# CKKS encode 3

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 <- 4
    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)
    p                   <- sigma_inverse(xi, M, rounded_scale_pi_zi)
    coef                <- as.vector(round(Re(p)))
    p                   <- polynomial(coef)

Let’s view the result.

    print(p)
    #> 10 + 5*x + 10*x^2 + 2*x^3

Let’s decode to obtain the original number:

    rescaled_p <- coef(p) / scale
    z          <- sigma_function(xi, M, rescaled_p)
    decoded_z  <-pi_function(M, z)

    print(decoded_z)
    #> [1] 3.03033+3.737437i 1.96967-1.262563i

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

Next, work through the CKKS-encode-2 vignette, which breaks down the encode and decode functions into the individual steps.