Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP

We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically (B → B · poly(n)) with every multiplication (before “refreshing”), our noise only grows linearly (B → B · poly(n)). We use this technique to construct a scale-invariant fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus q and the initial noise level B, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for “modulus switching”), and this modulus can take arbitrary form. In addition, security can be classically reduced from the worst-case hardness of the GapSVP problem (with quasi-polynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction from GapSVP.