Toward Basing Fully Homomorphic Encryption on Worst-Case Hardness

Joël Alwen, Manuel Barbosa, Pooya Farshim +4 more

Sonam Mittal, K. R. Ramkumar

Guangsheng Ma, Hongbo Li

We present a novel quantum fully homomorphic encryption (QFHE) scheme, which allows to perform conditional rotations with the control bit in encrypted form. In our scheme, any quantum circuit can be directly evaluated with no need to decompose into Clifford/non-Clifford gates, nor to be transformed into real representation. Our scheme is able to evaluate quantum Fourier transform algorithm faster than previous QFHE schemes in the worst case. The security of our scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.

Pedro Costa Branco, Giulio Malavolta, Zayd Maradni

Zvika Brakerski, Adeline Roux-Langlois, Chris Peikert +2 more

We show that the Learning with Errors (LWE) problem is classically at least as hard as standard worst-case lattice problems, even with polynomial modulus. Previously this was only known under quantum reductions. Our techniques capture the tradeoff between the dimension and the modulus of LWE instances, leading to a much better understanding of the landscape of the problem. The proof is inspired by techniques from several recent cryptographic constructions, most notably fully homomorphic encryption schemes.

Carlos Aguilar Melchor, Slim Bettaieb, Philippe Gaborit +1 more

Abstract. In CRYPTO 2010, Aguilar et al. proposed a somewhat homomorphic encryption scheme, i.e. an encryption scheme allowing to compute a limited amount of sums and products over encrypted data, with a security reduction from LWE over general lattices. General lattices (as opposed to ideal lattices) do not have an inherent multiplicative structure but, using a tensorial product, Aguilar et al. managed to obtain a scheme allowing to compute products with a polylogarithmic amount of operands. In this paper we present an alternative construction allowing to compute products with polynomiallymany operands while preserving the security reductions of the initial scheme. Unfortunately, despite this improvement our construction seems to be incompatible with Gentry’s seminal transformation allowing to obtain fully-homomorphic encryption schemes. Recently, Brakerski et al. used the tensorial product approach introduced by Aguilar et al. in a new alternative way which allows to radically improve the performance of the obtained scheme. Based on this approach, and using two nice optimizations, their scheme is able to evaluate products with exponentially-many operands and can be transformed into an efficient fully-homomorphic encryption scheme while being based on general lattice problems. However, even if these results outperform the construction presented here, we believe the modifications we suggest for Aguilar et al.’s schemes are of independent interest.