# Is there a partially homomorphic quantum secure public key cryptosystem with IND-CCA1 security?

I recentely asked "IND-CCA1 RSA padding?" about whether there is a IND-CCA1 secure variant of RSA. The original version of the question also allowed usage of ECC which would allow usage of ElGamal, providing IND-CPA and somewhat IND-CCA1.

Now I have the same setting as in the above question, meaning I need a cryptosystem with some homomorphic property and maximal security notion (IND-CCA1 would be optimal, but IND-CPA would also be OK). But this time I'm asking for a post-quantum scheme having a homomorphic property and being IND-CCA1 secure.

So finally the question:
Is there any post-quantum public key encryption algorithm providing IND-CCA1/CPA security and a partial homomorphic property?

Please exclude fully homomorphic encryption schemes from the answers. I know most of them are PQ safe, but they're too slow. And of course please also exclude schemes being even slower than FHE schemes.

Essentially any IND-CPA-secure lattice-based cryptosystem offers additive homomorphism, up to a predetermined number of operations.

I don't know of any IND-CCA1-secure post-quantum candidate that offers any homomorphic property, except Loftus-May-Smart-Vercauteren SAC'11, which is based on a nonstandard "knowledge of error" lattice assumption.

• At first I thought that your CCA1-secure scheme from EUROCRYPT 2012 may be additively homomorphic, but then I realized that the tag matrices $\mathbf H$ are random for each ciphertext, so it doens't work straightaway. Is it possible to fix these matrices without impacting security? Also, I don't know if the encoding of the messages is homomorphic too. Aug 24 '15 at 8:50
• Now that I see it, you mention in your paper the possibility of setting $\mathbf H = \mathbf I$. In that case, and given that it seems that the encoding is additively homomorphic too, then this scheme could be an example of CCA1-secure lattice-based PKE that is additively homomorphic. Aug 24 '15 at 9:22
• Unfortunately no, the CCA1 proof breaks down entirely if you fix the tag matrices to be the same for all ciphertexts. The simulator needs to produce a challenge ciphertext at the tag on which the public key is "punctured," and needs decryption queries need to have different tags so the simulator can decrypt them. Also, I suspect that the scheme is actually insecure under CCA1 for the usual reasons, but haven't verified carefully. Aug 24 '15 at 12:54

There should be plenty of them. Off the top of my head, I'm thinking of the provable secure version of NTRU by Stehlé and Steinfeld [1], which is IND-CPA secure. In this scheme, ciphertexts are of the form:

$$c = pk \cdot s + p\cdot e + \operatorname{encode}(m)$$

where $s$ and $e$ are random polynomials, $p$ is a small prime, and $pk$ is the public key. It can be shown that this scheme is additively homomorphic: $$c + c' = pk \cdot (s+s') + p\cdot (e+e') + \operatorname{encode}(m + m')$$

In fact, this cryptosystem has been used as the basis of some fully homomorphic encryption schemes, such as [2].

In the response to this question I precisely tackle with the correctness condition for the addition of ciphertexts in this cryptosystem.

References

[1] Stehlé, D., & Steinfeld, R. (2011). Making NTRU as secure as worst-case problems over ideal lattices. In Advances in Cryptology–EUROCRYPT 2011 (pp. 27-47). Springer Berlin Heidelberg.

[2] López-Alt, A., Tromer, E., & Vaikuntanathan, V. (2012, May). On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing (pp. 1219-1234). ACM.