NTT-learning/notebooks/foundations/02_negative_wrapped_ntt/studio.ipynb

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        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "orientation",
          "title": "Studio Goals"
        }
      },
      "source": "## META | difficulty 1 | Studio Goals\n\nThe studio compares direct positive-wrapped and negative-wrapped transforms so the learner stops treating \u201cNTT\u201d as one unqualified object.\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "explanation",
          "title": "Same Input, Different Transform Story"
        }
      },
      "source": "## MANDATORY | difficulty 3 | Same Input, Different Transform Story\n\nThe same coefficient vector can be sent through two different transform stories depending on the quotient ring.\nThat difference is not implementation noise. It is structural.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "demo",
          "title": "Compare Positive-Wrapped And Negative-Wrapped Views"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 3 | Compare Positive-Wrapped And Negative-Wrapped Views\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import find_primitive_root, find_psi, forward_ntt, forward_ntt_psi\nfrom ntt_learning.visuals import plot_vector_comparison\n\nsignal = [1, 2, 3, 4]\nmodulus = 17\nomega = find_primitive_root(4, modulus)\npsi = find_psi(4, modulus)\n\npositive = forward_ntt(signal, modulus, omega)\nnegative = forward_ntt_psi(signal, modulus, psi)\n\nprint(\"positive-wrapped NTT:\", positive)\nprint(\"negative-wrapped NTT_psi:\", negative)\ndisplay(\n    plot_vector_comparison(\n        positive,\n        negative,\n        left_label=\"positive\",\n        right_label=\"negative\",\n        title=\"Same signal, different transform story\",\n    )\n)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "Debug Checklist"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Debug Checklist\n\nIf a direct transform result looks suspicious, inspect:\n\n1. the chosen modulus\n2. the order of the root\n3. whether you are using `\u03c9` or `\u03c8`\n4. whether the inverse includes `n^-1`\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "See A Wrong-Root Failure"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 2 | See A Wrong-Root Failure\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import find_primitive_root, find_psi, forward_ntt_psi\nfrom ntt_learning.visuals import plot_vector_comparison\n\nsignal = [1, 2, 3, 4]\nmodulus = 17\nomega = find_primitive_root(4, modulus)\npsi = find_psi(4, modulus)\n\ncorrect = forward_ntt_psi(signal, modulus, psi)\nwrong = forward_ntt_psi(signal, modulus, omega)\n\nprint(\"correct psi-based transform:\", correct)\nprint(\"wrongly using omega as if it were psi:\", wrong)\ndisplay(\n    plot_vector_comparison(\n        wrong,\n        correct,\n        left_label=\"wrong_root\",\n        right_label=\"correct\",\n        title=\"Wrong root vs correct root\",\n    )\n)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "reflection",
          "title": "Reflection"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Reflection\n\nExplain why \u201cpick any root of unity\u201d is not an acceptable habit in this subject.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "facultative",
          "difficulty": 4,
          "kind": "exploration",
          "title": "Optional: Matrix Comparison"
        }
      },
      "outputs": [],
      "source": "# FACULTATIVE | difficulty 4 | Optional: Matrix Comparison\n\nfrom ntt_learning.toy_ntt import find_primitive_root, find_psi, ntt_psi_matrix\n\nmodulus = 17\nomega = find_primitive_root(4, modulus)\npsi = find_psi(4, modulus)\n\nprint(\"omega:\", omega)\nprint(\"psi:\", psi)\nfor row in ntt_psi_matrix(4, modulus, psi):\n    print(row)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "handoff",
          "title": "Next Notebook"
        }
      },
      "source": "## META | difficulty 1 | Next Notebook\n\nNext notebook: `../../../butterfly_mechanics/03_fast_forward_ct/lecture.ipynb`\n"
    }
  ],
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      "title": "Studio: Negative-Wrapped NTT",
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        "notebooks/butterfly_mechanics/04_fast_inverse_gs/studio.ipynb",
        "notebooks/kyber_mapping/05_kyber_ntt_and_base_multiplication/lecture.ipynb",
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