GSW encryption

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

Import numpy package and eclib.gsw module.

import numpy as np

from eclib import gsw

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 gsw.keygen() requires to specify m, n, q, and sigma for creating public and secret keys, where n is the dimension of a lattice, which equals to the dimension of secret key, m is the subdimension of the lattice, q is the modulus of plaintext and ciphertext spaces, and sigma is the standard deviation of the discrete Gaussian distribution with mean zero used as an error distribution. The parameter m is optional and is set to 2 * n * ceil(log2(q)) if not given. This example omits m and uses n = 10, q = 2**64, and sigma = 3.2.

sec_params = (10, 2**64, 3.2)
params, pk, sk = gsw.keygen(*sec_params)

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

s = 0.01
A_enc = gsw.enc(params, pk, A, s)
x_enc = gsw.enc(params, pk, x, s)
y_enc = gsw.mult(params, A_enc, x_enc)

Note that A_enc, x_enc, and y_enc have the form

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

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 gsw.dec() function with \(s^2\).

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

Code

import numpy as np

from eclib import gsw

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

sec_params = (10, 2**64, 3.2)
params, pk, sk = gsw.keygen(*sec_params)

s = 0.01
A_enc = gsw.enc(params, pk, A, s)
x_enc = gsw.enc(params, pk, x, s)
y_enc = gsw.mult(params, A_enc, x_enc)

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