# Homomorphic Encryption

May 20, 2023

Homomorphic encryption is a cryptographic technique that allows computation to be performed on ciphertext, while preserving the confidentiality of the data. In other words, homomorphic encryption enables data to be processed without ever revealing its contents. This makes it an attractive solution for applications such as cloud computing, where data is outsourced to third-party servers for processing.

The main purpose of homomorphic encryption is to enable computation on encrypted data, without the need to decrypt it first. This is particularly useful in scenarios where sensitive data needs to be processed by third-party servers, but the data owner does not want to reveal the data to the server. Homomorphic encryption can also be used to perform secure computations on data that is stored in untrusted environments.

Some common use cases for homomorphic encryption include:

- Secure computation on data stored in the cloud
- Secure data sharing between multiple parties
- Privacy-preserving machine learning
- Secure computation on data stored on mobile devices

## Brief History and Development

The concept of homomorphic encryption was first introduced by Rivest, Adleman, and Dertouzos in 1978, but it was not until the mid-2000s that practical homomorphic encryption schemes were proposed. The first fully homomorphic encryption (FHE) scheme was proposed by Craig Gentry in 2009. Since then, several other FHE schemes have been proposed, each with different trade-offs between security and efficiency.

## Key Concepts and Principles

A homomorphic encryption scheme consists of three algorithms:

- Key generation: A key pair is generated consisting of a public key and a private key. The public key is used to encrypt data, while the private key is used to decrypt the data.
- Encryption: Data is encrypted using the public key, resulting in ciphertext.
- Decryption: Ciphertext is decrypted using the private key, resulting in plaintext.

Homomorphic encryption can be either partially homomorphic or fully homomorphic.

- Partially homomorphic encryption allows only one type of operation to be performed on ciphertext. For example, a partially homomorphic encryption scheme might allow addition to be performed on ciphertext, but not multiplication.
- Fully homomorphic encryption allows arbitrary computations to be performed on ciphertext.

In order for homomorphic encryption to be practical, it must be both secure and efficient. There are several security properties that a homomorphic encryption scheme must satisfy:

- Computational indistinguishability: It should be computationally infeasible to distinguish between two ciphertexts that encrypt different plaintexts.
- Semantic security: It should be computationally infeasible to recover the plaintext from the ciphertext, even if an attacker has access to multiple ciphertexts.
- Correctness: The decryption algorithm should correctly recover the plaintext from the ciphertext.

## Pseudocode and Implementation Details

The exact details of a homomorphic encryption scheme can be quite complex, but here is a high-level overview of the encryption and decryption algorithms for a partially homomorphic encryption scheme that supports addition:

```
# Key Generation
public_key, private_key = generate_key_pair()
# Encryption
plaintext = "Hello World"
ciphertext = encrypt(public_key, plaintext)
# Decryption
decrypted_text = decrypt(private_key, ciphertext)
# Homomorphic Addition
ciphertext1 = encrypt(public_key, "10")
ciphertext2 = encrypt(public_key, "20")
sum_ciphertext = add(ciphertext1, ciphertext2)
```

In this example, `generate_key_pair()`

generates a public key and a private key, `encrypt()`

encrypts a plaintext using the public key, `decrypt()`

decrypts a ciphertext using the private key, and `add()`

performs homomorphic addition on two ciphertexts.

## Examples and Use Cases

Homomorphic encryption has numerous potential use cases, some of which include:

- Secure computation on data stored in the cloud: By allowing computations to be performed on encrypted data, homomorphic encryption can enable secure computation on data that is stored in the cloud, without the need to decrypt the data first.
- Privacy-preserving machine learning: Homomorphic encryption can be used to enable machine learning algorithms to be performed on encrypted data, without revealing the data to the machine learning model. This has several potential applications, such as enabling healthcare data to be used for research without compromising patient privacy.
- Secure data sharing between multiple parties: Homomorphic encryption can be used to enable multiple parties to perform computations on encrypted data, without the need to reveal the data to each other.
- Secure computation on data stored on mobile devices: Homomorphic encryption can be used to enable computations to be performed on data that is stored on mobile devices, without the need to reveal the data to third-party servers.

## Advantages and Disadvantages

Homomorphic encryption has several advantages, including:

- Privacy: Homomorphic encryption enables computations to be performed on encrypted data, without the need to decrypt the data first. This can help to protect the privacy of sensitive data.
- Security: Homomorphic encryption can enable secure computation on data that is stored in untrusted environments, such as the cloud or mobile devices.
- Flexibility: Homomorphic encryption can be used to enable a wide range of computations to be performed on encrypted data, without the need to reveal the data to third-party servers.

However, homomorphic encryption also has several disadvantages, including:

- Performance: Homomorphic encryption is computationally intensive, which can make it impractical for some use cases.
- Complexity: Homomorphic encryption is a complex cryptographic technique that requires specialized knowledge to implement and use.
- Trade-offs: There is a trade-off between security and efficiency in homomorphic encryption schemes. Some schemes may be more secure but less efficient, while others may be less secure but more efficient.

## Related Algorithms or Variations

Homomorphic encryption has several variations, including:

- Partially homomorphic encryption: A homomorphic encryption scheme that supports only one type of operation, such as addition or multiplication.
- Fully homomorphic encryption: A homomorphic encryption scheme that supports arbitrary computations to be performed on ciphertext.
- Somewhat homomorphic encryption: A homomorphic encryption scheme that supports a limited set of computations to be performed on ciphertext, but not arbitrary computations.
- Multiparty homomorphic encryption: A homomorphic encryption scheme that enables multiple parties to perform computations on encrypted data, without the need to reveal the data to each other.