feat: merge-train/barretenberg#21318
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resolves AztecProtocol/barretenberg#1585 tldr: we make the offset generator a free witness while performing `batch_mul` ### Current Way of Masking Points While performing `batch_mul` we create ROM tables of linear combinations of the group elements. If we have an MSM of the form $[(s_1, s_2, \dots, s_n), (G_1, G_2, \dots, G_n)]$ then the ROM tables are of the form: | Index | Element | | --- | --- | | 0 | $G_1 + G_2 + \dots + G_5 + G_6$ | | 1 | $G_1 + G_2 + \dots + G_5 - G_6$ | | 2 | $G_1 + G_2 + \dots - G_5 - G_6$ | | $\vdots$ | $\vdots$ | | $2^6 - 1$ | $- G_1 - G_2 - \dots - G_5 - G_6$ | To avoid any entries of the ROM table to be a point at infinity, we add a multiple of an offset generator: $G'_i := G_i + 2^{i-1}\delta \cdot G\_{\textsf{offset}} \quad \forall i \in [0, 6).$ and then we create a ROM table with $(G'_1, G'_2, \dots, G'_6)$. Here, $\delta$ is a verifier-sent challenge so the prover cannot know it beforehand. ### Proposed Masking of Points Instead of sampling a challenge $\delta$, we make the offset generator $G\_{\textsf{offset}}$ a free witness in the circuit, so the masked generators become: $G'_i := G_i + 2^{i-1}\cdot G\_{\textsf{offset}} \quad \forall i \in [0, 6).$ We only constrain $G\_{\textsf{offset}}$ to be a valid point on the curve. This approach should also be safe because if a malicious prover tries to exploit the fact that he can freely choose $G\_{\textsf{offset}}$ the circuit will be unsatisfiable. An honest prover must choose a random point on the curve as $G\_{\textsf{offset}}$.
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### 🧾 Audit Context Stdlib curve audit ### 🛠️ Changes Made Only refactoring: remove some unused aliases in the structs. ### ✅ Checklist - [X] Audited all methods of the relevant module/class - [ ] Audited the interface of the module/class with other (relevant) components - [ ] Documented existing functionality and any changes made (as per Doxygen requirements) - [ ] Resolved and/or closed all issues/TODOs pertaining to the audited files - [ ] Confirmed and documented any security or other issues found (if applicable) - [ ] Verified that tests cover all critical paths (and added tests if necessary) - [X] Updated audit tracking for the files audited (check the start of each file you audited) ### 📌 Notes for Reviewers
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BEGIN_COMMIT_OVERRIDE
chore: use free witness for offset generator in
batch_mul(#21040)chore: Stdlib curves audit (#21177)
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