Paillier encrytpion

This example illustrates how to compute a matrix-vector product using the Paillier encryption.

Import numpy package and eclib.paillier module.

import numpy as np

from eclib import paillier

Define a matrix \(A\) and a vector \(x\) as

\[\begin{split} A = \begin{bmatrix} 1.1 & 2.2 \\ -3.3 & 4.4 \end{bmatrix}, \quad x = \begin{bmatrix} -5.5 \\ 6.6 \end{bmatrix}, \end{split}\]

and compute \(y = Ax\).

A = [
    [1.1, 2.2],
    [-3.3, 4.4],
]
x = [-5.5, 6.6]
y = np.dot(A, x)
print(y)

The key generation function paillier.keygen() requires to specify a key length for creating public and secret keys. This example uses a key length of 128 bits.

key_length = 128
params, pk, sk = paillier.keygen(key_length)

The matrix A and vector x are encoded and encrypted to A_ecd and x_enc, respectively, and y_enc is computed.

s = 0.01
A_ecd = paillier.encode(params, A, s)
x_enc = paillier.enc(params, pk, x, s)
y_enc = paillier.int_mult(params, A_ecd, x_enc)

Note that A_ecd, x_enc, and y_enc have the form

\[\begin{split} A_\mathrm{ecd} &= \begin{bmatrix} \bar{A}_{11} & \bar{A}_{12} \\ \bar{A}_{21} & \bar{A}_{22} \end{bmatrix}, \\ x_\mathrm{enc} &= \begin{bmatrix} \mathsf{encrypt}(\bar{x}_1) \\ \mathsf{encrypt}(\bar{x}_2) \end{bmatrix}, \\ y_\mathrm{enc} &= \begin{bmatrix} \mathsf{encrypt}(\bar{A}_{11} \bar{x}_1 + \bar{A}_{12} \bar{x}_2) \\ \mathsf{encrypt}(\bar{A}_{21} \bar{x}_1 + \bar{A}_{22} \bar{x}_2) \end{bmatrix}, \end{split}\]

where \(\bar{A}_{ij} = \mathsf{encode}(A_{ij} / s)\) and \(\bar{x}_j = \mathsf{encode}(x_j / s)\). Similar to the ElGamal encryption, the computation result can be recovered by the paillier.dec() function with \(s^2\).

y_ = paillier.dec(params, sk, y_enc, s**2)
print(y_)

Code

import numpy as np

from eclib import paillier

A = [
    [1.1, 2.2],
    [-3.3, 4.4],
]
x = [5.5, 6.6]
y = np.dot(A, x)
print(y)

key_length = 128
params, pk, sk = paillier.keygen(key_length)

s = 0.01
A_ecd = paillier.encode(params, A, s)
x_enc = paillier.enc(params, pk, x, s)
y_enc = paillier.int_mult(params, A_ecd, x_enc)

y_ = paillier.dec(params, sk, y_enc, s**2)
print(y_)