Multi-user Security Bound for Filter Permutators in the Random Oracle Model

At EUROCRYPT 2016, Méaux et al. introduced a new design
strategy for symmetric ciphers for Fully Homomorphic Encryption (FHE),
which they dubbed filter permutators. Although less efficient than classical
stream ciphers, when used in conjunction with an adequate FHE scheme,
they allow constant and small noise growth when homomorphically evaluating
decryption circuit. In this article, we present a security proof up to the birthday
bound (with respect to the size of the IV and the size of the key space) for this
new structure in the random oracle model and in the multi-user setting. In
particular, this result justifies the theoretical soundness of filter permutators.
We also provide a related-key attack against all instances of FLIP, a stream
cipher based on this design.