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Implementation of the Paillier cryptosystem

Project description

TNO MPC Lab - Encryption Schemes - Paillier

The TNO MPC lab consists of generic software components, procedures, and functionalities developed and maintained on a regular basis to facilitate and aid in the development of MPC solutions. The lab is a cross-project initiative allowing us to integrate and reuse previously developed MPC functionalities to boost the development of new protocols and solutions.

The package tno.mpc.encryption_schemes.paillier is part of the TNO Python Toolbox.

Implementation of the Paillier encryption scheme. This library was designed to facilitate both developers that are new to cryptography and developers that are more familiar with cryptography.


  • Positive and negative numbers, as well as fixed point encoding of numbers.
  • Homomorphic addition of ciphertexts, negation of ciphertexts, and multiplication of ciphertexts with integral scalars.
  • Precomputation of the randomness needed for refreshing / generating fresh encryptions.
  • Custom class of warnings for making cryptographic suggestions to the developer.

Limitations in (end-)use: the content of this software package may solely be used for applications that comply with international export control laws.
This implementation of cryptographic software has not been audited. Use at your own risk.


Documentation of the tno.mpc.encryption_schemes.paillier package can be found here.


Easily install the tno.mpc.encryption_schemes.paillier package using pip:

$ python -m pip install tno.mpc.encryption_schemes.paillier


A significant performance improvement can be achieved by installing the GMPY2 library.

$ python -m pip install 'tno.mpc.encryption_schemes.paillier[gmpy]'

If you wish to use the tno.mpc.communication module you can use:

$ python -m pip install 'tno.mpc.encryption_schemes.paillier[communication]'

If you wish to run the tests you can use:

$ python -m pip install 'tno.mpc.encryption_schemes.paillier[tests]'

Basic usage

Basic usage is as follows:

from tno.mpc.encryption_schemes.paillier import Paillier

if __name__ == "__main__":
    # initialize Paillier with key_length of 2048 bits and fixed point precision of 3 decimals
    paillier_scheme = Paillier.from_security_parameter(key_length=2048, precision=3)
    # encrypt the number 8.1
    ciphertext1 = paillier_scheme.encrypt(8.1)
    # add 0.9 to the original plaintext
    ciphertext1 += 0.9
    # multiply the original plaintext by 10
    ciphertext1 *= 10
    # encrypt the number 10
    ciphertext2 = paillier_scheme.encrypt(10)
    # add both encrypted numbers together
    encrypted_sum = ciphertext1 + ciphertext2
    # ...communication...
    # decrypt the encrypted sum to 100
    decrypted_sum = paillier_scheme.decrypt(encrypted_sum)
    assert decrypted_sum == 100

Running this example will show several warnings. The remainder of this documentation explains why the warnings are issued and how to get rid of them depending on the users' preferences.

Fresh and unfresh ciphertexts

An encrypted message is called a ciphertext. A ciphertext in the current package has a property is_fresh that indicates whether this ciphertext has fresh randomness, in which case it can be communicated to another player securely. More specifically, a ciphertext c is fresh if another user, knowledgeable of all prior communication and all current ciphertexts marked as fresh, cannot deduce any more private information from learning c.

The package understands that the freshness of the result of a homomorphic operation depends on the freshness of the inputs, and that the homomorphic operation renders the inputs unfresh. For example, if c1 and c2 are fresh ciphertexts, then c12 = c1 + c2 is marked as a fresh encryption (no rerandomization needed) of the sum of the two underlying plaintexts. After the operation, ciphertexts c1 and c2 are no longer fresh.

The fact that c1 and c2 were both fresh implies that, at some point, we randomized them. After the operation c12 = c1 + c2, only c12 is fresh. This implies that one randomization was lost in the process. In particular, we wasted resources. An alternative approach was to have unfresh c1 and c2 then compute the unfresh result c12 and only randomize that ciphertext. This time, no resources were wasted. The package issues a warning to inform the user this and similar efficiency opportunities.

The package integrates naturally with tno.mpc.communication and if that is used for communication, its serialization logic will ensure that all sent ciphertexts are fresh. A warning is issued if a ciphertext was randomized in the process. A ciphertext is always marked as unfresh after it is serialized. Similarly, all received ciphertexts are considered unfresh.

Tailor behavior to your needs

The crypto-neutral developer is facilitated by the package as follows: the package takes care of all bookkeeping, and the serialization used by tno.mpc.communication takes care of all randomization. The warnings can be disabled for a smoother experience.

The eager crypto-youngster can improve their understanding and hone their skills by learning from the warnings that the package provides in a safe environment. The package is safe to use when combined with tno.mpc.communication. It remains to be safe while you transform your code from 'randomize-early' (fresh encryptions) to 'randomize-late' (unfresh encryptions, randomize before exposure). At that point you have optimized the efficiency of the library while ensuring that all exposed ciphertexts are fresh before they are serialized. In particular, you no longer rely on our serialization for (re)randomizing your ciphertexts.

Finally, the experienced cryptographer can turn off warnings / turn them into exceptions, or benefit from the is_fresh flag for own purposes (e.g. different serializer or communication).


By default, the warnings package prints only the first occurrence of a warning for each location (module + line number) where the warning is issued. The user may easily change this behaviour to never see warnings:

from tno.mpc.encryption_schemes.paillier import EncryptionSchemeWarning

warnings.simplefilter("ignore", EncryptionSchemeWarning)

Alternatively, the user may pass "once", "always" or even "error".

Finally, note that some operations issue two warnings, e.g. c1-c2 issues a warning for computing -c2 and a warning for computing c1 + (-c2).

Advanced usage

The basic usage can be improved upon by explicitly randomizing at late as possible.

from tno.mpc.encryption_schemes.paillier import Paillier

if __name__ == "__main__":
    paillier_scheme = Paillier.from_security_parameter(key_length=2048, precision=3)
    # unsafe_encrypt does NOT randomize the generated ciphertext; it is deterministic still
    ciphertext1 = paillier_scheme.unsafe_encrypt(8.1)
    ciphertext1 += 0.9
    ciphertext1 *= 10
    ciphertext2 = paillier_scheme.unsafe_encrypt(10)
    # no randomness can be wasted by adding the two unfresh encryptions
    encrypted_sum = ciphertext1 + ciphertext2
    # randomize the result, which is now fresh
    # ...communication...
    decrypted_sum = paillier_scheme.decrypt(encrypted_sum)
    assert decrypted_sum == 100

As explained above, this implementation avoids wasted randomization for encrypted_sum and therefore is more efficient.

Speed-up encrypting and randomizing

Encrypting messages and randomizing ciphertexts is an involved operation that requires randomly generating large values and processing them in some way. This process can be sped up which will boost the performance of your script or package. The base package tno.mpc.encryption_schemes.templates provides several ways to more quickly generate randomness and we will show two of them below.

Generate randomness with multiple processes on the background

The simplest improvement gain is to generate the required amount of randomness as soon as the scheme is initialized (so prior to any call to randomize or encrypt):

from tno.mpc.encryption_schemes.paillier import Paillier

if __name__ == "__main__":
    paillier_scheme = Paillier.from_security_parameter(key_length=2048, precision=3)
    # Possibly do some stuff here
    for msg in range(5):
        # The required randomness for encryption is already prepared, so this operation is faster.

Calling Paillier.boot_randomness_generation will generate a number of processes that is each tasked with generating some of the requested randomness. By default, the number of processes equals the number of CPUs on your device.

Share Paillier scheme and generate randomness a priori

A more advanced approach is to generate the randomness a priori and store it. Then, if you run your main protocol, all randomness is readily available. This looks as follows. First, the key-generating party generates a public-private keypair and shares the public key with the other participants. Now, every player pregenerates the amount of randomness needed for her part of the protocol and stores it in a file. For example, this can be done overnight or during the weekend. When the main protocol is executed, every player uses the same scheme (public key) as communicated before, configures the scheme to use the pregenerated randomness from file, and runs the main protocol without the need to generate randomness for encryption at that time. A minimal example is provided below.

from pathlib import Path
from typing import List

from tno.mpc.communication import Serialization
from tno.mpc.encryption_schemes.templates.random_sources import FileSource

from tno.mpc.encryption_schemes.paillier import Paillier, PaillierCiphertext

def initialize_and_store_scheme() -> None:
    # Generate scheme
    scheme = Paillier.from_security_parameter(key_length=2048, precision=3)

    # Store without secret key for others
    with open(Path("scheme_without_secret_key"), "wb") as file:
        file.write(Serialization.pack(scheme, msg_id="", use_pickle=False))

    # Store with secret key for own use
    scheme.share_secret_key = True
    with open(Path("scheme_with_secret_key"), "wb") as file:
        file.write(Serialization.pack(scheme, msg_id="", use_pickle=False))

    # Tidy up to simulate real environment (program terminates)

def load_scheme(path: Path) -> Paillier:
    # Load scheme from disk
    with open(path, "rb") as file:
        scheme_raw =
    return Serialization.unpack(scheme_raw)[1]

def pregenerate_randomness_in_weekend(
    scheme: Paillier, amount: int, path: Path
) -> None:
    # Generate randomness
    # Save randomness to comma-separated csv
    with open(path, "w") as file:
        for _ in range(amount):
    # Shut down processes gracefully

def show_pregenerated_randomness(scheme: Paillier, amount: int, path: Path) -> None:
    # Configure file as randomness source
    # Consume randomness from file
    for i in range(amount):
        print(f"Random element {i}: {scheme.get_randomness()}")

def use_pregenerated_randomness_in_encryption(
    scheme: Paillier, amount: int, path: Path
) -> List[PaillierCiphertext]:
    # Configure file as randomness source
    # Consume randomness from file
    ciphertexts = [scheme.encrypt(_) for _ in range(amount)]
    return ciphertexts

def decrypt_result(scheme: Paillier, ciphertexts: List[PaillierCiphertext]) -> None:
    # Show result
    for i, ciphertext in enumerate(ciphertexts):
        print(f"Decryption of ciphertext {i}: {scheme.decrypt(ciphertext)}")

if __name__ == "__main__":
    AMOUNT = 5
    RANDOMNESS_PATH = Path("randomness.csv")

    # Alice initializes, stores and distributes the Paillier scheme

    # Tidy up to simulate real environment (second party doesn't yet have the Paillier instance)

    # Bob loads the Paillier scheme, pregenerates randomness and encrypts the values 0,...,AMOUNT-1
    scheme_without_secret_key = load_scheme("scheme_without_secret_key")
    assert (
        scheme_without_secret_key.secret_key is None
    ), "Loaded paillier scheme contains secret key! This is not supposed to happen."
        scheme_without_secret_key, AMOUNT, RANDOMNESS_PATH
    show_pregenerated_randomness(scheme_without_secret_key, AMOUNT, RANDOMNESS_PATH)
    # Prints the following to screen (numbers will be different):
    # Random element 0: 663667452419034735381232312860937013...
    # Random element 1: ...
    # ...
    ciphertexts = use_pregenerated_randomness_in_encryption(
        scheme_without_secret_key, AMOUNT, RANDOMNESS_PATH

    # Tidy up to simulate real environment (first party should use own Paillier instance)

    # Alice receives the ciphertexts from Bob and decrypts them
    scheme_with_secret_key = load_scheme("scheme_with_secret_key")
    decrypt_result(scheme_with_secret_key, ciphertexts)
    # Prints the following to screen:
    # Decryption of ciphertext 0: 0.000
    # Decryption of ciphertext 1: 1.000
    # ...

Statistical security for the one-time pad without modular reduction

The mathematics behind the statistical security of the sample_mask implementation in this library can be found here.

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