Encrypted Operator Computing: a novel scheme for computation on encrypted data

Abstract: We introduce a new approach to computation on encrypted data -- Encrypted Operator Computing (EOC) -- as an alternative to Fully Homomorphic Encryption (FHE). Given a plaintext vector $|{x}\rangle$, $x\in {0,1}^n$, and a function $F(x)$ represented as an operator $\hat F$, $\hat F\;|{x}\rangle = |{F(x)}\rangle$, the EOC scheme is based on obfuscating the conjugated operator (circuit) $\hat{F}^E = \hat E\;\hat F\;\hat{E}^{-1}$ that implements computation on encrypted data, $\hat E |{x}\rangle$. The construction of EOC hinges on the existence of a two-stage NC$^1$ reversible-circuit-based IND-CCA2 cipher $\hat{E} = \hat{N} \hat{L}$, where $\hat{L}$ and $\hat{N}$ represent, respectively, linear and non-linear NC$^1$ tree-structured circuits of 3-bit reversible gates. We make and motivate security assumptions about such a NC$^1$ cipher. Furthermore, we establish the polynomial complexity of the obfuscated circuit, the evaluator $O(\hat{F}^E)$, by proving that: (a) conjugation of each gate of $F$ with $\hat{L}$ yields a polynomial number of gates; and (b) the subsequent conjugation with $\hat{N}$ yields a polynomial number of ``chips,'' $n$-input/$n$-output reversible functions, with outputs expressed as polynomial-sized ordered Binary Decision Diagrams (OBDDs). The security of individual chips is connected to the notion of Best Possible Obfuscators which relies on poly-size OBDDs and the fact that OBDDs are normal forms that expose the functionality but hide the gate implementation of the chip. We conjecture that the addition of random pairs of NOTs between layers of $\hat{N}$ during the construction of $F^E$, a device analogous to the AddRoundKey rounds of AES, ensures the security of the the evaluator. We also present a generalization to asymmetric encryption.