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

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "theme",
          "title": "Notebook Theme"
        }
      },
      "source": "## META\n\n<style>\n.jp-RenderedHTMLCommon h2 {\n  display: none;\n}\n\n.jp-RenderedHTMLCommon .ntt-cell-head {\n  position: relative;\n  margin: 0 0 0.8rem 0;\n  padding: 0.95rem 1rem 0.95rem 1rem;\n  border-radius: 16px;\n  border: 1px solid rgba(16,42,67,0.12);\n  box-shadow: 0 8px 24px rgba(16,42,67,0.06);\n  font-family: \"Avenir Next\", \"Trebuchet MS\", sans-serif;\n}\n\n.jp-RenderedHTMLCommon .ntt-cell-row {\n  display: flex;\n  flex-wrap: wrap;\n  gap: 0.45rem;\n  align-items: center;\n  margin-bottom: 0.55rem;\n}\n\n.jp-RenderedHTMLCommon .ntt-role-pill,\n.jp-RenderedHTMLCommon .ntt-kind-pill,\n.jp-RenderedHTMLCommon .ntt-difficulty-pill {\n  display: inline-flex;\n  align-items: center;\n  padding: 0.18rem 0.58rem;\n  border-radius: 999px;\n  font-size: 0.72rem;\n  font-weight: 800;\n  letter-spacing: 0.04em;\n  text-transform: uppercase;\n}\n\n.jp-RenderedHTMLCommon .ntt-role-meta {\n  background: linear-gradient(135deg, #e2f3ff 0%, #f5fbff 100%);\n}\n\n.jp-RenderedHTMLCommon .ntt-role-mandatory {\n  background: linear-gradient(135deg, #fff0b3 0%, #ffd58a 100%);\n}\n\n.jp-RenderedHTMLCommon .ntt-role-facultative {\n  background: linear-gradient(135deg, #ffe1d6 0%, #ffc9b9 100%);\n}\n\n.jp-RenderedHTMLCommon .ntt-role-pill {\n  color: #102a43;\n  background: rgba(16,42,67,0.1);\n}\n\n.jp-RenderedHTMLCommon .ntt-kind-pill {\n  color: #486581;\n  background: rgba(255,255,255,0.72);\n}\n\n.jp-RenderedHTMLCommon .ntt-difficulty-pill {\n  position: absolute;\n  top: 0.95rem;\n  right: 1rem;\n  color: #102a43;\n  background: rgba(255,255,255,0.88);\n  border: 1px solid rgba(16,42,67,0.08);\n  box-shadow: 0 4px 12px rgba(16,42,67,0.08);\n}\n\n.jp-RenderedHTMLCommon .ntt-cell-title {\n  font-size: 1.25rem;\n  line-height: 1.25;\n  font-weight: 850;\n  color: #102a43;\n  padding-right: 5.3rem;\n}\n\n.jp-RenderedHTMLCommon .ntt-cell-body-hint {\n  margin-top: 0.3rem;\n  font-size: 0.84rem;\n  color: #486581;\n}\n\n.jupyter-widgets.widget-container,\n.jupyter-widgets.widget-box,\n.jupyter-widgets.widget-vbox,\n.jupyter-widgets.widget-hbox {\n  width: 100% !important;\n  max-width: 100% !important;\n  min-width: 0 !important;\n}\n\n.widget-html-content svg {\n  display: block !important;\n  width: 100% !important;\n  max-width: 100% !important;\n  height: auto !important;\n}\n</style>\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "route_nav",
          "title": "Route Guardrails"
        }
      },
      "source": "## META\n\n<div class=\"ntt-cell-head ntt-role-meta\">\n  <span class=\"ntt-difficulty-pill\">Level 1</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">META</span>\n    <span class=\"ntt-kind-pill\">route nav</span>\n  </div>\n  <div class=\"ntt-cell-title\">Route Guardrails</div>\n</div>\n\nYou are at **step 7 of 27** in the only supported route.\n\nNever choose the next notebook manually from the file tree. Use only the links in this cell and the final handoff cell.\n\n**Immediate navigation**\n- Next notebook: [Foundations / 02 Negative Wrapped NTT / Lab](lab.ipynb)\n- Previous notebook: [Foundations / 01 Convolution To Toy NTT / Studio](../01_convolution_to_toy_ntt/studio.ipynb)\n- Restart route: [Start Here](../../START_HERE.ipynb)\n\n**Official route chain**\n1. [Start Here](../../START_HERE.ipynb)\n2. [Course Blueprint](../../COURSE_BLUEPRINT.ipynb)\n3. [Foundations / 01 Convolution To Toy NTT / Lecture](../01_convolution_to_toy_ntt/lecture.ipynb)\n4. [Foundations / 01 Convolution To Toy NTT / Lab](../01_convolution_to_toy_ntt/lab.ipynb)\n5. [Foundations / 01 Convolution To Toy NTT / Problems](../01_convolution_to_toy_ntt/problems.ipynb)\n6. [Foundations / 01 Convolution To Toy NTT / Studio](../01_convolution_to_toy_ntt/studio.ipynb)\n7. **Foundations / 02 Negative Wrapped NTT / Lecture** <- you are here\n8. [Foundations / 02 Negative Wrapped NTT / Lab](lab.ipynb)\n9. [Foundations / 02 Negative Wrapped NTT / Problems](problems.ipynb)\n10. [Foundations / 02 Negative Wrapped NTT / Studio](studio.ipynb)\n11. [Butterfly Mechanics / 03 Fast Forward CT / Lecture](../../butterfly_mechanics/03_fast_forward_ct/lecture.ipynb)\n12. [Butterfly Mechanics / 03 Fast Forward CT / Lab](../../butterfly_mechanics/03_fast_forward_ct/lab.ipynb)\n13. [Butterfly Mechanics / 03 Fast Forward CT / Problems](../../butterfly_mechanics/03_fast_forward_ct/problems.ipynb)\n14. [Butterfly Mechanics / 03 Fast Forward CT / Studio](../../butterfly_mechanics/03_fast_forward_ct/studio.ipynb)\n15. [Butterfly Mechanics / 04 Fast Inverse GS / Lecture](../../butterfly_mechanics/04_fast_inverse_gs/lecture.ipynb)\n16. [Butterfly Mechanics / 04 Fast Inverse GS / Lab](../../butterfly_mechanics/04_fast_inverse_gs/lab.ipynb)\n17. [Butterfly Mechanics / 04 Fast Inverse GS / Problems](../../butterfly_mechanics/04_fast_inverse_gs/problems.ipynb)\n18. [Butterfly Mechanics / 04 Fast Inverse GS / Studio](../../butterfly_mechanics/04_fast_inverse_gs/studio.ipynb)\n19. [Kyber Mapping / 05 Kyber NTT And Base Multiplication / Lecture](../../kyber_mapping/05_kyber_ntt_and_base_multiplication/lecture.ipynb)\n20. [Kyber Mapping / 05 Kyber NTT And Base Multiplication / Lab](../../kyber_mapping/05_kyber_ntt_and_base_multiplication/lab.ipynb)\n21. [Kyber Mapping / 05 Kyber NTT And Base Multiplication / Problems](../../kyber_mapping/05_kyber_ntt_and_base_multiplication/problems.ipynb)\n22. [Kyber Mapping / 05 Kyber NTT And Base Multiplication / Studio](../../kyber_mapping/05_kyber_ntt_and_base_multiplication/studio.ipynb)\n23. [Professional / 06 Debugging NTT Failures / Lecture](../../professional/06_debugging_ntt_failures/lecture.ipynb)\n24. [Professional / 06 Debugging NTT Failures / Lab](../../professional/06_debugging_ntt_failures/lab.ipynb)\n25. [Professional / 06 Debugging NTT Failures / Problems](../../professional/06_debugging_ntt_failures/problems.ipynb)\n26. [Professional / 06 Debugging NTT Failures / Studio](../../professional/06_debugging_ntt_failures/studio.ipynb)\n27. [Course Complete](../../COURSE_COMPLETE.ipynb)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "orientation",
          "title": "Objectives"
        }
      },
      "source": "## META\n\n<div class=\"ntt-cell-head ntt-role-meta\">\n  <span class=\"ntt-difficulty-pill\">Level 1</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">META</span>\n    <span class=\"ntt-kind-pill\">orientation</span>\n  </div>\n  <div class=\"ntt-cell-title\">Objectives</div>\n</div>\n\nThis bundle introduces the transform itself in its negacyclic form.\n\nFocus:\n\n- the difference between `\u03c9` and `\u03c8`\n- direct NTT\u03c8 and INTT\u03c8\n- the direct convolution theorem in the negacyclic setting\n- why this is still too slow at `O(n^2)` without butterflies\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "explanation",
          "title": "Why \u03c8 Shows Up"
        }
      },
      "source": "## MANDATORY\n\n<div class=\"ntt-cell-head ntt-role-mandatory\">\n  <span class=\"ntt-difficulty-pill\">Level 2</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">MANDATORY</span>\n    <span class=\"ntt-kind-pill\">explanation</span>\n  </div>\n  <div class=\"ntt-cell-title\">Why \u03c8 Shows Up</div>\n</div>\n\nFor negative-wrapped convolution, the clean transform formula uses a `2n`-th root `\u03c8` with:\n\n- `\u03c8^2 = \u03c9`\n- `\u03c8^n = -1`\n\nThat is what bakes the negacyclic sign rule into the transform itself.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "demo",
          "title": "Inspect \u03c9, \u03c8, And The Direct Transform Matrix"
        }
      },
      "outputs": [],
      "source": "# MANDATORY\n# title: Inspect \u03c9, \u03c8, And The Direct Transform Matrix\n# kind: demo | level: 2\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import find_primitive_root, find_psi, ntt_psi_exponent_grid, ntt_psi_matrix\nfrom ntt_learning.visuals import direct_ntt_player, plot_ntt_psi_exponent_heatmap, plot_ntt_psi_matrix_heatmap\n\nmodulus = 17\nn = 4\nomega = find_primitive_root(n, modulus)\npsi = find_psi(n, modulus)\n\nprint(\"omega:\", omega)\nprint(\"psi:\", psi)\nprint(\"exponent grid:\")\nfor row in ntt_psi_exponent_grid(n):\n    print(row)\nprint(\"NTT_psi matrix:\")\nfor row in ntt_psi_matrix(n, modulus, psi):\n    print(row)\n\ndisplay(direct_ntt_player([1, 2, 3, 4], modulus, psi))\ndisplay(plot_ntt_psi_exponent_heatmap(n, title=\"Exponents 2ij + i for n=4\"))\ndisplay(plot_ntt_psi_matrix_heatmap(n, modulus, psi, title=\"Concrete NTT_psi matrix in Z_17\"))\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "explanation",
          "title": "Direct NTT\u03c8 Is Mechanically Clear But Still Quadratic"
        }
      },
      "source": "## MANDATORY\n\n<div class=\"ntt-cell-head ntt-role-mandatory\">\n  <span class=\"ntt-difficulty-pill\">Level 2</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">MANDATORY</span>\n    <span class=\"ntt-kind-pill\">explanation</span>\n  </div>\n  <div class=\"ntt-cell-title\">Direct NTT\u03c8 Is Mechanically Clear But Still Quadratic</div>\n</div>\n\nThe direct transform is useful because every coefficient and every exponent is visible.\nIt is not yet efficient. It still performs the full matrix multiplication.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "demo",
          "title": "Run A Direct NTT\u03c8 / INTT\u03c8 Round Trip"
        }
      },
      "outputs": [],
      "source": "# MANDATORY\n# title: Run A Direct NTT\u03c8 / INTT\u03c8 Round Trip\n# kind: demo | level: 2\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import find_psi, forward_ntt_psi, inverse_ntt_psi\nfrom ntt_learning.visuals import direct_ntt_player\n\nsignal = [1, 2, 3, 4]\nmodulus = 17\npsi = find_psi(len(signal), modulus)\nspectrum = forward_ntt_psi(signal, modulus, psi)\n\nprint(\"signal:\", signal)\nprint(\"spectrum:\", spectrum)\nprint(\"inverse recovery:\", inverse_ntt_psi(spectrum, modulus, psi))\ndisplay(direct_ntt_player(signal, modulus, psi))\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 3,
          "kind": "demo",
          "title": "Use Direct NTT\u03c8 For Negacyclic Multiplication"
        }
      },
      "outputs": [],
      "source": "# MANDATORY\n# title: Use Direct NTT\u03c8 For Negacyclic Multiplication\n# kind: demo | level: 3\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import find_psi, forward_ntt_psi, inverse_ntt_psi, negacyclic_multiply, pointwise_multiply\nfrom ntt_learning.visuals import plot_transform_pipeline\n\nleft = [1, 2, 3, 4]\nright = [5, 6, 7, 8]\nmodulus = 17\npsi = find_psi(4, modulus)\n\nleft_hat = forward_ntt_psi(left, modulus, psi)\nright_hat = forward_ntt_psi(right, modulus, psi)\nproduct_hat = pointwise_multiply(left_hat, right_hat, modulus)\n\nprint(\"NTT_psi(left):\", left_hat)\nprint(\"NTT_psi(right):\", right_hat)\nprint(\"pointwise product:\", product_hat)\nprint(\"inverse of pointwise product:\", inverse_ntt_psi(product_hat, modulus, psi))\nprint(\"schoolbook negacyclic:\", negacyclic_multiply(left, right, n=4, modulus=modulus))\ndisplay(plot_transform_pipeline(left, right, modulus=modulus, psi=psi, title=\"Direct negacyclic multiply pipeline\"))\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "quiz",
          "title": "Retrieval Check"
        }
      },
      "source": "## MANDATORY\n\n<div class=\"ntt-cell-head ntt-role-mandatory\">\n  <span class=\"ntt-difficulty-pill\">Level 2</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">MANDATORY</span>\n    <span class=\"ntt-kind-pill\">quiz</span>\n  </div>\n  <div class=\"ntt-cell-title\">Retrieval Check</div>\n</div>\n\n1. Why is `\u03c8` stronger than `\u03c9` in the negacyclic story?\n2. What exact property does the inverse add that the forward transform does not?\n3. Why are we still dissatisfied after seeing the direct transform work correctly?\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "facultative",
          "difficulty": 4,
          "kind": "exploration",
          "title": "Optional: Compare Positive And Negative Wrapped Transforms"
        }
      },
      "outputs": [],
      "source": "# FACULTATIVE\n# title: Optional: Compare Positive And Negative Wrapped Transforms\n# kind: exploration | level: 4\n\nfrom ntt_learning.toy_ntt import find_primitive_root, find_psi, forward_ntt, forward_ntt_psi\n\nsignal = [1, 2, 3, 4]\nmodulus = 17\nomega = find_primitive_root(4, modulus)\npsi = find_psi(4, modulus)\n\nprint(\"positive-wrapped NTT:\", forward_ntt(signal, modulus, omega))\nprint(\"negative-wrapped NTT_psi:\", forward_ntt_psi(signal, modulus, psi))\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "handoff",
          "title": "Next Notebook"
        }
      },
      "source": "## META\n\n<div class=\"ntt-cell-head ntt-role-meta\">\n  <span class=\"ntt-difficulty-pill\">Level 1</span>\n  <div class=\"ntt-cell-row\">\n    <span class=\"ntt-role-pill\">META</span>\n    <span class=\"ntt-kind-pill\">handoff</span>\n  </div>\n  <div class=\"ntt-cell-title\">Next Notebook</div>\n</div>\n\nYou finished **Foundations / 02 Negative Wrapped NTT / Lecture**.\n\n**Primary next action**\n- Next notebook: [Step 8 of 27 - Foundations / 02 Negative Wrapped NTT / Lab](lab.ipynb)\n\n**Recovery links if you get lost**\n- Previous notebook: [Foundations / 01 Convolution To Toy NTT / Studio](../01_convolution_to_toy_ntt/studio.ipynb)\n- Restart route: [Start Here](../../START_HERE.ipynb)\n"
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    },
    "ntt_learning": {
      "title": "Lecture: Negative-Wrapped NTT",
      "contract_version": "0.2",
      "sequence": [
        "notebooks/START_HERE.ipynb",
        "notebooks/COURSE_BLUEPRINT.ipynb",
        "notebooks/foundations/01_convolution_to_toy_ntt/lecture.ipynb",
        "notebooks/foundations/01_convolution_to_toy_ntt/lab.ipynb",
        "notebooks/foundations/01_convolution_to_toy_ntt/problems.ipynb",
        "notebooks/foundations/01_convolution_to_toy_ntt/studio.ipynb",
        "notebooks/foundations/02_negative_wrapped_ntt/lecture.ipynb",
        "notebooks/foundations/02_negative_wrapped_ntt/lab.ipynb",
        "notebooks/foundations/02_negative_wrapped_ntt/problems.ipynb",
        "notebooks/foundations/02_negative_wrapped_ntt/studio.ipynb",
        "notebooks/butterfly_mechanics/03_fast_forward_ct/lecture.ipynb",
        "notebooks/butterfly_mechanics/03_fast_forward_ct/lab.ipynb",
        "notebooks/butterfly_mechanics/03_fast_forward_ct/problems.ipynb",
        "notebooks/butterfly_mechanics/03_fast_forward_ct/studio.ipynb",
        "notebooks/butterfly_mechanics/04_fast_inverse_gs/lecture.ipynb",
        "notebooks/butterfly_mechanics/04_fast_inverse_gs/lab.ipynb",
        "notebooks/butterfly_mechanics/04_fast_inverse_gs/problems.ipynb",
        "notebooks/butterfly_mechanics/04_fast_inverse_gs/studio.ipynb",
        "notebooks/kyber_mapping/05_kyber_ntt_and_base_multiplication/lecture.ipynb",
        "notebooks/kyber_mapping/05_kyber_ntt_and_base_multiplication/lab.ipynb",
        "notebooks/kyber_mapping/05_kyber_ntt_and_base_multiplication/problems.ipynb",
        "notebooks/kyber_mapping/05_kyber_ntt_and_base_multiplication/studio.ipynb",
        "notebooks/professional/06_debugging_ntt_failures/lecture.ipynb",
        "notebooks/professional/06_debugging_ntt_failures/lab.ipynb",
        "notebooks/professional/06_debugging_ntt_failures/problems.ipynb",
        "notebooks/professional/06_debugging_ntt_failures/studio.ipynb",
        "notebooks/COURSE_COMPLETE.ipynb"
      ]
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}