Optimize multi-threaded CKKS bootstrapping

[yspolyakov]

Currently, we see a very modest speed-up. E.g. , when we run on a machine with 8 cores, the speed-up is only 2.2x (as compared to the single-threaded execution) vs the theoretical maximum of 8x.

[yspolyakov]

I ran some experiments for CKKS bootstrapping.

Results for 8 threads (on a 16-core machine):

CKKS scheme is using ring dimension 65536

Input: (0.25, 0.5, 0.75, 1, 2, 3, 4, 5,  ... ); Estimated precision: 59 bits

Initial number of levels remaining: 1
Encoding time: 5764
Reduce time: 11974
Decode time: 2695
Total time:     22573 ms
Number of levels remaining after bootstrapping: 10

Output after bootstrapping
        (0.249857, 0.499905, 0.749801, 1, 2.00029, 2.99998, 4.00012, 5.00001,  ... ); Estimated precision: 13 bits

Results for 1 core

CKKS scheme is using ring dimension 65536

Input: (0.25, 0.5, 0.75, 1, 2, 3, 4, 5,  ... ); Estimated precision: 59 bits

Initial number of levels remaining: 1
Encoding time: 11847
Reduce time: 25371
Decode time: 5278
Total time:     47302 ms
Number of levels remaining after bootstrapping: 10

Output after bootstrapping
        (0.249984, 0.499862, 0.749969, 1.00005, 2.00009, 3.00001, 4.00006, 5.00008,  ... ); Estimated precision: 13 bits

So I only observed a speed-up of 2.1x

[yspolyakov]

In most places we rely on OMP parallelization at the DCRTPoly level. I ran a polynomial benchmark (also for N = 64K, just like for CKKS bootstrapping; see the table below) and also got a modest speed-up (8 towers vs 1):

Add: 2.82x
Mul: 3.51x
NTT: 3.47x
iNTT: 3.34x

The fact that we get a low speed-up at DCRTPoly (and we have many other operations as well) leads to a small speed-up for CKKS bootstrapping.

Running bin/benchmark/poly-benchmark-64k
Run on (16 X 2496 MHz CPU s)
Load Average: 0.52, 0.58, 0.59
----------------------------------------------------------------------------------
Benchmark                                        Time             CPU   Iterations
----------------------------------------------------------------------------------
Native_AddEq                                  39.9 us         40.1 us        17920
DCRT_AddEq/towers:1                           40.5 us         40.8 us        17231
DCRT_AddEq/towers:2                           44.0 us         44.5 us        15448
DCRT_AddEq/towers:4                           70.0 us         69.8 us         8960
DCRT_AddEq/towers:8                            113 us          112 us         6400
DCRT_AddEq/towers:16                           241 us          240 us         2800
Native_SubEq                                   203 us          204 us         3446
DCRT_SubEq/towers:1                            201 us          200 us         3446
DCRT_SubEq/towers:2                            216 us          218 us         3446
DCRT_SubEq/towers:4                            257 us          256 us         2987
DCRT_SubEq/towers:8                            327 us          330 us         2133
DCRT_SubEq/towers:16                           652 us          645 us          896
Native_MulEq                                   131 us          132 us         4978
DCRT_MulEq/towers:1                            130 us          131 us         5600
DCRT_MulEq/towers:2                            147 us          148 us         4978
DCRT_MulEq/towers:4                            179 us          180 us         3733
DCRT_MulEq/towers:8                            299 us          298 us         2358
DCRT_MulEq/towers:16                           598 us          600 us         1120
Native_ntt                                     984 us          977 us          640
DCRT_ntt/towers:1                              982 us          983 us          747
DCRT_ntt/towers:2                             1101 us         1099 us          640
DCRT_ntt/towers:4                             1425 us         1412 us          498
DCRT_ntt/towers:8                             2268 us         2247 us          299
DCRT_ntt/towers:16                            4767 us         4719 us          149
Native_intt                                    970 us          952 us          640
DCRT_intt/towers:1                             987 us          952 us          640
DCRT_intt/towers:2                            1041 us         1050 us          640
DCRT_intt/towers:4                            1457 us         1443 us          498
DCRT_intt/towers:8                            2323 us         2299 us          299
DCRT_intt/towers:16                           4814 us         4849 us          145