Fully Homomorphic Encryption (FHE) Schemes

• FHE allows for arbitrary computation over encrypted data.
• In a FHE scheme, the relationship between encryption and decryption is "homomorphic."
• This means that we can do something like: Encrypt(3xy + x) = Encrypt(3xy) + Encrypt(x) = [3 * Encrypt(x) * Encrypt(y)] + Encrypt(x)
• This allows us to do complex mathematical operations on encrypted data.
• FHE schemes are "homomorphic" because there is a special relationship between encryption and decryption.
• FHE is a type of cryptography that uses a specific type of post-quantum cryptography called lattice cryptography.
• Lattice cryptography sits inside the real vector space and can be represented using vectors and matrices.
• FHE schemes can be broken down into 3 types depending on how they model computation (see this presentation if interested in more advanced technical details).
• The first type models computations as boolean circuits (i.e. bits).
• The second type models computation as modular arithmetic (i.e. "clock" arithmetic).
• The third and final type models computations as floating point arithmetic.
• Private Transactions on a Public Ledger (e.g. blockchain) can be implemented using an FHE scheme that models computation as modular arithmetic.
• Private machine learning is becoming more popular as people try to learn about their data without sharing it with others.
• There are a variety of different schemes that need to be understood in order to use them successfully.
• Having a lot of choices is great, but it can be difficult to choose the right scheme for a particular use case.
• Currently, there is little research explaining the relationship between different schemes.
• FHE schemes can be categorized into 3 types based on the computational model used: boolean, modular arithmetic, and floating point arithmetic.
• Each type of FHE scheme has its own tradeoffs to consider when choosing between them.
• Performance can be improved by choosing a scheme that fits the needs of the application.
• The size of the plaintext space and the number of operations that will be performed are important factors to consider.
• FHE schemes that model computation as modular arithmetic can take advantage of "SIMD" style operations to perform the same operation on multiple plaintexts or ciphertexts simultaneously.

Test your knowledge about Fully Homomorphic Encryption (FHE) schemes, which enable arbitrary computation over encrypted data. Explore the different types of FHE schemes and their relationships with encryption and decryption, as well as their applications in private transactions and machine learning.