Symmetric Key Ciphers for FHE

In recent years, small devices with limited storage and computing power have acquired more and more popularity. This phenomenon, together with the appearance of cloud services that offer extensive storage and high computing power, has made computation outsourcing a key application to study. This setting, where a user with limited resources stores data on a remote server and also asks this server to make some computation on his behalf, comes together with new challenges and concerns related to privacy and security.

A simple scenario

The user Alice encrypts her data under a homomorphic encryption scheme and sends the ciphertexts to the server (eg. cloud). Now the server computes some function (eg. mean value) on Alice's encrypted data and sends back an encryption of the result, which can be decrypted by Alice with the appropriate key. Despite its simplicity, this model still requires a lot of effort from the user. In fact, all the FHE schemes that we know of have very large keys and heavy encryption overhead. The most natural approach is then trying to free the user from the burden of complex homomorphic computations, by asking her to do only the key generation part, the symmetric encryption and the final decryption, while leaving all the rest to the powerful server.

A hybrid framework

The user Alice generates a secret key $sk^S$ for a symmetric encryption scheme, a pair $(pk^H, sk^H)$ for a homomorphic public key encryption scheme, and sends $pk^H$ to the server together with an encryption of $sk^S$ under $pk^H$. Then, Alice encrypts her data under the symmetric encryption scheme and sends the resulting ciphertexts to the server, which homomorphically evaluates the decryption circuit of the symmetric encryption scheme, thus obtaining ciphertexts under the homomorphic scheme. This last operation can be seen as a sort of bootstrapping (from one scheme to another one), but does not involve any circular security assumption. At this point, the server can homomorphically compute a certain function $f$ on the resulting ciphertexts and send back the result, encrypted under the homomorphic encryption scheme. Alice can finally recover the result thanks to her $sk^H$.

Despite numerous improvements in recent years FHE remains highly unpractical, and one of the main problems is limited homomorphic capacity. In fact, all the FHE schemes we know of contain a noise term that grows with homomorphic operations. The homomorphic capacity corresponds to the amount of computation that can be performed before having to ``refresh'' the ciphertext via the expensive operation of bootstrapping. It is thus of vital importance to manage the error growth during homomorphic operations.

State-of-the-art ciphers

AES is a natural benchmark for FHE implementations. The homomorphic evaluations of some lightweight block ciphers such as SIMON have been investigated by Lepoint and Naehrig as well. It turns out that there is much room for improvement in optimizing the multiplicative depth. In this direction, many dedicated symmetric key ciphers with low-noise ciphertexts have been proposed.

• In 2015, Albrecht et al presented a flexible block cipher LowMC, which is based on an SPN structure with $80,128$ and $256$-bit key variants. The design philosophy is to minimize the multiplicative complexity, which is more relevant in the multi-party computation context. It also has very low multiplicative depth compared with existing block ciphers.

  Later, Dinur et al mounted optimized interpolation attacks against some instances of LowMC. More precisely, some $80$-bit key instances could be broken $2^{23}$ times faster than exhaustive search and all instances that are claimed to provide $128$-bit security could be broken about $1000$ times faster than exhaustive search. These attacks show that the ciphers do not provide the expected security level. The cryptanalysis also led to a revised version of LowMC.
• Keyvrium was designed by Canteaut et al and is a stream cipher aiming at $128$-bit security. It is based on the lightweight stream cipher Trivium, which has $80$-bit security. Trivium and Keyvrium both achieve similar multiplicative depth compared to instances of LowMC. They have a smaller latency than LowMC, but have a slightly smaller throughput.
• The FLIP family stream ciphers proposed by Méaux et al have the lowest depth compared with previous ciphers. Several versions are provided including $80$-bit and $128$-bit security instantiations. The main design principal is to filter a constant key register with a time-varying public bit permutation, which allows for small and constant noise growth. However, FLIP has a much higher number of ANDs per encrypted bit than the above ciphers.

To sum up, blending symmetric ciphers and FHE is definitely an interesting idea that could prove fundamental for bringing homomorphic encryption in the real world.

The other fellows also made contributions to this blog post.