experimental wnaf type change + try flatten MultiExp input

[protolambda]

This is experimental, just for illustration, it needs review, and maybe some of the changes are worthwhile.

The wnaf type:

• Didn't allocate enough when creating it, and keeps appending and re-allocating (from empty, up to 128, doubling allocation etc.)
• Uses int as element type, even though for the default window size, it could fit in a int8, a 8x size reduction
  • Note: the benchmarks tried different window sizes. With int8, and the way the half size and full size works, you can't go higher than 6 if it needs to fit in int8, but that's fine.
  • If the window size was declared as constant (it's not only because of benchmarks), some allocations of slices here and there may just be declared as small fixed-size arrays that could fit in the stack? (Could maybe improve performance)

And MultiExp has a []*PointG1 input, while I much prefer []PointG1, since we allocate large arrays of these, and there is no use for an unnecessary extra layer of indirection. Or maybe an alternative function could be exposed, with this input type?

Benchmark:

goos: linux
goarch: amd64
pkg: github.com/kilic/bls12-381
cpu: AMD Ryzen 9 5950X 16-Core Processor
                                                     old      new      old              new
BenchmarkG1MulWNAF/Naive-32                         9961     8464   120897 ns/op     121713 ns/op
BenchmarkG1MulWNAF/Fr,_window:_1-32                10000    11935   119168 ns/op     104580 ns/op
BenchmarkG1MulWNAF/Big,_window:_1-32                7494    10000   148903 ns/op     140881 ns/op
BenchmarkG1MulWNAF/Fr,_window:_2-32                 9423    11120   113026 ns/op      97312 ns/op
BenchmarkG1MulWNAF/Big,_window:_2-32               10000    10000   133106 ns/op     120360 ns/op
BenchmarkG1MulWNAF/Fr,_window:_3-32                10000    12624   118624 ns/op      98030 ns/op
BenchmarkG1MulWNAF/Big,_window:_3-32                8514     9373   133294 ns/op     124872 ns/op
BenchmarkG1MulWNAF/Fr,_window:_4-32                 9913    10000   129724 ns/op     109758 ns/op
BenchmarkG1MulWNAF/Big,_window:_4-32                8151     9626   129339 ns/op     121394 ns/op
BenchmarkG1MulWNAF/Fr,_window:_5-32                 9234    10000   128689 ns/op     106181 ns/op
BenchmarkG1MulWNAF/Big,_window:_5-32                8271    10033   140273 ns/op     133795 ns/op
BenchmarkG1MulWNAF/Fr,_window:_6-32                 7934            161776 ns/op
BenchmarkG1MulWNAF/Big,_window:_6-32                7244            186107 ns/op
BenchmarkG1MulWNAF/Fr,_window:_7-32                 5818            202863 ns/op
BenchmarkG1MulWNAF/Big,_window:_7-32                4736            219887 ns/op

BenchmarkG1MulGLV/Naive-32                          9211     9111   126705 ns/op     129577 ns/op
BenchmarkG1MulGLV/Fr,_window:_1-32                 12975    14316    87115 ns/op      77555 ns/op
BenchmarkG1MulGLV/Big,_window:_1-32                10797    10000   124148 ns/op     105504 ns/op
BenchmarkG1MulGLV/Fr,_window:_2-32                 15717    16918    78684 ns/op      71145 ns/op
BenchmarkG1MulGLV/Big,_window:_2-32                10000    13036   103570 ns/op      99024 ns/op
BenchmarkG1MulGLV/Fr,_window:_3-32                 15278    16030    87883 ns/op      73374 ns/op
BenchmarkG1MulGLV/Big,_window:_3-32                10928    13880   110459 ns/op      91182 ns/op
BenchmarkG1MulGLV/Fr,_window:_4-32                 14187    15295   107709 ns/op      80290 ns/op
BenchmarkG1MulGLV/Big,_window:_4-32                 9690    11877   113151 ns/op     105051 ns/op
BenchmarkG1MulGLV/Fr,_window:_5-32                 10059    12541   123242 ns/op      95641 ns/op
BenchmarkG1MulGLV/Big,_window:_5-32                 7377    11188   152681 ns/op     103277 ns/op
BenchmarkG1MulGLV/Fr,_window:_6-32                  7101            163839 ns/op
BenchmarkG1MulGLV/Big,_window:_6-32                 6588            154301 ns/op
BenchmarkG1MulGLV/Fr,_window:_7-32                  5584            204235 ns/op
BenchmarkG1MulGLV/Big,_window:_7-32                 5283            221057 ns/op

BenchmarkG1MultiExp/100-32                           399     393   3038978 ns/op    3118516 ns/op
BenchmarkG1MultiExp/1000-32                           69      75  16212074 ns/op   16841531 ns/op

BenchmarkG2MulWNAF/Naive-32                         4137     4191   296644 ns/op     280547 ns/op
BenchmarkG2MulWNAF/Fr,_window:_1-32                 5106     5712   240345 ns/op     203272 ns/op
BenchmarkG2MulWNAF/Big,_window:_1-32                5184     5575   265715 ns/op     244774 ns/op
BenchmarkG2MulWNAF/Fr,_window:_2-32                 5048     5533   255389 ns/op     210939 ns/op
BenchmarkG2MulWNAF/Big,_window:_2-32                4590     5036   269946 ns/op     224165 ns/op
BenchmarkG2MulWNAF/Fr,_window:_3-32                 5312     5796   224518 ns/op     211165 ns/op
BenchmarkG2MulWNAF/Big,_window:_3-32                4630     5412   239869 ns/op     224360 ns/op
BenchmarkG2MulWNAF/Fr,_window:_4-32                 5008     5554   239132 ns/op     203044 ns/op
BenchmarkG2MulWNAF/Big,_window:_4-32                4561     5437   242594 ns/op     234347 ns/op
BenchmarkG2MulWNAF/Fr,_window:_5-32                 5698     6247   228788 ns/op     221313 ns/op
BenchmarkG2MulWNAF/Big,_window:_5-32                4977     5979   277129 ns/op     237378 ns/op
BenchmarkG2MulWNAF/Fr,_window:_6-32                 3754            334291 ns/op
BenchmarkG2MulWNAF/Big,_window:_6-32                4237            345307 ns/op
BenchmarkG2MulWNAF/Fr,_window:_7-32                 2660            413984 ns/op
BenchmarkG2MulWNAF/Big,_window:_7-32                2678            461477 ns/op

BenchmarkG2MulGLV/Naive-32                          3782     4016   299864 ns/op     289786 ns/op
BenchmarkG2MulGLV/Fr,_window:_1-32                  6756     7719   193997 ns/op     153801 ns/op
BenchmarkG2MulGLV/Big,_window:_1-32                 5253     6594   203329 ns/op     167729 ns/op
BenchmarkG2MulGLV/Fr,_window:_2-32                  6775     7752   154446 ns/op     152584 ns/op
BenchmarkG2MulGLV/Big,_window:_2-32                 5881     8366   177652 ns/op     174940 ns/op
BenchmarkG2MulGLV/Fr,_window:_3-32                  7868     9421   160579 ns/op     133884 ns/op
BenchmarkG2MulGLV/Big,_window:_3-32                 6894     6538   188744 ns/op     163645 ns/op
BenchmarkG2MulGLV/Fr,_window:_4-32                  7194     7750   180463 ns/op     149795 ns/op
BenchmarkG2MulGLV/Big,_window:_4-32                 5620     6684   198709 ns/op     167899 ns/op
BenchmarkG2MulGLV/Fr,_window:_5-32                  6284     5954   233635 ns/op     186982 ns/op
BenchmarkG2MulGLV/Big,_window:_5-32                 4880     8649   215989 ns/op     204473 ns/op
BenchmarkG2MulGLV/Fr,_window:_6-32                  7910            290345 ns/op
BenchmarkG2MulGLV/Big,_window:_6-32                 3234            316511 ns/op
BenchmarkG2MulGLV/Fr,_window:_7-32                  5815            394321 ns/op
BenchmarkG2MulGLV/Big,_window:_7-32                 2563            431117 ns/op

BenchmarkG2MultiExp/100-32                           169      160  6877217 ns/op    7044235 ns/op
BenchmarkG2MultiExp/1000-32                           30       30 40613382 ns/op   41251641 ns/op
PASS

[protolambda]

Hey @kilic, can we productionize this improvement? I can polish the code and make the PR ready if you can give some initial feedback?

[kilic]

@protolambda Thanks! Sure we can add this improvement.

I'm happy with switching to []PointG1/2. Can you also apply the same to the MultiExpBig function?

Constant window size and smaller type also sound good to me. It seems like window size 4 gives the best result for G1 and both wnaf and glv multiplications. And again for both wnaf and glv in G2 there is a tiny difference between 3 and 4. Shared my results below and includes Fr type.

G1 wnaf

BenchmarkG1MulWNAF/Fr,_window:_1-8          8690            134234 ns/op
BenchmarkG1MulWNAF/Fr,_window:_2-8          8943            127914 ns/op
BenchmarkG1MulWNAF/Fr,_window:_3-8          9138            125774 ns/op
BenchmarkG1MulWNAF/Fr,_window:_4-8          9255            122877 ns/op
BenchmarkG1MulWNAF/Fr,_window:_5-8          9462            126541 ns/op

G1 glv

BenchmarkG1MulGLV/Fr,_window:_1-8          10000            100515 ns/op
BenchmarkG1MulGLV/Fr,_window:_2-8          13446             89177 ns/op
BenchmarkG1MulGLV/Fr,_window:_3-8          13623             87922 ns/op
BenchmarkG1MulGLV/Fr,_window:_4-8          13941             85936 ns/op
BenchmarkG1MulGLV/Fr,_window:_5-8          13285             89931 ns/op
BenchmarkG1MulGLV/Fr,_window:_6-8          10000            103667 ns/op

G2 wnaf

BenchmarkG2MulWNAF/Fr,_window:_1-8          3534            333858 ns/op
BenchmarkG2MulWNAF/Fr,_window:_2-8          3535            327158 ns/op
BenchmarkG2MulWNAF/Fr,_window:_3-8          3783            309761 ns/op
BenchmarkG2MulWNAF/Fr,_window:_4-8          3913            302329 ns/op
BenchmarkG2MulWNAF/Fr,_window:_5-8          3908            303272 ns/op
BenchmarkG2MulWNAF/Fr,_window:_6-8          3643            322695 ns/op

G1 glv

BenchmarkG2MulGLV/Fr,_window:_1-8           4539            232194 ns/op
BenchmarkG2MulGLV/Fr,_window:_2-8           5492            211677 ns/op
BenchmarkG2MulGLV/Fr,_window:_3-8           5928            195555 ns/op
BenchmarkG2MulGLV/Fr,_window:_4-8           6051            195011 ns/op
BenchmarkG2MulGLV/Fr,_window:_5-8           5682            205829 ns/op
BenchmarkG2MulGLV/Fr,_window:_6-8           4831            241982 ns/op