FHE: Put on Harry Potter's Invisibility Cloak

FHE (Fully Homomorphic Encryption) is an advanced encryption technology that allows computation to be performed directly on encrypted data. This means that data can be processed while protecting privacy. FHE has multiple potential application scenarios, especially in fields requiring privacy-preserving data processing and analysis, such as finance, healthcare, cloud computing, machine learning, voting systems, the Internet of Things, and blockchain privacy protection. However, its commercialization still requires time, with the main challenges being the significant computational and memory overhead brought by the algorithms, as well as poor scalability. Next, we will briefly introduce the basic principles of the algorithm and focus on the issues faced by this cryptographic algorithm.

Basic Principles

To perform calculations on encrypted data and obtain the same results, FHE uses polynomials to conceal the original information. Polynomials can be transformed into linear algebra problems or vector computation problems, facilitating highly optimized operations by modern computers, such as parallel computing (.

Taking the encrypted number 2 as an example, in a simplified HE system, it may:

1. Select the key polynomial s)x( = 3x^2 + 2x + 1
2. Generate the random polynomial a)x( = 2x^2 + 5x + 3
3. Generate a small "error" polynomial e)x( = -x + 2 c)x( = 2 + a)x(*s)x( + e)x(

This is done to protect the confidentiality of s)x(. As long as you know s)x( and ignore the small errors in c)x(, you can obtain the plaintext m.

When selecting a polynomial, you need to consider:

• The degree of a polynomial is usually a power of 2, such as 1024/2048.
• Coefficients are randomly selected from the finite field q, such as mod 10000.
• Different plans have different coefficient selection requirements.

Introducing noise e)x( is to confuse attackers, preventing them from inferring the relationship between s)x( and c)x( through repeated inputs of plaintext m. The noise budget )Noise Budget( is an important parameter that determines the number of computations that can be performed.

To represent operations such as c)x( * d)x(, it needs to be transformed into a "circuit." The circuit model can accurately track and manage the noise introduced by each operation, making it easier to accelerate calculations on dedicated hardware such as ASICs and FPGAs in the future. Circuits can be divided into two types: arithmetic circuits and Boolean circuits.

Noise is the main reason that restricts the HE algorithm from expressing arbitrary computations. To address this issue, various solutions have been proposed:

1. LHE: Suitable for executing any function within a specified depth.
2. Key switching: compress ciphertext, but will introduce a small amount of noise
3. Modulus Switching: Reduce noise by decreasing the modulus q
4. Bootstrap: Reset the noise to the original level and maintain the system's computational capacity.

Currently, mainstream FHE schemes use Bootstrap technology, including BGV, BFV, CKKS, TFHE, etc.

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Issues Facing FHE

The main challenge of FHE lies in its enormous computational overhead. Taking AES-128 decryption as an example, the computational time of the FHE version is about 500 million times that of the ordinary version.

To address this challenge, DARPA launched the Dprive program in 2021, aiming to increase the speed of FHE computations to 1/10th of normal computing. The program focuses on several aspects:

1. Increase the processor word length to 1024 bits or larger
2. Build specialized ASIC processors
3. Build MIMD Parallel Architecture

Despite the slow progress, FHE technology still holds unique significance, especially in handling sensitive data. It is particularly applicable to critical sensitive data in sectors such as defense, healthcare, and finance, and becomes even more important in the post-quantum era.

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The Combination of Blockchain

In blockchain, FHE is primarily used to protect data privacy, with application areas including on-chain privacy, AI training data privacy, on-chain voting privacy, and on-chain privacy transaction auditing, among others. FHE is also considered one of the potential on-chain MEV solutions.

However, fully encrypted transactions can also bring some issues, such as the disappearance of positive externalities brought by MEV bots, and validators and builders need to operate in an FHE environment, significantly increasing the operational requirements for nodes and reducing network throughput.

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Main Projects

Most of the current FHE projects use technology from Zama, such as Fhenix, Privasea, Inco Network, and Mind Network. The main difference among these projects lies in their business models:

• Fhenix: Building a Privacy-First Optimism Layer 2
• Privasea: Using FHE for LLM data operations
• Inco Network: Build Layer 1
• Arcium: Integrating FHE, MPC, and ZK technologies
• Mind Network: Choose Restaking Track

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) Zama

Zama is based on the TFHE scheme and uses Bootstrap technology, suitable for handling Boolean operations and low-word-length integer calculations. Its main tasks include:

1. Rewrite TFHE using Rust
2. Develop Concrate tool to convert Python into equivalent code for rs-TFHE.
3. Develop fhEVM, supporting end-to-end encrypted smart contracts based on Solidity compilation.

Zama, as a To B product, has built a relatively complete blockchain + AI development stack based on TFHE.

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) Octra

Octra uses hypergraph technology to implement bootstrap, believing that it can achieve more efficient FHE. Its features include:

1. Build a new smart contract language
2. Develop Hypergraph FHE Library
3. Build the mainnet and subnets architecture
4. Develop a machine learning-based ML-consensus consensus protocol

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Looking Forward

FHE technology is still in its early stages, developing slower than ZK technology. Major challenges include high costs, engineering difficulties, and unclear commercialization prospects. As more funding and attention pour in, it is expected that more FHE projects will emerge.

The implementation of FHE chips is an important prerequisite for commercialization. Currently, several manufacturers such as Intel, Chain Reaction, and Optalysys are exploring this field.

Despite facing technical resistance, FHE, as a highly promising technology with definite demand, could bring profound changes to industries such as defense, finance, and healthcare. With the implementation of FHE chips and the potential to combine with privacy data and future quantum algorithms, FHE is expected to usher in a moment of explosion.

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