# BGV 2

    library(polynom)
    library(HomomorphicEncryption)

Set some parameters.

    d  =   4
    n  =   2^d
    p  =   (n/2)-1
    t  =   p
    q  = 868

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.

    # generate a secret key
    s = polynomial( sample.int(3, n, replace=TRUE)-2 )
    print(s)
    #> 1 + x + x^2 + x^4 + x^8 - x^9 - x^12 + x^14 - x^15

Create a (part of the public key)

    # generate a
    a = polynomial(sample.int(q, n, replace=TRUE))
    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 + 373*x^14 + 844*x^15

Create the error term `e` to be used to generate the public key.

    # generate the error
    e = polynomial( coef=round(stats::rnorm(n, 0, n/3)) )
    print(e)
    #> -6 - x - 5*x^2 - 4*x^3 - 3*x^4 - 9*x^5 + 4*x^6 + x^7 - 6*x^8 + 7*x^9 + 2*x^10 -
    #> 2*x^11 + 5*x^12 + 5*x^13 + 4*x^14 + 4*x^15

Generate Part 1 of the Public Key.

    pk1 = -(a*s + p*e)
    pk1 = pk1 %% pm
    pk1 = CoefMod(pk1, q)
    print(pk1)
    #> 577 + 764*x + 467*x^2 + 395*x^3 + 537*x^4 + 201*x^5 + 372*x^6 + 401*x^7 +
    #> 733*x^8 + 255*x^9 + 642*x^10 + 37*x^11 + 818*x^12 + 830*x^13 + 65*x^14 +
    #> 405*x^15

Generate Part 2 of the Public Key (which is actually just equal to a).

    pk2 = a

Create a polynomial message

    # create a message
    m = polynomial( coef=c(2, 3, 4) )

Create polynomials for the encryption of the message. Since e1 and e2 are constructed the same way as e, we don’t print them, we just print u.

    # 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 )
    print(u)
    #> x^3 - x^5 + x^9 + x^11 + x^13 - x^14

Generate Part 1 of the ciphertext version of the message.

    ct1 = pk1*u + p*e1 + m
    ct1 = ct1 %% pm
    ct1 = CoefMod(ct1, q)
    print(ct1)
    #> 436 + 377*x + 95*x^2 + 818*x^3 + 820*x^4 + 695*x^5 + 61*x^6 + 620*x^7 + 86*x^8
    #> + 392*x^9 + 533*x^10 + 420*x^11 + 701*x^12 + 159*x^13 + 572*x^14 + 788*x^15

Generate Part 2 of the ciphertext version of the message.

    ct2 = pk2*u + p*e2
    ct2 = ct2 %% pm
    ct2 = CoefMod(ct2, q)
    print(ct2)
    #> 745 + 352*x + 194*x^2 + 35*x^3 + 741*x^4 + 420*x^5 + 488*x^6 + 655*x^7 +
    #> 511*x^8 + 241*x^9 + 796*x^10 + 149*x^11 + 530*x^12 + 264*x^13 + 476*x^14 +
    #> 306*x^15

Decrypt

    decrypt = (ct2 * s) + ct1
    decrypt = decrypt %% pm
    decrypt = CoefMod(decrypt, q)
    decrypt = CoefMod(round(decrypt), p)
    print(decrypt)
    #> 2 + 3*x + 4*x^2