# Why are the parameters (such as modulus and dimension) of homomorphic encryption so large?

Compared with the common lattice-based PQC schemes, the modulus $$q$$ and dimension $$n$$ of homomorphic encryption are so large. For example, in Kyber, $$n=256, n \times k = \{512,768,1024\}$$, $$q = 12289$$ or $$3329$$, while in FV or other HE schemes, $$n$$ could be $$2^{14}$$, and $$q$$ might reach $$2^{744}$$. Both of them are based on lattice problem. Why are the parameters so different?

But for homomorphic schemes, you also want to operate with the ciphertexts and these operations increase the noise, so you have to start with small noise (for example, if fresh ciphertexts have noise close to q/4, then one single addition brings the noise to q/2 and no further homomorphic operation is possible). Usually, the schemes choose the initial noise to be bounded by some small constant, and increase q to have more room for the noise growth generated by the homomorphic operations. But then, the LWE problem becomes easier, so, to reach the desired security level, they have to increase the dimension, $$n$$.
You will notice that schemes whose noise growth is smaller are able to choose much smaller parameter (e.g., TFHE uses $$n = 2^{10}$$ and $$q = 2^{32}$$).