Public input permutation

[adr1anh]

Fixes #28, #22

• comment-out work queue
• change sumcheck initialization round to check that the last element is 0 (not necessary now since we don't do ZK, but creates the least amount of change)
• reinstore gamma challenge

Clang-tidy related

There are many changes that silence warnings from clang-tidy.

• Initializing class members when possible
• Explicitly deleting copy constructors/assignment operators for non-copyable classes
• Using = default for trivial destructors
• Adding noexcept to move constructors/assignment
• Using auto after static_cast to avoid duplication.
• Using std::move when copying an argument like a string or vector to help the compiler (references may alias, so passing by value tells the compiler that this value is really constant)
• Removing unused headers, and adding #pragma once to prevent import clashes.
• Reorder some imports, and use the module-relative paths
• Replace some typedef with using
• Use const auto& element : container to prevent unnecessary copies.

Circuit Constructor

• Replace n by num_gates to prevent clashes with n we use to refer to the subgroup_size
• Added comments about things that seemed fishy, (all marked with TODO(Adrian)
• Removed NUM_RESERVED_GATES and WireType since they are not used/duplicated
• Simplified set_public_input, though not sure if the removal of the ASSERT is correct.

Composer

• Refactor compute_proving_key_base
  • Take the CircuitConstructor by const reference and modify the behavior so we only modify the proving_key. We no longer add any padding or dummy gates to the constructor, and instead do the padding entirely over the polynomials.
  • Remove unnecessary zero-ing out of selector values by using the fact that polynomials are initialized to 0.
  • Explicitly state where the public inputs are stored.
• Refactor compute_witness_base
  • Use explicit types when referencing the circuit_constructor and ensure we only get objects that we don't modify.
  • Create array of wires to handle the program_width more generally.
  • No longer modify the circuit_constructor wires to add padding, and instead add 0-padding to the wire polynomials.
  • Remove fr::__copy
• Ensure all calls to circuit_constructor are const
• PermutationHelper: Big refactor to clarify handling of public inputs.
  • Simplify cycle_node behavior, and more generically handle different number of columns, and remove confusing WireType enum.
  • Make compute_wire_copy_cycles return a vector of CyclicPermutation for clarity.
  • Remove resize by noting that the number of cycles is equal to the number of variables
  • Reverse the order in which we were applying each CyclicPermutation to the sigma polynomials. Now, each cycle_node will map to the next one in the list.
  • Add comments to explain what parts of the function are necessary for our public input handling.
  • Changed the composer_test to also test for public inputs.
    • Add many more tests to ensure the permutation polynomials that we create have the expected form.
    • Compare results with the public_input_delta

Prover & Verifier

• Comment-out the work_queue related members.
• Restore gamma challenge that was removed due to a misunderstanding
• Remove default argument beta = 1 for the grand_product computation.

Sumcheck

• Add #pragma once to headers
• Modify the GrandProductComputationRelation to work with Z_perm that has the first coefficient equal to 0.
• Handle public_input_delta in GrandProductComputationRelation
• Modify GrandProductInitializationRelation to instead check that the last "real" value of Z_perm_shift is 0. In the current ZK-less situation, this is not necessary since it will be guaranteed by Gemini shift opening. But leaving it here for later.
• Renamed LAGRANGE_1 to LAGRANGE_FIRST to be consistent with other parts of the code.

[adr1anh]

Explanations for some changes

[Rumata888]

Everything seems great