Double-Precision towering

[mratsim]

This PR reworks the towering to introduce double-precision finite fields.

This moderately accelerate G2 operations and significantly improves Fp12 and pairings computations.

• Double-precision towering: now uses lazy reduction (related to Implement lazy carries and reductions #15 and continues FpDbl revisited #144)
• Consistent naming diffUnr for unrediced Fp operations instead of diffNoReduce
• double-width -> double-precision
• Consistent naming diff2xMod and diff2xUnr for double-precision operations.
  • prod2x, square2x and redc2x for double-precision modular arithmetic
• Removal of canUseNoCarryMontyMul and canUseNoCarryMontySquare to instead have getSpareBits to access the number of extra spare bits in the field element representation.
• Slightly improved assembly for add/sub via interleaving copies and operations and dependency chain latency hiding.
• Small addition chains for Bigint, Fp, FpDbl, extension fields and double-precision extension fields use less temporaries and copies
• Alternate formulae for Fp4[BN254_Snarks] squaring which is 25%, fix BN254-Snarks: bad performance on Fp4 squaring #154

[mratsim]

Some benches:

Master Fp2 BLS12-381

---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                                               Fp2[BLS12_381]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Substraction                                                           Fp2[BLS12_381]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Negation                                                               Fp2[BLS12_381]       200000000.000 ops/s             5 ns/op            17 CPU cycles (approx)
Conditional Copy                                                       Fp2[BLS12_381]       333333333.333 ops/s             3 ns/op            10 CPU cycles (approx)
Division by 2                                                          Fp2[BLS12_381]                 inf ops/s             0 ns/op             0 CPU cycles (approx)
Multiplication                                                         Fp2[BLS12_381]         9174311.927 ops/s           109 ns/op           328 CPU cycles (approx)
Squaring                                                               Fp2[BLS12_381]        13513513.514 ops/s            74 ns/op           223 CPU cycles (approx)
Inversion (constant-time default impl)                                 Fp2[BLS12_381]           68956.006 ops/s         14502 ns/op         43507 CPU cycles (approx)
Square Root + isSquare (constant-time default impl)                    Fp2[BLS12_381]           34209.086 ops/s         29232 ns/op         87700 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

Tower 2x Fp2 BLS12-381

---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                                               Fp2[BLS12_377]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Substraction                                                           Fp2[BLS12_377]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Negation                                                               Fp2[BLS12_377]       200000000.000 ops/s             5 ns/op            15 CPU cycles (approx)
Conditional Copy                                                       Fp2[BLS12_377]       333333333.333 ops/s             3 ns/op            10 CPU cycles (approx)
Division by 2                                                          Fp2[BLS12_377]                 inf ops/s             0 ns/op             0 CPU cycles (approx)
Multiplication                                                         Fp2[BLS12_377]         7751937.984 ops/s           129 ns/op           389 CPU cycles (approx)
Squaring                                                               Fp2[BLS12_377]         8547008.547 ops/s           117 ns/op           353 CPU cycles (approx)
Inversion (constant-time default impl)                                 Fp2[BLS12_377]           70264.193 ops/s         14232 ns/op         42698 CPU cycles (approx)
Square Root + isSquare (constant-time default impl)                    Fp2[BLS12_377]            8640.205 ops/s        115738 ns/op        347223 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                                               Fp2[BLS12_381]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Substraction                                                           Fp2[BLS12_381]       125000000.000 ops/s             8 ns/op            24 CPU cycles (approx)
Negation                                                               Fp2[BLS12_381]       200000000.000 ops/s             5 ns/op            15 CPU cycles (approx)
Conditional Copy                                                       Fp2[BLS12_381]       333333333.333 ops/s             3 ns/op            10 CPU cycles (approx)
Division by 2                                                          Fp2[BLS12_381]                 inf ops/s             0 ns/op             0 CPU cycles (approx)
Multiplication                                                         Fp2[BLS12_381]        10204081.633 ops/s            98 ns/op           294 CPU cycles (approx)
Squaring                                                               Fp2[BLS12_381]        14084507.042 ops/s            71 ns/op           213 CPU cycles (approx)
Inversion (constant-time default impl)                                 Fp2[BLS12_381]           69386.622 ops/s         14412 ns/op         43237 CPU cycles (approx)
Square Root + isSquare (constant-time default impl)                    Fp2[BLS12_381]           34511.320 ops/s         28976 ns/op         86932 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

So 10% improvement in multiplication and squaring

Master GT / Fp12 BLS12-381

---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                                               Fp12[BLS12_381]       22222222.222 ops/s            45 ns/op           136 CPU cycles (approx)
Substraction                                                           Fp12[BLS12_381]       22727272.727 ops/s            44 ns/op           133 CPU cycles (approx)
Negation                                                               Fp12[BLS12_381]       32258064.516 ops/s            31 ns/op            93 CPU cycles (approx)
Multiplication                                                         Fp12[BLS12_381]         389408.100 ops/s          2568 ns/op          7704 CPU cycles (approx)
Squaring                                                               Fp12[BLS12_381]         590318.772 ops/s          1694 ns/op          5082 CPU cycles (approx)
Inversion (constant-time default impl)                                 Fp12[BLS12_381]          52162.120 ops/s         19171 ns/op         57516 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

Tower 2x GT / Fp12 BLS12-381

---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition                                                               Fp12[BLS12_381]       22727272.727 ops/s            44 ns/op           134 CPU cycles (approx)
Substraction                                                           Fp12[BLS12_381]       22727272.727 ops/s            44 ns/op           132 CPU cycles (approx)
Negation                                                               Fp12[BLS12_381]       34482758.621 ops/s            29 ns/op            89 CPU cycles (approx)
Multiplication                                                         Fp12[BLS12_381]         465332.713 ops/s          2149 ns/op          6449 CPU cycles (approx)
Squaring                                                               Fp12[BLS12_381]         648508.431 ops/s          1542 ns/op          4628 CPU cycles (approx)
Inversion (constant-time default impl)                                 Fp12[BLS12_381]          54153.580 ops/s         18466 ns/op         55399 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------

So a 16% improvement on multiplication and 9% on squarings

Master Pairing BLS12-381

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Line double                                                  BLS12_381            931966.449 ops/s          1073 ns/op          3216 CPU cycles (approx)
Line add                                                     BLS12_381            654450.262 ops/s          1528 ns/op          4583 CPU cycles (approx)
Mul 𝔽p12 by line xy000z                                      BLS12_381            586166.471 ops/s          1706 ns/op          5116 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Final Exponentiation Easy                                    BLS12_381             40466.170 ops/s         24712 ns/op         74134 CPU cycles (approx)
Final Exponentiation Hard BLS12                              BLS12_381              2409.360 ops/s        415048 ns/op       1245168 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Miller Loop BLS12                                            BLS12_381              3311.302 ops/s        301996 ns/op        906004 CPU cycles (approx)
Final Exponentiation BLS12                                   BLS12_381              2273.296 ops/s        439890 ns/op       1319693 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Pairing BLS12                                                BLS12_381              1295.107 ops/s        772137 ns/op       2316448 CPU cycles (approx)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Tower 2x Pairing BLS12-381

Line double                              BLS12_381            960614.793 ops/s          1041 ns/op          3121 CPU cycles (approx)
Line add                                 BLS12_381            709219.858 ops/s          1410 ns/op          4229 CPU cycles (approx)
Mul 𝔽p12 by line xy000z                  BLS12_381            682128.240 ops/s          1466 ns/op          4396 CPU cycles (approx)
------------------------------------------------------------------------------------------------------------------------------------
Final Exponentiation Easy                BLS12_381             43402.778 ops/s         23040 ns/op         69118 CPU cycles (approx)
Final Exponentiation Hard BLS12          BLS12_381              2621.177 ops/s        381508 ns/op       1144545 CPU cycles (approx)
------------------------------------------------------------------------------------------------------------------------------------
Miller Loop BLS12                        BLS12_381              3676.984 ops/s        271962 ns/op        815898 CPU cycles (approx)
Final Exponentiation BLS12               BLS12_381              2472.396 ops/s        404466 ns/op       1213422 CPU cycles (approx)
------------------------------------------------------------------------------------------------------------------------------------
Pairing BLS12                            BLS12_381              1419.436 ops/s        704505 ns/op       2113557 CPU cycles (approx)
------------------------------------------------------------------------------------------------------------------------------------

So a 10% improvement overall.

Future perspectives

There are some low-hanging fruits that can help accelerate further:

• Use double-width in Cubic extension fields, right now we are only using it on Fp->Fp2 and Fp2->Fp4 but not on Fp4->Fp12 and we might actually get even larger savings than this PR.
• There are 3 inversions needed for pairings, one on G1/Fp, one on G2/Fp2, one on GT/Fp12, this represents ~150000 cycles or about 50ms which is 7% of running time that can be squashed at least 5x with new (constant-time) inversion techniques from this summer and G1 and G2 inversion can be fused using Montgomery simultaneous inversion Simultaneous inversion #49.

[mratsim]

testutils brings chronos which brings BearSSL

   Success: testutils installed successfully.

 Installing chronos@any version

Downloading https://github.com/status-im/nim-chronos using git

  Verifying dependencies for chronos@2.5.2

      Info: Dependency on stew@any version already satisfied

  Verifying dependencies for stew@0.1.0

 Installing bearssl@any version

Downloading https://github.com/status-im/nim-bearssl using git

       Tip: 136 messages have been suppressed, use --verbose to show them.

     Error: Execution failed with exit code 1

        ... Command: git clone --recursive --shallow-submodules --depth 1 https://github.com/status-im/nim-bearssl /tmp/nimble_5788/githubcom_statusimnimbearssl

        ... Output: Cloning into '/tmp/nimble_5788/githubcom_statusimnimbearssl'...

        ... Submodule 'bearssl/csources' (https://www.bearssl.org/git/BearSSL) registered for path 'bearssl/csources'

        ... Cloning into '/tmp/nimble_5788/githubcom_statusimnimbearssl/bearssl/csources'...

        ... error: Server does not allow request for unadvertised object dda1f8a0c46e15b4a235163470ff700b2f13dcc5

        ... Fetched in submodule path 'bearssl/csources', but it did not contain dda1f8a0c46e15b4a235163470ff700b2f13dcc5. Direct fetching of that commit failed.