BFV relinearization
In HomomorphicEncryption: BFV, BGV, CKKS Schema for Fully Homomorphic Encryption

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

This vignette follows the same structure as BFV multiply, but before decryption, instead of decrypting the terms C1, C2, and C3, this vignette implements relinearization of those terms to: C1hat and C2hat. The term C3 includes the terms s^2 (or s*s), by removing this exponential term, the whole of C is linear again (i.e. only including s^1 terms).

Load libraries that will be used.

library(polynom)
library(HomomorphicEncryption)

Set some parameters.

d  =      4     # n and d need to be renamed throughout the package
n  =      2^d
p  =      11
q  =  p * 15000
pm = GenPolyMod(n)

Set a working seed for random numbers

set.seed(123)

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

# generate a secret key
s = GenSecretKey(n)

# generate a
a = GenA(n, q)

# generate the error
e = GenError(n/10) # need to figure out how this division can be removed, by scaling q/p

Generate the public key.

# generate the public key
pk0 = GenPubKey0(a, s, e, pm, q)
pk1 = GenPubKey1(a)

Generate the evaluation key (EvalKey, EK).

ek0 = GenEvalKey0(a, s, e)
ek1 = a

Create polynomials for the encryption

# polynomials for encryption
e1 = GenError(n)
e2 = GenError(n)
u  = GenU(n)

Now create to messages to multiply.

m1 = polynomial(c(3, 2, 2))
m2 = polynomial(c(0, 2   ))

Encrypt the two messages (i.e. genete the ct0 and ct1 part for each m1 and m2).

m1_ct0 = EncryptPoly0(m1, pk0, u, e1, p, pm, q)
m1_ct1 = EncryptPoly1(    pk1, u, e2,    pm, q)
m2_ct0 = EncryptPoly0(m2, pk0, u, e1, p, pm, q)
m2_ct1 = EncryptPoly1(    pk1, u, e2,    pm, q)

Multiply the encrypted messages.

multi_ct0 = m1_ct0 * m2_ct0 * (p/q)
multi_ct0 = multi_ct0 %% pm
multi_ct0 = CoefMod(multi_ct0, q)
multi_ct0 = round(multi_ct0) # the rounding should come before the mod (both of the mods)

multi_ct1 = (m1_ct0 * m2_ct1 + m1_ct1 * m2_ct0) * (p/q)
multi_ct1 = multi_ct1 %% pm
multi_ct1 = CoefMod(multi_ct1, q)
multi_ct1 = round(multi_ct1)

multi_ct2 = (m1_ct1 * m2_ct1) * (p/q)
multi_ct2 = multi_ct2 %% pm
multi_ct2 = CoefMod(multi_ct2, q)
multi_ct2 = round(multi_ct2)

Relinearize:

ct0hat = CoefMod(multi_ct0 + ek0 * multi_ct2 %% pm, q)
ct1hat = CoefMod(multi_ct1 + ek1 * multi_ct2 %% pm, q)

Decrypt the multiple

decrypt = ct0hat + ct1hat * s
decrypt = decrypt %% pm
decrypt = CoefMod(decrypt, q)

# rescale
decrypt = decrypt * p/q

# round then mod p
decrypt = CoefMod(round(decrypt), p)
print(decrypt)