# A Universal Privacy-Preserving Multi-Blockchain Aggregated Identity Scheme
## 3 System Design
### Management of the Authorized Addresses
#### 3.4.2 Construction of the Accumulator
1: if (\(n==2\bigm{|}\bigm{|}n==3\)) 2: return true; 3: if (\(n\pmod{2}==0\bigm{|}\bigm{|}n<2\)) 4: return false; // Lines 5-9 compute \(d\) and \(s\), such that \(n-1=2^{s}*d\), where \(d\) is odd. This is done so that we can write \(n\)-1 as a power of 2 times an odd number. 5: \(s=0\), \(d=n-1\); 6: while (\(d\pmod{2}==0\)) {
7:\(d=d/2\); 8:\(++s\); 9:} //Line 10 initializes an array of base values to test. These are predefined values that have been shown to be effective in practice. 10: \(bases=[2,325,9375,28,178,450,775,9,780,504,17,955,265,022]\); //The following loop (lines 11-22) is the main body of the test. It checks if \(n\) is prime. The outer loop (// line 11) repeats \(k\) times, using a different base value each time. 11: WitnessLoop: do{ //Line 12 selects the base value to use for this iteration of the test. 12:\(a=bases[k-1]\); 13:\(//\) if \(n\) is prime, \(n\) must satisfy A or B // Lines 14-15 handle the case A. If this is true, skip to the next iteration of the outer loop. 14: if (\(a^{d}\pmod{n}==1\))//A 15: continue; // Lines 16-19 handle the case B. If this is true, skip to the next iteration of the outer loop. 16: for(\(j=0\) to \(s-1\)) do (\(//\) B 17: if (\(a^{2j}d\pmod{n}==

## 4 Analysis and Discussion

In this scheme, the identity provider only has information about each user's identity, but does not know which wallet account is authorized for that user. On the other hand, the VASP (virtual asset service provider) only knows the user's wallet account, but does not know the user's real-world identity. Therefore, neither party can link the user's identity to their wallet account, allowing the wallet to remain anonymous. An identity NFT contains information such as the hash of the user's identity data and the scope of business in which the user is allowed to participate. These data are stored on IPFS which ensures that the identity data cannot be tampered with or denied. The NFT does not contain sensitive information, protecting the user's privacy. The user cannot transfer their identity NFT. If the private key of the user's original admin address is lost, the user can simply contact the identity provider for brief authentication and designate a new admin address. If the user believes that their identity may be used fraudulently, they can contact the identity provider and use the contract to lock their identity. If the private key of a wallet account is lost or stolen, the user can revoke the authorization of that wallet account by updating the accumulator value in the contract data. VASPs include NFT markets, OTC (over-the-counter) platforms, etc. By requiring users to show proof that their wallet accounts for deposits and withdrawals on the OTC platform have been bound to an identity NFT, the conversion of criminal funds between fiat and cryptocurrencies can be reduced. By requiring virtual asset service providers to verify the identity of users associated with wallet accounts, we can reduce the occurrence of crimes such as fraud and protect the rights and interests of investors. This requirement also helps to prevent money laundering, as criminals would need to use verified accounts to receive funds. By verifying the identities of users, we can ensure the security and integrity of the virtual asset ecosystem. Regulatory agencies and criminal investigation agencies, acting as regulators, can access the user's identity data held by the identity provider and the user's wallet account held by the VASP to link real identities with wallet accounts for the purpose of sanctions or criminal investigations. This helps to ensure the security and integrity of the virtual asset ecosystem. Note that the regulator must provide evidence of malicious behavior in the account to be investigated before the identity provider grants access to its corresponding identity data. This approach ensures that innocent users are not subject to investigations by regulatory agencies.