What is Fully Homomorphic Encryption

The definition and basic principle of Fully Homomorphic Encryption (FHE)

Fully Homomorphic Encryption (FHE) is a special encryption technique that allows arbitrary calculations to be performed on encrypted data without decrypting the data, and the results of the calculations are exactly the same as those of plaintext data. This property makes full homomorphic encryption have important application value in data privacy protection and secure computing.
$$\text{Dec}(f(\text{Enc}(m_1), \text{Enc}(m_2), \ldots, \text{Enc}(m_k))) = f(m_1, m_2, \ldots, m_k)$$

Definition

FHE is an encryption algorithm that satisfies the following conditions:

1. Additive homomorphism：For any two ciphertexts C1 and C2, the result of calculating C1+C2 can correspond to the plaintext f(A)+f(B)f(A)+f(B).
2. Multiplicative homomorphism：For any two ciphertexts C1 and C2, the result of calculating C1×C2 can correspond to the plaintext f(A)×f(B)f(A)× f*(B).

If a encryption algorithm satisfies both additive homomorphism and multiplicative homomorphism, it is called a fully homomorphic encryption algorithm

Technical Points

1. Support arbitrary calculation : Through the combination of addition and multiplication, it can theoretically construct arbitrary functions, such as logical operations, statistical analysis, etc.
2. Key generation and Noise management : Generate key pairs (public key encryption, private key decryption), and introduce noise mechanism to ensure security; The accumulation of noise with the operation needs to be reset by “Bootstrapping” technology.
3. Difference from partial homomorphic encryption : Partial homomorphism supports only a single operation (e.g. RSA supports multiplication only), while FHE supports infinite addition and multiplication.

Development course

The concept of holomorphic encryption was first proposed by Rivest, Adleman, and Dertouzos in 1978, but it was not until 2009 that Craig Gentry first proposed a holomorphic encryption scheme based on ideal lattices. Since then, Brakerski et al. have further optimized the scheme to make it more efficient and practical

Advantages of full homomorphic encryption

1. Privacy Protection and Data security

   • Data is always stored and transmitted in ciphertext and cannot be obtained in plain text even when a third party processes data. This method applies to sensitive scenarios such as medical and financial services.
   • In the blockchain, private information such as transaction amount and smart contract logic can be protected.

2. Flexible data processing capability

   • Supports ciphertext retrieval, statistical analysis and machine learning model training without decrypting data. For example, the cloud can perform diagnostic analysis directly on encrypted medical records.
   • Reduce multi-round interaction: Traditional multi-party computation requires multiple interactive decryption, FHE only needs a single encryption and result return, reducing communication overhead.

3. Compatible with existing computing architectures

   • It can be combined with cloud computing and AI technology to use third-party computing resources to process encrypted data and improve data utilization efficiency.

4. Potential against quantum attack

   • FHE schemes based on lattice cryptography, such as the LWE problem, are theoretically safe for quantum computers.

   In short, in the medical field, this technology is working wonders. A collaborative study at Boston Children’s Hospital last year allowed five hospitals to train more accurate disease prediction models without sharing raw patient data. Through full-homomorphic encryption technology, the encrypted data of each hospital is like a transparent building block, which can be combined and built without exposing the internal structure.

   The financial industry has also found a new key. Jp Morgan recently tested the Joint Credit rating system, which allows banks to complete credit scores across institutions while customer data is encrypted at all times. This protects customer privacy and breaks down data silos, resulting in a 40% increase in risk assessment accuracy.

Disadvantages and challenges

1. High computational overhead

   • Ciphertext expansion problem: 1 bit of plain text may be expanded to thousands of bits after encryption, and the size of the public key reaches GB.
   • Low efficiency: simple multiplication operations take millions of times longer than plaintext operations, and complex operations take hours or even days.

2. Noise management and accuracy limitation

   • Each operation produces noise and requires frequent Bootstrapping, with a delay 3-4 orders of magnitude higher than normal multiplication.
   • Only addition and multiplication are supported, and square root and logical judgment need to be approximated, resulting in loss of precision.

3. Standardization and practicality bottleneck

   • Lack of a unified standard, different schemes (such as BGV, CKKS) different application scenarios, has not been large-scale commercial.
   • Multi-user support is difficult: the key management and anti-collusion mechanism is complex, and it is difficult to deal with massive user data.

   Despite its strong encryption capabilities, the technology is not flawless. Just as early mobile phones needed a charging bank to carry around, the current full-homomorphic encryption consumes a lot of computing resources. Processing a simple encrypted data query can take up to 1,000 times longer than traditional methods. The optimization plan announced by IBM Research last year successfully increased the computing speed by 80 times, but it is still far from practical application.

   Another challenge comes from lack of standardization. At present, there are more than 20 different implementation schemes in the world, just like the mobile phone charging interface is not unified, which brings many obstacles to practical applications. According to an industry survey released earlier this year by the European Union’s Cyber Security Agency, 75% of companies are hesitant to adopt the technology, with the lack of uniform standards the main concern.

Conclusion

Full homomorphic encryption is a revolutionary technology in the field of privacy computing, and although it is currently limited by efficiency issues, its potential for data security, cross-institutional collaboration, and resistance to quantum attacks is irreplaceable. With algorithm optimization, hardware acceleration and ecological construction, it is expected to achieve large-scale application in cloud computing, AI and blockchain in the future, and become one of the infrastructure in the era of data economy.