# Additive homomorphic encryption over small fields

Are there encryption schemes that are additively homomorphic with respect to small fields such as $\mathbb{F}_{2^4}$ or $\mathbb{F}_{2^8}$?

• Do you mean where ciphertexts are in those fields or the plaintext would be constrained to being within the field and all mathematical operations are in it as well? – mikeazo Jun 8 '16 at 18:15
• I meant the plaintexts are in a small field. – crypto wonderer Jun 8 '16 at 20:08
• That is equivalent to xor-homomorphic encryption of bits - Just let the ciphertext be the ordered tuple, such as a 4-tuple or an 8-tuple, of encryptions of the plaintext's bits. ​ ​ – user991 Jun 8 '16 at 20:17
• Suppose that Player 1 wants to compute $f(ab)$, where he holds $f(a)$ and $f(b); a,b \in \mathbb{F_{2^k}}$. Player 1 doesn't have the private key. But he can use the help of Player 2 who posses the private key and also he Player 1 needs to prevent from Player 2 to know the values $a$ nor $b$. @RickyDemer is it possible to implement this scheme with your suggestion? Thanks. – user1387682 Jun 4 '17 at 23:19