blocksight 2021-04-26 16:16:40 阅读数:25

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non
member
proof
mathematical
rsa

** In the last introduction **** Accumulator and RSA Accumulator****, The accumulator can implement the set member proof , It can also be used as non member certification , This section continues with RSA The nonmember proof part of the accumulator .**

** This paper is based on the above , All the same symbols have the same meaning , It is suggested to read first ！**

** Non member certification （Non-Membership Witness） Is to prove that an element is not in the set . As a general rule , Positive proof （ In the assembly ） Relatively easy , Reverse proof （ Not in the assembly ） It's relatively difficult .**

** principle ：****** Suppose there are three elements in the set , when ,root = mod N**** To prove that an element is not in a set , You need to prove that it's not pi = Of （ plain ） factor , namely pi, Coprime .****** Bezu's theorem (Bézout's identity) It's useful . On bezu's Theorem , It's also mentioned in previous historical articles , Here's a brief introduction ：**

Bezu's theorem ：****ax + by = m (x, y) If and only if there is an integer solution m yes (a, b) Multiple of the greatest common factor of . in other words ：ax + by = 1 ,(x, y) If and only if there is an integer solution (a, b) Coprime . On the contrary, relative to a.b also ！

** that , Our goal is to , Find a set of numbers <a,b> bring a + b pi = 1 You can prove that you are not root It means accumulator Inside .**

** Illustrate with examples ：******* Make =5, = 13, = 17,pi = 5 13 * 17 = 1105, root = mod N**

** Make = 7, Generate 7 Proof that is not in the accumulator set ：******* 158 7 + （-1）*1105 = 1 namely （a = 158, b = -1）**

** prove w =**

** Validation phase ：**

** = g**

** The blockchain system has the feature that data only increases but not decreases , The problem of data storage becomes more and more obvious as time goes by , There are only all nodes in the current blockchain system (Full node) All the data is stored , Light node （Light Client） To verify whether the transaction is legal, we need to check the status of the whole blockchain (State) Understanding , So at present, all consensus and verification are completed by the whole node .**

**Stateless Client early （2013） It is a research and development direction of blockchain , The bitcoin forum is also known as Storageless Client, It's right SPV Light node is an improvement , In short, one doesn't need to store all State, But can participate in the role of transaction verification （ Not like it SPV Light nodes can only be used for probabilistic validity verification ）.**

** Blockchain existing merkle tree There are some problems with the scheme Stateless Client The obstacles . Such as Ethereum , Three merkle state root Record all state, Let's say there's only one left header Stateless client for , We can provide more than one merkle proof To prove to him that many storage Value , Indirectly proving that a transaction is legal . but merkle root did not stateless update Characteristics of , Can't in don't know all state Under the circumstances , Rely on others to provide proof To make the State root Update . in short , Even if you know the deal's legal , It doesn't update state root In order to verify the next transaction .**

** Switch to RSA Accumulator, In theory, you can have stateless client The ability to participate in the verification of transactions ：****** Submit the transaction to a stateless client, Also attach all the State Corresponding proof （ You can also use proof aggregation , Reduce proof size ）.****** When the stateless node receives a request , Take advantage of these witness To verify the transaction . After accepting the deal , Update local accumulator.**

** Theory belongs to theory , Reality is thin ！ In practical engineering , There are many problems in application , For example, the generation of proof increases the amount of calculation , Trade off required .**

** Stateless client maintains a RSA Accumulator. This needs to constantly monitor the transactions of Ethereum or let the whole node broadcast the transaction time , It will be attached with witness, Increased traffic .**

** It adds complexity to the role division of nodes , All nodes may act as Data Provider&witness provider , There are also nodes dedicated to verification , Even specialized stateless client Required for interaction Accumulator node . It's looks good , It needs to be verified step by step ！**

**RSA Accumulator Non member certification , Can carry on if use Accumulator Record a UTXO aggregate , Prove something UTXO There's no such thing as a scene .******** There are other types of accumulators , The application in blockchain is also worth looking forward to ！.**

** Okay , Next, we'll go on to zero knowledge proof ！.**

** Welcome to your attention & Looking at , If you have any questions, please leave a message ！**

** Mathematics in blockchain （ seventy-one )**** Accumulator and RSA Accumulator**

** Mathematics in blockchain （ Sixty-nine )**** Kate Promise volume Certification **

** Mathematics in blockchain （ sixty-seven ）**** Knowledge and commitment **

** Mathematics in blockchain （ sixty-six ）**** Pedersen Key sharing **

** Mathematics in blockchain （ Sixty five ）**** Cryptography promises --Pedersen promise **

** Mathematics in blockchain （ sixty-three ）**** Oblivious transport protocol **

** Mathematics in blockchain （ Twelve ）**** RSA Encryption and decryption algorithm **

** Mathematics in blockchain （ sixty one ）**** BLS m of n Threshold signature **

** Mathematics in blockchain （ fifty-nine ）**** BLS Key aggregation **

**Schnorr Signature and elliptic curve **** Schnorr Signature and elliptic curve **

** Mathematics in blockchain （ Thirty-seven ）**** Uniwap Core algorithm analysis （ in ）**

版权声明：本文为[blocksight]所创，转载请带上原文链接，感谢。 https://netfreeman.com/2021/04/20210426102605272b.html

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