feat(avm): ToRadix gadget

[sirasistant]

Implements a gadget to decompose fields in LE radixes, and integrates this gadget into scalar mul.
The tricky part about the gadget is avoiding overflows in the last non padding limb. In order to avoid that we have precomputed the 255 possible decompositions of P (radix goes from 2 to 256 inclusive) and we compare that the accumulator must be under P if we reach the unsafe limb.

+-------+-------+------+--------+------------+------+-------------+----------+------------------+-------+------------+-----------+-------+-----+-----+---------+
| value | radix | limb | p_limb | limb_index | acc  | acc_under_p | exponent | not_padding_limb | found | safe_limbs | is_unsafe | start | end | sel | not_end |
+-------+-------+------+--------+------------+------+-------------+----------+------------------+-------+------------+-----------+-------+-----+-----+---------+
|     0 |     0 |    0 |      0 |          0 |    0 |           0 | 0        |                0 |     0 |          0 |         0 |     0 |   0 |   0 |       0 |
|  1337 |    10 |    7 |      7 |          0 |    7 |           0 | 1        |                1 |     0 |         76 |         0 |     1 |   0 |   1 |       1 |
|  1337 |    10 |    3 |      1 |          1 |   37 |           0 | 10       |                1 |     0 |         76 |         0 |     0 |   0 |   1 |       1 |
|  1337 |    10 |    3 |      6 |          2 |  337 |           1 | 100      |                1 |     0 |         76 |         0 |     0 |   0 |   1 |       1 |
|  1337 |    10 |    1 |      5 |          3 | 1337 |           1 | 1000     |                1 |     1 |         76 |         0 |     0 |   0 |   1 |       1 |
|  1337 |    10 |    0 |      9 |          4 | 1337 |           1 | 10000    |                1 |     1 |         76 |         0 |     0 |   0 |   1 |       1 |
|  1337 |    10 |    0 |      2 |         76 | 1337 |           1 | 10**76   |                1 |     1 |         76 |         1 |     0 |   0 |   1 |       1 |
|  1337 |    10 |    0 |      0 |         77 | 1337 |           0 | 0        |                0 |     1 |         76 |         0 |     0 |   1 |   1 |       0 |
|     0 |     0 |    0 |      0 |          0 |    0 |           0 | 0        |                0 |     0 |          0 |         0 |     0 |   0 |   0 |       0 |
+-------+-------+------+--------+------------+------+-------------+----------+------------------+-------+------------+-----------+-------+-----+-----+---------+

Bench

Running bin/relations_acc_bench
Run on (192 X 3599.39 MHz CPU s)
CPU Caches:
  L1 Data 32 KiB (x96)
  L1 Instruction 32 KiB (x96)
  L2 Unified 512 KiB (x96)
  L3 Unified 32768 KiB (x12)
Load Average: 18.30, 23.96, 21.02
-------------------------------------------------------------------------------
Benchmark                                     Time             CPU   Iterations
-------------------------------------------------------------------------------
alu_acc                                   0.139 us        0.139 us      5053009
bc_decomposition_acc                       13.4 us         13.4 us        51950
bc_decomposition_interactions_acc          1.24 us         1.24 us       563554
bc_hashing_acc                            0.848 us        0.848 us       831879
bc_hashing_interactions_acc                1.54 us         1.54 us       456115
bc_retrieval_acc                          0.024 us        0.024 us     21634002
bc_retrieval_interactions_acc             0.936 us        0.935 us       765095
bitwise_acc                                1.54 us         1.54 us       446026
bitwise_interactions_acc                  0.930 us        0.930 us       760097
class_id_derivation_acc                   0.176 us        0.176 us      3988480
class_id_derivation_interactions_acc       1.04 us         1.04 us       678844
ecc_acc                                    2.60 us         2.60 us       268246
execution_acc                             0.522 us        0.522 us      1341997
instr_fetching_acc                         7.15 us         7.15 us        97965
instr_fetching_interactions_acc            3.94 us         3.94 us       177737
poseidon2_hash_acc                         2.04 us         2.04 us       346984
poseidon2_hash_interactions_acc           0.744 us        0.744 us       938393
poseidon2_perm_acc                         46.3 us         46.3 us        15488
range_check_acc                            2.69 us         2.69 us       249636
range_check_interactions_acc               3.58 us         3.58 us       197211
scalar_mul_acc                             2.15 us         2.15 us       325391
scalar_mul_interactions_acc                2.12 us         2.12 us       331354
sha256_acc                                 6.36 us         6.36 us       112293
sha256_interactions_acc                   0.406 us        0.406 us      1736291
to_radix_acc                               3.57 us         3.56 us       195897
to_radix_interactions_acc                  1.90 us         1.90 us       369766

[fcarreiro]

I'll do a deeper review on Monday but I guess @IlyasRidhuan will check the PIL.

Could you run the benchmarks and add it to the description?

[IlyasRidhuan]

PIL LGTM, just one comment but that doesnt change functionality.

[fcarreiro]

I guess this is fine for now but in reality, this conversion will have to probably be either done and/or constrained by the caller. (someone has to constrain it)

[sirasistant]

The pil ensures that limb will be less than radix, so to radix pil constrains that for radix = 2, limb is boolean. A different topic is memory tagging to u1, that needs to be done by the opcode, but it's not gadget business hehehe