Add option for algebraic batching to build DEEP polynomial

[Al-Kindi-0]

We probably need to sync the next branch so that we can change the base branch for this PR.
Once #356 is merged, I can update the security estimator in this PR.

[irakliyk]

Looks good! Thank you! I left some comments inline.

Also, shouldn't proof security estimates be affected by the type of batching we do? Or is this too difficult to incorporate in the current structure?

[irakliyk]

Also, let's resolve the merge conflict against the current main.

[Al-Kindi-0]

Also, shouldn't proof security estimates be affected by the type of batching we do? Or is this too difficult to incorporate in the current structure?

Exactly! The only thing we need is the the widths of each of the committed traces or the total number of committed columns. If this number is say N, then there will be a (potential) loss of log2(N-1).

[irakliyk]

Looks good! Thank you! I left a couple more comments inline.

[irakliyk]

Exactly! The only thing we need is the the widths of each of the committed traces or the total number of committed columns. If this number is say N, then there will be a (potential) loss of log2(N-1).

Would we do it in this PR? Or should we leave it for a follow-up?

[Al-Kindi-0]

Some clippy warnings related to div_ceil are appearing, think we should just supress them as we are not really doing div_ceil

[Al-Kindi-0]

Exactly! The only thing we need is the the widths of each of the committed traces or the total number of committed columns. If this number is say N, then there will be a (potential) loss of log2(N-1).

Would we do it in this PR? Or should we leave it for a follow-up?

I propose we do it in a subsequent PR

[irakliyk]

Some clippy warnings related to div_ceil are appearing, think we should just supress them as we are not really doing div_ceil

I guess these are functionally equivalent to div_ceil but, yeah - if that's not what we are doing, I'd suppress them.

[irakliyk]

Looks good! Thank you! I left some comments inline (mostly about rename BatchType into BatchMethod). After these are addressed, we can merge.