bench: verifying N poseidon permutation over goldilocks field using `spartan_parallel`

[kunxian-xia]

[kunxian-xia]

The output of this issue will fill the missing numbers in this table.

proof system	#(permutations)	recursion proof size	proving time	recursion proof verification time
SP1 poseidon2 chip with deg = 3	2^20	501305 (489KB)	327s	tbd
SP1 poseidon2 chip with deg = 9	2^20	487397 (476KB)	223s	tbd
SP1 poseidon2 chip with deg = 3	2^17	421385 (411KB)	41s	tbd
SP1 poseidon2 chip with deg = 9	2^17	407477 (398KB)	28s	tbd
spartan-parallel

Notes

1. poseidon2 chip with deg = 3 in SP1: uses 313 columns and 49 preprocessed columns.
2. poseidon2 chip with deg = 9 in SP1: uses 172 columns and 49 preprocessed columns.

[kunxian-xia]

kunxian-xia/sp1#7

[kunxian-xia]

According to @Jiangkm3, to verify 2^17 poseidon permutations using spartan-parallel.

• The spartan proof (without mpcs opening proof) size is around 32KB.
• And the witness assignment's MLE has 27 variables.

And according to @yczhangsjtu, the WHIR opening proof for MLE with 27 variables has 113KB.

However, we cannot get the proving time based on current impl because most of the routines in spartan-parallel are single threaded right now. We need to resolve #55 first.