# CKKS encode encrypt 2

Load libraries that will be used.

    library(polynom)
    library(HomomorphicEncryption)

Set a working seed for random numbers (so that random numbers can be replicated exactly).

    set.seed(123)

Set some parameters.

    M     <-   8
    N     <-   M / 2
    scale <- 200
    xi    <- complex(real = cos(2 * pi / M), imaginary = sin(2 * pi / M))

Create the (complex) numbers we will encode.

    z <- c(complex(real=3, imaginary=4), complex(real=2, imaginary=-1))
    print(z)
    #> [1] 3+4i 2-1i

Now we encode the vector of complex numbers to a polynomial.

    pi_z                <- pi_inverse(z)
    scaled_pi_z         <- scale * pi_z
    rounded_scale_pi_zi <- sigma_R_discretization(xi, M, scaled_pi_z)
    m                   <- sigma_inverse(xi, M, rounded_scale_pi_zi)
    coef                <- as.vector(round(Re(m)))
    m                   <- polynomial(coef)

Let’s view the result.

    print(m)
    #> 500 + 283*x + 500*x^2 + 142*x^3

Set some parameters:

    n  =  16
    p  =   7
    q  = 874
    pm = polynomial( coef=c(1, rep(0, n-1), 1 ) )

Create the secret key and the polynomials a and e, which will go into the public key:

    # generate a secret key
    s = polynomial( sample.int(3, n, replace=TRUE)-2 )

    # generate a
    a = polynomial(sample.int(q, n, replace=TRUE))

    # generate the error
    e = polynomial( coef=round(stats::rnorm(n, 0, n/3)) )

Generate the public key:

    pk0 = CoefMod(-(a*s +e)%%pm,q)
    pk1 = a

Create polynomials for the encryption:

    # polynomials for encryption
    e1 = polynomial( coef=round(stats::rnorm(n, 0, n/3)) )
    e2 = polynomial( coef=round(stats::rnorm(n, 0, n/3)) )
    u  = polynomial( coef=sample.int(3, (n-1), replace=TRUE)-2 )

Generate the ciphertext (encryption):

    ct0 = CoefMod((pk0*u + e1 + m) %% pm, q)
    ct1 = CoefMod((pk1*u + e2    ) %% pm, q)

Decrypt:

    decrypt <- (ct1 * s) + ct0
    decrypt <- decrypt %% pm
    decrypt <- CoefMod(decrypt, q)
    print(decrypt[1:length(coef(m))])
    #> [1] 450 254 515 119

Let’s decode to obtain the original numbers:

    rescaled_p <- decrypt[1:length(m)] / scale
    z          <- sigma_function(xi, M, rescaled_p)
    decoded_z  <- pi_function(M, z)

    print(decoded_z)
    #> [1] 2.727297+3.893754i 1.772703-1.256246i

The decoded z is indeed very close to the original z, we round the result to make the clearer.

    round(decoded_z)
    #> [1] 3+4i 2-1i