# Homomorphic Encryption - A Quick Intro

## Some Definitions

"ὁμός" (homos) => "same"
"μορφή" (morphe) => "shape"

homomorphism => a map/function preserving structure
homomorphic encryption => operations on encrypted data map to same operations on the unencrypted data

#### Weak Analogy - Gold Ring in a Locked Box

## Some Applications

Machine Learning Models
Private Queries
"Outsourced Computation"

## Some Types

### Partially Homomorphic Encryption (PHE)
Only one type of operation and no limit on number of operations

### Somewhat Homomorphic Encryption (SWE or SWHE)
More than one type of operation, but limited number of operations

### Fully Homomorphic Encryption (FHE)
All types of operation and no limit on number of operations

## Some History


## Some Cryptography

Symmetric vs Public Key

Circuits - AND and XOR

All functions can be reduced to a series of additions and multiplications - AND and XOR

## Some Math

Fun with Abstract Algebra / Number Theory

Groups => Rings => Fields

- Closure
- Associativity
- Identity
- Inverse

Abelian Group add Commutativity

### Hard Problems

Factorization - RSA
Discrete Logarithm - ElGamal
Elliptic Curve - Bitcoin, Ethereum BLS
Lattices - FHE (post quantum)

Shortest Vector Problem (SVP)
Closest Vector Problem (CVP)
Short Integer Solution Problem (SIS)
Learning With Errors Problem (LWE)

## Some Details

### Noise

Makes the encryption secure
Too much noise prevent decryption
Operations add noise - exponentially

### Bootstrapping

Reset the noise


### Some Implementations

https://homomorphicencryption.org
https://fhe.org

Implementation schemes (BGV, BFV, CKKS, etc.) focus on particular classes of computation support

https://github.com/Microsoft/SEAL (`brew install seal`)

https://github.com/openfheorg/openfhe-development

https://github.com/tuneinsight/lattigo

## Some Alternatives

### MPC

Multi Party Computation

### TEE

Trusted Execution Envirnment
SGX (Intel) and other "Secure Enclave" solutions