Lookups can take array values

[kmerrill18]

Create a version of value_in_array that works with arrays so we can do lookups of tuples. This implementation "hashes" (computes a random linear combination based on a verifier challenge) each array element into one field element, and then it hooks into the regular Habock lookup infrastructure.

[kmerrill18]

Constraint count comes out to (k - 2) + Ak + N:

• (k -2): compute k powers of the random challenge. The 0th and 1st power are given, so no constraints needed there, hence the -2.
• A(k - 1): the random linear combo of the tuple/array to lookup (the "hash"). (k - 1) constraints because the first value is multiplied by the challenge the 0th power, which is not actually a multiplication.
• A: regular inverse check per lookup with standard Habock lookup infrastructure
• N: regular table setup with standard Habock lookup infrastructure

So in total: (k - 2) + A(k - 1) + A + N = (k - 2) + Ak + N.

The (k - 2) term is perhaps unnecessary. Theoretically, the challenge is public, so the prover does not need to prove that it computed those powers correctly. However, circ puts this on the prover side instead of the verifier side. In practice, for our applications, N >> (k - 2), so this term should not matter, and I think it is fine to leave as-is.