Feat: Reduced operation via karatsuba multiplication algo

[nlok5923]

Benchmark comparisons

Before Karatsuba Multiplication benchmark

After Karatsuba Multiplication benchmark

[Janmajayamall]

I benched the update and apparently mul_lazy without karatsuba multiplication performs worse (albeit by few percent) than normal tensoring.

Benched on m6i.8xlarge:

Command: cargo bench bfv/mul_lazy

Before -

bfv/mul_lazy/n=32768/logq=600
                        time:   [73.258 ms 73.440 ms 73.682 ms]
bfv/mul_lazy/n=32768/logq=700
                        time:   [92.830 ms 93.008 ms 93.139 ms]
bfv/mul_lazy/n=32768/logq=750
                        time:   [99.239 ms 99.371 ms 99.593 ms]

After -

bfv/mul_lazy/n=32768/logq=600
                        time:   [78.425 ms 78.873 ms 79.511 ms]
                        change: [+7.2978% +7.8368% +8.3112%] (p = 0.00 < 0.05)
bfv/mul_lazy/n=32768/logq=700
                        time:   [97.026 ms 97.054 ms 97.078 ms]
                        change: [+4.2935% +4.4167% +4.5578%] (p = 0.00 < 0.05)
bfv/mul_lazy/n=32768/logq=750
                        time:   [104.38 ms 104.48 ms 104.58 ms]
                        change: [+4.6334% +4.8922% +5.1423%] (p = 0.00 < 0.05)

I think it is probably due to many non-assign operations required for karatsuba multiplication.

I think before going ahead with merging this, we shall figure out how to improve non-assign vector modular arithmetic operations.

[nlok5923]

This way of implementation introduced an extra assignment and i think we can get rid of it.

Do you think this might hindering the performance ?

[Janmajayamall]

I don't think replacing one add with 2 sub_assigns will make up for 4-5% in performance degradation. Just a reminder for me to look into vector modular operations to check what can be done.