Implementation of the Paillier cryptosystem

# 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 with support for the following:

• 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

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

## Install

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

$python -m pip install tno.mpc.encryption_schemes.paillier  ### Note: 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 proces. 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).

### Warnings

By default, the warnings package prints only the first occurence 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).

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
encrypted_sum.randomize()
# ...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.

## Project details

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