Improve CKKS bootstrapping precision for FLEXIBLEAUTOEXT about openfhe-development HOT 2 CLOSED

@sarojaduality implemented a logic for automatically computing the best correction factor and observed that the correction factor and precision are the same for FLEXIBLEAUTO and FLEXIBLEAUTO.

@yspolyakov examined this further and ran experiments for FLEXIBLEAUTOEXT, FLEXIBLEAUTO, and FIXEDAUTO. There is no difference in the bootstrapping precision between FLEXIBLEAUTOEXT and FLEXIBLEAUTO. In this case, the deterministic error due to the approximation $sin(x) \approx x$ is much higher than the nondeterministic error due to RLWE noise. All FLEXIBLEAUTOEXT does is reduces the fresh encryption error to the rescaling error, but the accumulated nondeterministic error turns out to be lower than then the deterministic sine wave approximation error incurred in bootstrapping.

Note that FIXEDAUTO has a significantly lower precision than both FLEXIBLEAUTO and FLEXIBLEAUTOEXT because the former has a deterministic error due to approximate scaling in the rescaling operation.

from openfhe-development.

Additional note. The above is true for fresh ciphertexts, i.e., FLEXIBLEAUTO and FLEXIBLEAUTOEXT give roughly the same precision. After some computations (when the nondeterministic error becomes higher and comparable to the deterministic error), FLEXIBLEAUTOEXT should start providing better precision than FLEXIBLEAUTO.

from openfhe-development.