I'm reading "Fully Homomorphic Encryption over the Integers"——the first generation of FHE. However, this paper seems to detail a scheme to encrypt each bit in a message. I'm very confused about how to implement homomorphic encryption of the whole k-bits message. More specifically,
if I want to achieve homomorphic addition of ciphertexts, such as encryption(3)+encrpytion(3)=encrpytion(6), just using the one-bit FHE scheme seems insufficient.
My question is there any approach to implement that likes the function of carry bits. Or homomorphic encryption of boolean functions, which is capable of extending one-bit homomorphic arithmetic operations to k-bit homomorphic arithmetic operations.
Any references or answers will be much appreciated!