A cryptographic function is homomorphic on some operation if that operation is preserved in the encrypted data.
Such a function is homomorphic on addition modulo if the following holds for some cryptographic function $E$:
$ E(x)\cdot E(y) = E(x+y\mod N) $
and on xor:
$ E(x)\cdot E(y) = E(x \oplus y) $
This is called a partially homomorphic encryption.
Such schemes are described on the Wikipedia page:
However, all the partially homomorphic schemes on XOR and addition modulo are based on prime factorization.
Are there partially homomorphic encryption methods on XOR or addition modulo that do not are not based on prime factorization (other than the fully homomorphic system proposed by Gentry based on lattice cryptography).
This paper by Mykletun, Girao and Westhoff suggests the El-Gamal method: