In the paper Fully Homomorphic Encryption over the Integers, a method is shown for doing homomorphic encryption using only integers. The basic idea is that a bit m is encoded as a large integer c.
The principle behind homomorphic encryption through bootstrapping is that after some calculations, the cyphertext is recrypted by encrypting it under a new key and then performing the decryption on the first key homomorphically. This is explained in section 1.2 of Fully Homomorphic Encryption Using Ideal Lattices.
My question: since the scheme in the first paper encrypts a bit as an integer, how is this integer encrypted under the new key? Is each bit of c encrypted using the new key?