arXiv:2610.29983v1 Announce Type: new Abstract: Gate bootstrapping (GB) is a widely used technique in FHEW-like fully homomorphic encryption schemes, such as TFHE, for evaluating Boolean gates while simultaneously reducing the noise of the resulting ciphertext. However, the computational cost of GB is typically dominated by an expensive blind rotation (BR). \cite{chen2024free} proposed a gate bootstrapping framework in which $\txor$ gates can be evaluated simply by adding the two input ciphertexts, resulting in what they call \textit{free-XOR}. To obtain a functionally complete set of operations, they further designed a GB procedure for evaluating $\tand$ gates using two lightweight blind rotations. In this work, we propose a similar gate bootstrapping framework that supports free evaluation of $\txor$ gates while evaluating \textit{all symmetric binary Boolean gates}, i.e., the 8 gates whose output is identical for the inputs $(0,1)$ and $(1,0)$, using two blind rotations. In addition to providing broader gate support, our construction is substantially simpler than that of \cite{chen2024free}, both conceptually and in terms of implementation.