D-4.10 Homomorphic Rotation of Input Vector Slots

BGV’s homomorphic rotation scheme of input vector slots is exactly the same as BFV’s rotation scheme (Summary D-2.9 in §D-2.9).

Summary D-4.10 BGV’s Homomorphic Rotation of Input Vector Slots

Suppose we have a BGV ciphertext and a key-switching key as follows:

𝖱𝖫𝖶𝖤S,σ(M + ΔE) = (A,B),𝖱𝖫𝖾𝗏S,σβ,l(SJ(h))

Then, the procedure of rotating the first-half elements of the ciphertext’s original input vector v by h positions to the left (in a wrapping manner among them) and the second-half elements of v by h positions to the left (in a wrapping manner among them) is as follows:

1.
Update A(X), B(X) to A(XJ(h)), B(XJ(h)).
2.
Perform the following key switching (§D-3.8) from S(XJ(h)) to S(X):

𝖱𝖫𝖶𝖤S(X),σ(M(XJ(h)) + ΔE(XJ(h)))

= (0,B(XJ(h)))  +  𝖣𝖾𝖼𝗈𝗆𝗉β,l(A(XJ(h))), 𝖱𝖫𝖾𝗏S(X),σβ,l(S(XJ(h)))