FHE: Putting on Harry Potter's Invisibility Cloak

Fully Homomorphic Encryption (FHE) ( is an advanced encryption technology that allows for computation and processing of data while it is still encrypted. This means that data can be analyzed and 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, commercialization still requires some time, primarily due to the significant computational and memory overhead introduced by its algorithms, resulting in poor scalability. Below, we will briefly introduce the basic principles of this algorithm and the main issues it faces.

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Basic Principles

The fundamental goal of FHE is to perform computations on encrypted data and obtain results that are the same as those from plaintext computations. In cryptography, polynomials are commonly used to obscure the original information, as they can be transformed into linear algebra problems, making it easier for modern computers to perform highly optimized vector calculations.

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

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

The noise e)x( is introduced here to confuse attackers and prevent them from analyzing the relationship between s)x( and c)x( through repeated plaintext inputs. The magnitude of the noise is also referred to as the noise budget.

Transforming c)x( * d)x( and similar operations into "circuits" allows for precise tracking and management of the noise introduced by each operation, making it easier to accelerate calculations on specialized hardware such as ASICs and FPGAs in the future. Any complex operation can be mapped to simple addition and multiplication modules.

However, as the computational depth increases, the noise will grow exponentially, ultimately leading to an inability to recover the original text. To address this issue, the following solutions have been proposed:

• Key switching: Compress the ciphertext after each multiplication, but it will introduce a small amount of noise.
• Modulus Switching: Reducing the modulus q to decrease noise, but it will compress computing power.
• Bootstrap: Reset the noise to the original level, without reducing the modulus, but with a high computational cost.

The current FHE schemes mainly include:

• BGV: Based on RLWE, supports arbitrary depth circuits
• BFV: Based on RLWE, suitable for arithmetic operations
• TFHE: Based on LWE/TLWE, suitable for Boolean circuits
• CKKS: Based on RLWE, supports approximate arithmetic

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Issues Facing FHE

Due to the need to encrypt data and transform it into "circuits", and then introduce technologies such as Bootstrap to address noise issues, the computational overhead of FHE is several orders of magnitude higher than that of ordinary computation.

Taking AES-128 decryption as an example, the standard version requires about 67 nanoseconds on a 3 GHz processor, while the FHE version requires 35 seconds, which is about 500 million times longer than the standard version.

To address this issue, the US DARPA launched the Dprive program in 2021, aiming to increase the speed of FHE computations to one-tenth of that of ordinary computing. The focus is mainly on the following aspects:

1. Increase the processor word length to 1024 bits or larger to support larger modulus q.
2. Build specialized ASIC processors to run FHE algorithms
3. Adopting MIMD parallel architecture, supporting parallel processing of data with different instructions.

Although progress is slow, FHE technology still holds significant importance for protecting the privacy of sensitive data in the long term, especially in the post-quantum era.

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The Integration of Blockchain

In blockchain, FHE is mainly used to protect data privacy, with applications including on-chain privacy, AI training data privacy, on-chain voting privacy, and on-chain privacy transaction auditing. FHE is also seen as one of the potential solutions to the on-chain MEV problem.

However, fully encrypted transactions also bring some issues, such as the disappearance of positive externalities brought by MEV bots, validators need to run on the FHE virtual machine, significantly increasing node requirements 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, Mind Network, etc. These projects are built based on the libraries provided by Zama, with the main difference being the business models.

) Zama

Zama is based on the TFHE scheme, has rewritten TFHE using Rust, and provides a Python transpiler tool called Concrate. Its fhEVM product supports compiling end-to-end encrypted smart contracts on the EVM. Zama offers a well-developed FHE development stack for web3 projects.

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) Octra

Octra uses original technology based on hypergraphs to achieve FHE. It has built a new smart contract language and a machine learning-based ML-consensus protocol. Octra adopts a mainnet + subnet architecture design.

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Looking forward

FHE technology is still in its early stages, facing challenges such as high costs, engineering difficulties, and uncertain commercialization prospects. However, with more funding and attention pouring in, as well as the implementation of FHE-specific chips, this technology is expected to bring profound changes in fields such as defense, finance, and healthcare. Although its current application range is limited, FHE, as a highly promising technology, is still worth continuous attention and exploration in the future.

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