Binius counts

[sragss]

Implements the conversion to multiplicities for Binius counts as described in Section 4.4 of the Binius paper. Does not wire it into the offline memory checking system yet.

Runs a similar strategy to halo2curves::PrimeField and our fork of Arkworks to precompute some tables to reduce the cost of calls to from_count_index. Unfortunately does not move all the pre-computation work to compile time as the Binius library does not currently have a const fn impl. If this does get implemented we can revert back to 1241739007964a4e2c2ba71fc07e76082a0ca925 which is substantially simpler.

Thaler's optimization strategy:

• Split the count $x$ such that $x = a \cdot 2^{16} + b$ with $a$ and $b$ each ranging from $0$ to $2^{16}$.
• Maintain two lookup tables:
  • One for $2^{16}$ powers of $g^{2^{16}}$.
  • Another for $2^{16}$ powers of $g$.
• Any count up to $2^{32}$ can be computed using two lookups and one multiplication:
  $$g^x = (g^{2^{16}})^a \cdot g^b$$

2^32 may be overkill and we would get significant speedups on the initialization time of tables by reducing size.