Fix Frobenius implementation in GoldilocksExt2

[jsemldonado]

Summary

• Fixes incorrect Frobenius automorphism in GoldilocksExt2
• Old code computed frob(a0 + a1x) = -a0 + a1x (wrong)
• New code computes frob(a0 + a1x) = a0 - a1x (correct)
• The bug didn't break inversion because errors cancelled out, but Frobenius was mathematically wrong

Test plan

• Added 3 tests verifying Frobenius properties:
  • frob^2 == identity
  • frob fixes base field
  • frob(X) == -X
• All existing goldilocks tests pass
• All uni_kzg tests pass

[gemini-code-assist]

Summary of Changes

Hello @jsemldonado, I'm Gemini Code Assist¹! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request addresses a critical mathematical inaccuracy in the GoldilocksExt2 Frobenius automorphism implementation. The change ensures the Frobenius operation now correctly applies the transformation a0 - a1*x, which is essential for cryptographic primitives relying on finite field arithmetic. The fix is accompanied by new, specific tests to validate the corrected behavior and maintain the integrity of the field operations.

Highlights

• Frobenius Automorphism Correction: The implementation of the Frobenius automorphism in GoldilocksExt2 was corrected. The old code incorrectly computed frob(a0 + a1*x) = -a0 + a1*x, which has been updated to the mathematically accurate frob(a0 + a1*x) = a0 - a1*x.
• Mathematical Correctness Ensured: While the previous incorrect implementation did not break inversion due to error cancellation, the new implementation ensures the mathematical correctness of the Frobenius operation, which is critical for field arithmetic.
• Comprehensive Test Coverage Added: Three new dedicated tests were introduced to rigorously verify key Frobenius properties: frob^2 == identity, frob fixes base field elements, and frob(X) == -X.
• No Regressions Introduced: All existing Goldilocks and uni_kzg tests continue to pass, confirming that no regressions were introduced by these changes.

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[gemini-code-assist]

Code Review

This pull request correctly fixes the implementation of the Frobenius automorphism in GoldilocksExt2. The new logic is mathematically sound, and the added tests provide good coverage for the key properties of the Frobenius map. I have one suggestion to make the code more idiomatic and improve readability by removing a magic number.

[gemini-code-assist]

To improve readability and maintainability, you can replace the NEG_ONE constant and its usage with a direct negation. The Neg trait is implemented for Goldilocks, so you can use the - operator. This makes the code more idiomatic and removes the magic number 18446744069414584320.

        Self {
            v: [arr[0], -arr[1]],
        }