零知識證明

　　在最小泄露協議中零知識證明需要滿足下述兩個性質：

In a minimal disclosure agreement, zero knowledge proves that the following two characteristics need to be satisfied:

　　(1)正確性。P無法欺騙V。換言之，若P不知道一個定理的證明方法，則P使V相信他會證明定理的概率很低。

(1) Correctness. P can't deceive V. In other words, if P does not know the proof of a theory, P convinces V that he will prove that the theory is low.

　　(2)完備性。V無法欺騙P。若P知道一個定理的證明方法，則P使V以絕對優勢的概率相信他能證明。

(2) Completeness. V cannot deceive P. If P knows the proof of a theory, P makes it possible for V to prove the absolute advantage of .

　　在零知識協議中，除滿足上述兩個條件以外，還滿足下述的第三個性質：

In the Zero Knowledge Agreement, in addition to the two above-mentioned conditions, the following third condition is satisfied:

　　(3)零知識性。V無法獲取任何額外的知識。

(3) Zero knowledge. V cannot obtain any additional knowledge.

　　我們把性質(1)和(2)稱為零知識證明的正確性和完備性，而性質(3)稱為零知識性。

We call sex (1) and (2) correctness and integrity of proof of zero knowledge, and sex (3) zero knowledge.

　　零知識證明需要滿足三個屬性。

Zero knowledge proves that there are three attributes that need to be satisfied.

　　1、如果語句為真，誠實的驗證者（即：正確遵循協議的驗證者）將由誠實的證明者確信這一事實。

If the words are true, an honest tester (i.e., a testee who is correct in following the agreement) will be convinced by an honest attester.

　　2、如果語句為假，不排除有概率欺騙者可以說服誠實的驗證者它是真的。

2. If the phrase is false, it does not exclude that a person with a probability of deception can convince an honest witness that it is true.

　　3、如果語句為真，證明者的目的就是向驗證者證明並使驗證者相信自己知道或擁有某一消息，而在證明過程中不可向驗證者泄漏任何有關被證明消息的內容。

3. If the statement is true, the purpose of the witness is to prove to and convince the witness that he or she knows or possesses a certain information and, in the process, not to reveal to the witness anything about the proven information.

　　零知識證明並不是數學意義上的證明，因為它存在小概率的誤差，欺騙者有可能通過虛假陳述騙過證明者。換句話來說，零知識證明是概率證明而不是確定性證明。但是也存在有技術能將誤差降低到可以忽略的值。

Zero proof is not mathematical proof, because it has a small probability error, and fraudsters can lie through . In other words, zero knowledge is proof of probability rather than certainty. But there are technologies that can reduce errors to negligible values.

　　零知識的形式定義必須使用一些計算模型，最常見的是圖靈機的計算模型。

Zero knowledge forms must be defined using a number of computational models, most commonly computing models.

　　A要向B證明自己擁有某個房間的鑰匙，假設該房間只能用鑰匙打開鎖，而其他任何方法都打不開。這時有2個方法：

A has to prove to B that he owns the key to a room, assuming that it can only be opened with the key, and no other way can open it. At this point, there are two ways:

　　①A把鑰匙出示給B，B用這把鑰匙打開該房間的鎖，從而證明A擁有該房間的正確的鑰匙。

1A showed the key to B, which used the key to open the lock in the room, thereby proving that A had the right key to the room.

　　②B確定該房間內有某一物體，A用自己擁有的鑰匙打開該房間的門，然後把物體拿出來出示給B，從而證明自己確實擁有該房間的鑰匙。

2B determined that there was an object in the room and A opened the door of the room with the key he owned, then took it out and showed it to B, thereby proving that he did own the key to the room.

　　後面的②方法屬於零知識證明。它的好處在於，在整個證明的過程中，B始終不能看到鑰匙的樣子，從而避免了鑰匙的泄露。

The two methods that follow are proof of zero knowledge. The good thing about it is that, throughout the process of proof, B never sees the key and avoids it.

　　A擁有B的公鑰，A沒有見過B，而B見過A的照片，偶然一天兩個人見面了，B認出了A，但A不能確定面前的人是否是B，這時B要向A證明自己是B，也有2個方法：

A has the B key, A has never seen B, B has seen A's photo, two people meet one day, B recognizes A, but A is not sure if the person in front of him is B, and then B has to prove to A that he is B, and there are two ways:

　　① B把自己的私鑰給A，A用這個私鑰對某個數據加密，然後用B的公鑰解密，如果正確，則證明對方確實是B。

1B gives its private key to A, which encrypts a data message with this key, then decrypts it with the B key and, if correct, proves that it is.

　　② A給出一個隨機值，並使用B的公鑰對其加密，然後將加密後的數據交給B，B用自己的私鑰解密並展示給A，如果與A給出的隨機值相同，則證明對方是B。後面的方法屬於零知識證明。

2A gives a random value and encrypts it using the B key, then delivers the encrypted data to B, which decrypts it with its own private key and shows it to A, and if it is the same as the random value given by A, proves that it is B. The method behind it is a zero-knowledge proof.

　　一天，阿裡巴巴被強盜抓住了，強盜向阿裡巴巴拷問進入山洞的咒語。面對強盜，阿裡巴巴是這麼想的：如果我把咒語告訴了他們，他們就會認為我沒有價值了，就會殺了我省糧食；但如果我死活不說，他們也會認為我沒有價值而殺了我。怎樣才能做到既讓他們確信我知道咒語，但又一丁點咒語內容也不泄露給他們呢？

One day, Alibaba was captured by a bandit, who tortured him for entering the cave. In the face of a bandit, Alibaba thought: "If I told them the curse, they would think that I was worthless, and they would kill me, but if I didn't say it, they would think that I was worthless, and they would kill me. How can they believe that I know the curse, but not a single part of it is leaked to them?"

　　這的確是一個令人糾結的問題，但阿裡巴巴想了一個好辦法，當強盜向他拷問打開山洞石門的咒語時，他對強盜說：“你們在離開我一箭遠的地方，用弓箭指著我，當你們舉起右手我就念咒語打開石門，舉起左手我就念咒語關上石門，如果我做不到或逃跑，你們就用弓箭射死我。”

This is indeed a difficult question, but Aliba thought of a good way, and when the robbers tortured him to open the door of the cave, he said to the robbers, “If you leave me far away and point your arrows at me, then when you lift up your right hand, I will open the door of the stone, and if you lift your left hand, I will close the door of the stone, and if I cannot do it or run away, you will shoot me with an arrow.”

　　強盜們當然會同意，因為這個方案不僅對他們沒有任何損失，而且還能幫助他們搞清楚阿裡巴巴到底是不是真的知道咒語這個問題。阿裡巴巴也沒有損失，因為處於一箭之地的強盜們聽不到他念的咒語，不必擔心泄露了秘密，同時他又確信自己的咒語有效，也不會發生被射死的杯具。