零知識範圍證明是個很好用的基礎密碼學演算法。零知識範圍證明可以被用 來證明某些想要隱藏的機密在特別的範圍區間之中然而不會洩漏任何跟想要隱藏 的機密有關的資訊，但是剛剛提到的特別的範圍區間是公開資訊，這是一個頗嚴 重的問題，任何人都可以很輕鬆地選一個在範圍區間內的數字並且宣稱此數字是 屬於使用者本身的機密，因為零知識的特性，沒有任何人可以質疑零知識證明的 機密的真偽。為了解決這個嚴重的問題，我們整合零知識證明和簽章演算法，在 證明者產生證明之前，必須先請第三方可信任團體進行和機密相關的簽章，之後 驗證者在驗證零知識範圍證明之前，可以先驗證此簽章是否為證明者本人。我們 堅信著可驗證零知識範圍證明一定會對之後的應用非常的有所幫助。

Zero-Knowledge range proof is a useful cryptographic primitive. It can be used to show some secret lies in a specific range without leaking the secret itself. The problem is that the range is public information. Everyone can easily pick a number in the range and claim that the number belongs to the user. Because of the zero-knowledge property, no one can challenge the proof generated from a fake number. To solve this problem, we integrate a signature service with the zero-knowledge proof protocol. Before a prover generates a proof, a trusted-third party needs to create some authenticated primitives, which are related to the message, for the proof generation. So a verifier can check if the proof is authenticated before accepting the proof. We believe that proposed Authenticated Zero Knowledge Range Proof can be beneficial to many applications in the world.

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