---
title: "CKKS encode encrypt 2"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{CKKS encode encrypt 2}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

Load libraries that will be used.
```{r setup}
library(polynom)
library(HomomorphicEncryption)
```

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

```{r seed}
set.seed(123)
```

Set some parameters.

```{r params}
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.

```{r z}
z <- c(complex(real=3, imaginary=4), complex(real=2, imaginary=-1))
print(z)
```

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

```{r encode}
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.

```{r print-m}
print(m)
```

Set some parameters:

```{r params2}
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:

```{r seckey}
# 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:

```{r pubkey}
pk0 = CoefMod(-(a*s +e)%%pm,q)
pk1 = a
```

Create polynomials for the encryption:

```{r}
# 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):

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

Decrypt:

```{r}
decrypt <- (ct1 * s) + ct0
decrypt <- decrypt %% pm
decrypt <- CoefMod(decrypt, q)
print(decrypt[1:length(coef(m))])
```

Let's decode to obtain the original numbers:

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

print(decoded_z)
```

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

```{r round}
round(decoded_z)
```
