The Beginner's Textbook for Fully Homomorphic Encryption

ArXiv's PDF Link: https://arxiv.org/abs/2503.05136

Dynamic Website Version: https://fhetextbook.github.io

• Please post any bugs or errors regarding the draft to the Issues board or create a pull request.

Python FHE Demo Library Quickstart

We implemented this library for educational purposes. Because it is written in pure Python and does not use an RNS-optimized backend, computations are relatively slow. For this reason, the default parameters are intentionally small—for example, the ring dimension / polynomial modulus degree ($N$) and ciphertext modulus ($Q$) are set to small values (e.g., $N=4$, $Q = 2^{30}$).

In the source code, the global variable N denotes the polynomial modulus degree (i.e., computations are performed modulo $x^N + 1$. The variable Q denotes the ciphertext modulus (or modulus chain, depending on the scheme configuration) for BFV and TFHE. For CKKS and BGV (when using a modulus chain), Q_BIT and Q_LEVEL control the per-prime bit-size and the number of levels in the modulus chain.

You may adjust these globals to experiment with larger parameters. However, when increasing N, you should also increase the ciphertext modulus (Q or Q_BIT, and potentially Q_LEVEL) and, when selecting NTT-friendly primes, increase the max_span used in pick_ntt_primes. Otherwise, ciphertext noise may overflow the available modulus and decryption results can become incorrect.

Installation

cd 'source code'
pip3 install numpy fastcore sympy

TFHE Library

usage: tfhe_example.py [-h] [--encrypt] [--add-cipher-cipher] [--add-cipher-plain] [--mult-cipher-cipher] [--mult-cipher-plain] \
[--keyswitch] [--random] [--all]
options:
-h, --help show this help message and exit
--random A bulk of random tests
--encrypt Encrytion/decryption test
--add-cipher-cipher Cipher-cipher addition test
--add-cipher-plain Cipher-plain addition test
--mult-cipher-cipher Cipher-cipher multiplication test
--mult-cipher-plain Cipher-plain multiplication test
--keyswitch Key switch test
--all All test

Test Operations	Command
Encrypt	python3 tfhe_example.py --encryption
Add Cipher-Cipher	python3 tfhe_example.py --add-cipher-cipher
Add Cipher-Plain	python3 tfhe_example.py --add-cipher-plain
Multiply Cipher-Plain	python3 tfhe_example.py --mult-cipher-plain
Multiply Cipher-Cipher	python3 tfhe_example.py --mult-cipher-cipher
Key Switching	python3 tfhe_example.py --keyswitch
Random 1000 'Encrypt' Tests	python3 tfhe_example.py --random --encrypt
Random 1000 'Rotate' Tests	python3 tfhe_example.py --random --rotate
Random 1000 Any Tests	python3 tfhe_example.py --all

BFV Library

usage: bfv_example.py [-h] [--encode] [--encrypt] [--add-cipher-cipher] [--add-cipher-plain] [--mult-cipher-cipher] [--mult-cipher-plain] \
[--rotate] [--keyswitch] [--random] [--all]
options:
-h, --help show this help message and exit
--encode Encoding test
--random A bulk of random tests
--encrypt Encrytion/decryption test
--add-cipher-cipher Cipher-cipher addition test
--add-cipher-plain Cipher-plain addition test
--mult-cipher-cipher Cipher-cipher multiplication test
--mult-cipher-plain Cipher-plain multiplication test
--rotate Rotation test
--keyswitch Key switch test
--all All test

Test Operations	Command
Encode	python3 bfv_example.py --encode
Encrypt	python3 bfv_example.py --encryption
Add Cipher-Cipher	python3 bfv_example.py --add-cipher-cipher
Add Cipher-Plain	python3 bfv_example.py --add-cipher-plain
Multiply Cipher-Plain	python3 bfv_example.py --mult-cipher-plain
Multiply Cipher-Cipher	python3 bfv_example.py --mult-cipher-cipher
Key Switching	python3 bfv_example.py --keyswitch
Rotate	python3 bfv_example.py --rotate
Random 1000 'Encrypt' Tests	python3 bfv_example.py --random --encrypt
Random 1000 'Rotate' Tests	python3 bfv_example.py --random --rotate
Random 1000 Any Tests	python3 bfv_example.py --all

CKKS Library

usage: ckks_example.py [-h] [--encode] [--encrypt] [--add-cipher-cipher] [--add-cipher-plain] [--mult-cipher-cipher] [--mult-cipher-plain] \
[--rotate] [--conjugate] [--keyswitch] [--random] [--all]
options:
-h, --help show this help message and exit
--encode Encoding test
--random A bulk of random tests
--encrypt Encrytion/decryption test
--add-cipher-cipher Cipher-cipher addition test
--add-cipher-plain Cipher-plain addition test
--mult-cipher-cipher Cipher-cipher multiplication test
--mult-cipher-plain Cipher-plain multiplication test
--rotate Rotation test
--conjugate Conjugation test
--keyswitch Key switch test
--all All test

Test Operations	Command
Encode	python3 ckks_example.py --encode
Encrypt	python3 ckks_example.py --encryption
Add Cipher-Cipher	python3 ckks_example.py --add-cipher-cipher
Add Cipher-Plain	python3 ckks_example.py --add-cipher-plain
Multiply Cipher-Plain	python3 ckks_example.py --mult-cipher-plain
Multiply Cipher-Cipher	python3 ckks_example.py --mult-cipher-cipher
Key Switching	python3 ckks_example.py --keyswitch
Rotate	python3 ckks_example.py --rotate
Conjugate	python3 ckks_example.py --conjugate
Random 1000 'Encrypt' Tests	python3 ckks_example.py --random --encrypt
Random 1000 'Rotate' Tests	python3 ckks_example.py --random --rotate
Random 1000 Any Tests	python3 ckks_example.py --all

BGV Library

usage: ckks_example.py [-h] [--encode] [--encrypt] [--add-cipher-cipher] [--add-cipher-plain] [--mult-cipher-cipher] [--mult-cipher-plain] \
[--rotate] [--keyswitch] [--random] [--all]
options:
-h, --help show this help message and exit
--encode Encoding test
--random A bulk of random tests
--encrypt Encrytion/decryption test
--add-cipher-cipher Cipher-cipher addition test
--add-cipher-plain Cipher-plain addition test
--mult-cipher-cipher Cipher-cipher multiplication test
--mult-cipher-plain Cipher-plain multiplication test
--rotate Rotation test
--keyswitch Key switch test
--all All test

Test Operations	Command
Encode	python3 bgv_example.py --encoding
Encrypt	python3 bgv_example.py --encryption
Add Cipher-Cipher	python3 bgv_example.py --add-cipher-cipher
Add Cipher-Plain	python3 bgv_example.py --add-cipher-plain
Multiply Cipher-Plain	python3 bgv_example.py --mult-cipher-plain
Multiply Cipher-Cipher	python3 bgv_example.py --mult-cipher-cipher
Key Switching	python3 bgv_example.py --keyswitch
Rotate	python3 bgv_example.py --rotate
Random 1000 'Encrypt' Tests	python3 bgv_example.py --random --encrypt
Random 1000 'Rotate' Tests	python3 bgv_example.py --random --rotate
Random 1000 Any Tests	python3 bgv_example.py --all

Community Implementations

• For interested readers, a BFV-RNS implementation in SystemVerilog is available at https://github.com/alibillalhammoud/FHEforEEs.