Faster CKKS multiply

[fboemer]

Speeds up CKKS Multiply by ~1.2x - 2.1x. Several optimizations improve the default implementation:

• Avoid the add_poly_coeffmod step when not needed
• Use as much in-place computation as possible
• Process 256-coefficient tiles at a time rather than entire polynomials at a time.
• Also runs pre-commit for some whitespace changes

I imagine a similar approach could be used to optimize the BEHZ ciphertext product in BFV multiply.

If this approach turns out to improve performance on more evaluator functions, perhaps a better implementation would be to add another iterator layer beyond the CoeffIterator, e.g. "TileIterator", which performs the tiling iteration more cleanly.

On ICX compiled with gcc-10, running CKKS / EvaluateMulCt benchmark for 1000 iterations yields:

n=4096

HEXL	Time before (us)	Time after (us)	Speedup
OFF	148	120	1.23x
ON	52.6	39.1	1.34x

n=8192

HEXL	Time before (us)	Time after (us)	Speedup
OFF	640	493	1.30x.
ON	209	116	1.82x.

n=16384

HEXL	Time before (us)	Time after (us)	Speedup
OFF	2656	2008	1.32x
ON	891	414	2.15x

n=32768

HEXL	Time before (us)	Time after (us)	Speedup
OFF	10054	7634	1.31x
ON	3963	2509	1.57x

[fionser]

Typos in the comments

 // Given input pairs of polynomials x = (x[0], x[1], x[2]), y = (y[0], y[1])

Should it be x = (x[0], x[1]) ?

[fboemer]

Typos in the comments

 // Given input pairs of polynomials x = (x[0], x[1], x[2]), y = (y[0], y[1])

Should it be x = (x[0], x[1]) ?

The input to CKKS multiply would be x = (x[0], x[1]), but after encrypted1.resize(context_, context_data.parms_id(), dest_size);, x = (x[0], x[1], x[2]). So I suppose x = (x[0], x[1], x[2]) is a tuple, rather than a pair, of polynomials. Hopefully that makes it more clear.

[WeiDaiWD]

This is quite nice! Thanks a lot. I'm gonna make slight modifications before release (3.6.6).