ElixirPHE

ElixirPHE is a pure Elixir library for experimenting with homomorphic encryption on integer vectors (also supports floating-point numbers with fixed-point scaling). It offers a single, consistent API over multiple cryptosystems, with clear separation between additive and multiplicative homomorphism. The project is designed to stay on the BEAM runtime only: no NIF-based custom crypto code is required.

Different homomorphic schemes support different algebraic operations. In practice, that means your encrypted workflow depends on the scheme:

• additive schemes let you add encrypted values
• multiplicative schemes let you multiply encrypted values

ElixirPHE wraps this in one interface and rejects unsupported operations at runtime with explicit errors. A notable potential use of the library is to implement a private vector store, where homomorphically encrypted vectors can still undergo computation to yield encrypted similarity scores.

Supported cryptosystems

Scheme	Type	Supported homomorphic operations	Notes
:paillier	Additive	ciphertext addition, plaintext scalar multiplication	:damgard_jurik when s = 1
:damgard_jurik	Additive	ciphertext addition, plaintext scalar multiplication	s >= 1
:okamoto_uchiyama	Additive	ciphertext addition, plaintext scalar multiplication	Plaintexts in Z_p
:rsa	Multiplicative	ciphertext multiplication, plaintext exponentiation	Textbook RSA, for demonstration
:elgamal (:el_gamal)	Multiplicative	ciphertext multiplication, plaintext exponentiation	ElGamal over Z_p

Additive homomorphism

• :damgard_jurik (s >= 1, Paillier-compatible when s = 1)
• :okamoto_uchiyama

Multiplicative homomorphism

• :rsa (textbook RSA, for homomorphic demonstration)
• :elgamal (alias: :el_gamal)

Data model and Nx support

Plaintext vectors can be provided as either:

• numeric lists (integers or reals)
• rank-1 Nx tensors (integer or floating-point)

Real values are encoded using fixed-point scaling before encryption. Use the precision: option on ElixirPHE.encrypt_vector/3 to control decimal digits (default: 6).

Decryption can return:

• list output (default)
• tensor output via as: :tensor

Core API

Common methods:

• ElixirPHE.keygen/1
• ElixirPHE.encrypt_vector/2
• ElixirPHE.encrypt_vector/3
• ElixirPHE.decrypt_vector/2
• ElixirPHE.decrypt_vector/3

Additive operations:

• ElixirPHE.add_encrypted_vectors/2
• ElixirPHE.scalar_multiply_encrypted_vector/2

Multiplicative operations:

• ElixirPHE.multiply_encrypted_vectors/2
• ElixirPHE.power_encrypted_vector/2

For examples, refer to example/usage.exs.

Notes on crypto backend

RSA and ElGamal are backed by official OTP crypto/public_key primitives. Damgard-Jurik and Okamoto-Uchiyama do not have direct OTP scheme primitives, so scheme logic is implemented in this library while core arithmetic and randomness rely on official :crypto operations.

Important caveats

• RSA here is textbook RSA and intentionally uses raw mode to preserve multiplicative homomorphism.
• Homomorphic arithmetic is always modular and bounded by each scheme's plaintext space.
• This project is aimed at learning, prototyping, and controlled experiments.