# BFV

Load libraries that will be used.

    library(polynom)
    library(HomomorphicEncryption)

Set some parameters.

    d  =   4
    n  =   2^d
    p  =   (n/2)-1
    q  = 874

Set a working seed for random numbers

    set.seed(123)

Here we create the polynomial modulo.

    pm = polynomial( coef=c(1, rep(0, n-1), 1 ) )
    print(pm)
    #> 1 + x^16

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

    # generate a secret key
    s = GenSecretKey(n)
    print(s)
    #> 1 + x + x^2 + x^4 + x^8 - x^9 - x^12 + x^14 - x^15

    # generate a
    a = GenA(n, q)
    print(a)
    #> 91 + 348*x + 649*x^2 + 355*x^3 + 840*x^4 + 26*x^5 + 519*x^6 + 426*x^7 + 649*x^8
    #> + 766*x^9 + 211*x^10 + 590*x^11 + 593*x^12 + 555*x^13 + 871*x^14 + 373*x^15

Generate the error for the public key.

    e = GenError(n)
    print(e)
    #> -4 - x - 2*x^2 - 6*x^3 + 6*x^5 - x^6 - 6*x^7 - 4*x^8 + 4*x^9 - 2*x^10 - 7*x^11
    #> - 3*x^12 - x^13 + 5*x^14 - x^15

Generate the public key.

    pk0 = GenPubKey0(a, s, e, pm, q)
    print(pk0)
    #> 560 + 287*x + 70*x^2 + 788*x^3 + 534*x^4 + 150*x^5 + 43*x^6 + 331*x^7 + 328*x^8
    #> + 318*x^9 + 184*x^10 + 519*x^11 + 504*x^12 + 783*x^13 + 79*x^14 + 425*x^15

    pk1 = GenPubKey1(a)

Create a polynomial message

    # create a message
    m = polynomial( coef=c(6, 4, 2) )

Create polynomials for the encryption

    # polynomials for encryption
    e1 = GenError(n)
    e2 = GenError(n)
    u  = GenU(n)
    print(u)
    #> x^3 - x^5 + x^9 + x^11 + x^13 - x^14

Generate the ciphertext.

    ct0 = EncryptPoly0(m, pk0, u, e1, p, pm, q)
    print(ct0)
    #> 157 + 787*x + 337*x^2 + 236*x^3 + 454*x^4 + 575*x^5 + 87*x^6 + 14*x^7 + 448*x^8
    #> + 640*x^10 + 747*x^11 + 711*x^12 + 564*x^13 + 866*x^14 + 678*x^15

    ct1 = EncryptPoly1(   pk1, u, e2,    pm, q)
    print(ct1)
    #> 760 + 698*x + 679*x^2 + 477*x^3 + 329*x^4 + 414*x^5 + 487*x^6 + 165*x^7 +
    #> 111*x^8 + 642*x^9 + 409*x^10 + 565*x^11 + 660*x^12 + 644*x^13 + 469*x^14 +
    #> 297*x^15

Decrypt

    decrypt = (ct1 * s) + ct0
    decrypt = decrypt %% pm
    decrypt = CoefMod(decrypt, q)

    # rescale
    decrypt = decrypt * p/q

Round (remove the error) then mod p

    # round then mod p
    decrypt = CoefMod(round(decrypt), p)
    print(decrypt)
    #> 6 + 4*x + 2*x^2

Which is indeed the message that we first encrypted.

Next, look at the vignette BFV-2 which does the exact same process, but unpacks all the functions used here into basic mathematical operations.