよう当てはめておいた抽選結果を探せばよいです。これは簡単ですね。

なので、「透明性ある=(恣意的でないことを証明できる)抽選機能をつくる」には、「恣意的でないことを証明できる乱数をつくる」ことができれば、できるということになります。

従来の乱数

JavaScriptでは、

Math.random()

を使うと、[0,1)つまり0以上1未満のランダムな数字をつくることができますね。

当然これを使って抽選をつくることはできますが、

運営に都合の良いような抽選結果に変えていない

ということを証明する手立てがありません。

運営に都合の良いような抽選結果に変えていないことを証明するには、使った乱数を公開する必要があります。

だからといって、使った乱数をNEMブロックチェーンにアポスティーユしたところで、意味がありません。

アポスティーユした乱数を、運営に都合の良いような結果に変えていない

ことを証明できないからです。

じゃあどうするか

じゃあどうするか。ここで一つ思いつきました。

ハッシュ値を乱数の代わりに使えばいいやん。

と。

ハッシュ値とは、ハッシュ関数の結果のことなのですが、ハッシュ関数とはなにかを説明しておくと、

「任意の長さの入力を、衝突困難な固定長の出力に変換する関数」です。
イメージが湧きづらいので、256bitのハッシュ関数で具体的に考えましょう。

256bitのハッシュ関数の場合、どんなデータをハッシュ関数の入力に入れても、[0,2^256-1]の整数つまり0以上2^256-1以下の整数として出力します。

ハッシュ関数は、

• 入力が同じなら出力は常に同じ
• 入力が少しでも変わると出力は大きく変わる
• 出力が衝突する確率は限りなく低い(2^256-1か莫大であることからもまあわかるでしょう)

という性質を持つので、ブロックチェーンにおいても不正な取引記録でないかを検知できるために大いに使われています。

それに加えて、

• 0以上2^256-1以下の整数のうち、それぞれの数字が出力される確率は等しい

という「一様性」という性質を満たします。
この「一様性」は、乱数として使うためにも必須の性質となります。

使えるか検証してみる

NEMブロックチェーンの1900489ブロックから1901489ブロックまでのブロックハッシュをとってきました。

長いので折りたたみ。

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  "ae3ef02066e492831a7f10f97d61c1f1eb11e3bc94de70e25ce98e8cfc846334",
  "15e2df2c4005c58f20d8101b75dd9eb173ca07109b4609a3ed3b3942a1f53b0d",
  "f23d9022bdb498ec6e7d02a02708f7dd5337e65760537784a677ae5dd48e6324",
  "98a5d2929c1b09b95b7c3f2af2c1a7ca718570904d4731d33db58457b771d2d4",
  "2d5e6e94dca0e10ab87f9f7424b9010eef2343a8f5124a73b2ea349630823c1d",
  "716812f207a45f2e4b9715d2024c4de6b2f8d587170189dd04866adfa1542824",
  "3249ac2f58b94c9ad77971c8c1ac31467f364c3ba2d911a7c99188e462c76d3e",
  "51a4032006e4b7a3dbc907b17db394cc0299cf3cbaf413d416c1205534ea4cf0",
  "4c537529fd9ca762b91a723acab1308061991fcebe8467b2d03d6b5a57559dcb",
  "1d93f3a627136dd00f1328068e966de6854438b0342988180a765c045e0d3131",
  "de02eebd6e3d25bc0a16f8ac98fc5d44008bbf0dfe5059b391017e4fb3dad53b",
  "559735c7d93a66f76acfa9be0574e493579a9772528db907e3aa1f81305b8f7e",
  "55bab8f45ef96d95f5d1cd18c18ca3f9d429fcb945a17671c9dd45efe6926247",
  "0a1de26d3f8d8275f54c9e3db7dcf86f57c49e56836fb815e6193b1624a25d75"
]

ハッシュをそのまま乱数には使えないので、
「ハッシュの下32ビットを0xFFFFFFFF+1でわったもの」
をつくりました。
それが以下になります。

長いので折りたたみ。

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区間ごとに分ける

検証するために、[0,1)という値域を、
* [0,0.2)
* [0.2,0.4)
* [0.4,0.6)
* [0.6,0.8)
* [0.8,1)

という、5区分にわけ、それぞれの区分に入った乱数の数を数えてみます。

すると

[0,0.2)	[0.2,0.4)	[0.4,0.6)	[0.6,0.8)	[0.8,1)
193	216	181	207	203

となりました。
期待度数は200なはずなので、遠からずではあります。しかし181のように少し遠い値もあるため、ここらへんのばらつきが乱数として使うにあたって許容できるのかをみてみます。

カイ二乗検定

カイ二乗検定の説明はぱぱっとします。難しいですが、詳しい説明は省かせてください。

• とあるルールに従って計算された検定統計量が、χ^2分布に従う仮説をつくる
• カイ二乗分布の確率分布表をみると、確率が高いところ、確率が低いところがある
• その仮説のとおり検定統計量がχ^2分布に従っていて、検定統計量が確率が高い分布域に収まっているとまあ不思議はない
• その仮説のとおり検定統計量がχ^2分布に従っていて、検定統計量が確率が高い分布域に収まっていなければ、仮説が不自然であるとみなせる。つまり仮説は棄却される。

標本数をn、区間数をk、各区間の観測度数をxi、期待度数をF=n/kとすると、
n個の乱数が一様であれば、

\chi^2=Σ_{i=1}^{k}{\frac{(x_i-F)^2}{F}}

は自由度k-1のカイ2乗分布に従います。

ここではn=1000、k=5なので、
1000個の乱数が一様であれば、

\chi^2=3.620

が自由度4のカイ二乗検分布に従います。

検定結果

自由度4において、危険率5%とすると、棄却域は(9.48773,無限大)です。
3.620は棄却域に入っていないので、

1000個の乱数が一様であれば

\chi^2=3.620

が自由度4のカイ二乗検分布に従います。

という仮説は、棄却されません。つまり、

「1000個の乱数が一様である」は、正しい確率のほうが高い

ということです。

従来の乱数でも検定してみた

[0,0.2)	[0.2,0.4)	[0.4,0.6)	[0.6,0.8)	[0.8,1)
203	195	186	220	196

\chi^2=3.230

これも棄却域に入っていないので、「1000個の乱数が一様である」は、正しい確率のほうが高いと言えることになります。これはJavaScript標準のやつなので当然ですね笑。

結論

NEMのブロックハッシュのように、恣意性のない入力から作られたハッシュ値は、一様性があるので、乱数として十分使える。なので「恣意性が入ってないと証明できる乱数」になれる。ということになります。

ハッシュ値であればブロックハッシュでなく何でも良いということもいえます。マルチシグトランザクションのハッシュ値でも構いません。（マルチシグでない場合はオフラインでトランザクションつくりなおせるので要注意）

ということは

あらかじめ抽選に使うハッシュ値についてルールを決めておきさえすれば、

NEMブロックチェーンをつかって恣意性がないと証明できる乱数がつくれる、つまり、NEMブロックチェーンを使えば透明性ある抽選機能がつくれる！！！

といえます。

ゲーム用のガチャだとか　ガチャもぐとかにも　使えると思いますし、

弊社(LCNEM)の開発するちけっとピアツーピアにも応用したいと思います。

アイデアの種になれば良いですね。

では！