Vitalik's article on Fully Homomorphic Encryption (FHE) has once again sparked everyone's exploration and imagination of new encryption technologies. In my opinion, FHE's potential for fully homomorphic encryption is indeed a step up from ZKP technology, and it can help to enable more applications for AI+Crypto. How should we understand this?

1) Definition: FHE enables operations on encrypted data of specific forms without the concern of exposing the data and privacy. In contrast, ZKP can only address the consistency of data transmission in an encrypted state, where the receiving party can only verify that the submitted data from the sending party is authentic, serving as a point-to-point encryption scheme. Fully homomorphic encryption does not limit the range of entities performing operations, and can thus be considered a many-to-many encryption operation scheme.

2) How it works: Traditional computer operations are performed on plaintext data, and if the data is encrypted, it needs to be decrypted before computation, which inevitably exposes private data. Homomorphic encryption constructs a special encryption scheme that allows for "homomorphic" transformations of ciphertext, ensuring that the results of operations on ciphertext remain consistent with those on plaintext. In a homomorphic encryption system, plaintext addition corresponds to ciphertext multiplication (a rule), so performing addition on plaintext data only requires multiplication on ciphertext (equivalence).

In summary, homomorphic encryption uses special homomorphic transformations to enable operations on encrypted data to yield results identical to those on plaintext, as long as the homomorphic properties of the operation rules are preserved.

3) Application scenarios: In traditional internet domains, FHE's fully homomorphic encryption can be widely applied in areas such as cloud storage, biometric recognition, healthcare, finance, advertising, genetic sequencing, and many others.

Taking biometric recognition as an example, personal biometric data such as fingerprints, irises, and facial features are sensitive data, and FHE technology can be used to compare and verify this data in encrypted form on servers. Similarly, in the healthcare field, years of fragmented data can be unified using FHE, allowing different healthcare entities to conduct joint analysis and modeling without sharing original data.

In the Crypto domain, FHE's application space can also encompass gaming, DAO voting governance, MEV protection, private transactions, regulatory compliance, and other scenarios requiring strong privacy. For instance, in gaming scenarios, platforms can drive games without peeking at players' cards, making the games fairer. Similarly, in DAO voting, whales participating in governance can generate voting results through computation without exposing addresses and voting quantities. Additionally, users can submit encrypted transactions to the Mempool, thereby avoiding the exposure of private information such as target addresses and transfer amounts. In regulatory scenarios, governments can monitor fund pools and separate assets from illicit addresses without viewing the privacy data of legitimate transactions.

4) Shortcomings: It is worth noting that the computing environment for conventional plaintext operations is often complex, involving not only addition, subtraction, multiplication, and division, but also conditional loops, logical gate judgments, and more. Currently, partially homomorphic encryption and fully homomorphic encryption can only rapidly perform addition and multiplication, and more complex operations require combination and superposition, leading to increased computational power requirements.

Therefore, in theory, fully homomorphic encryption can support arbitrary computations, but due to performance bottlenecks and algorithm characteristics, the types and complexity of homomorphic computations that can be efficiently executed are very limited. Complex operations will require significant computational power consumption. Thus, the practical implementation of fully homomorphic encryption is actually a process of algorithm optimization and control of computational power costs, especially focusing on performance after hardware acceleration and computational power enhancement.

In my view, although it may be difficult for fully homomorphic encryption to mature and be applied in the short term, as an extension and complement to ZKP technology, it can provide significant value for AI large model privacy computation, AI data joint modeling, AI collaborative training, Crypto private compliance transactions, and the extension of Crypto scenarios.

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