{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "orientation",
          "title": "Problem Set Goals"
        }
      },
      "source": "## META | difficulty 1 | Problem Set Goals\n\nThis notebook checks whether the inverse flow, ordering, and scaling are now mechanically stable.\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "quiz",
          "title": "Multiple-Choice Retrieval"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Multiple-Choice Retrieval\n\nChoose one answer for each:\n\n1. GS is used here as the fast schedule for:\n   A. direct forward NTT\n   B. inverse NTT\n   C. schoolbook multiplication\n\n2. The final `n^-1` matters because:\n   A. each stage skipped local divisions that accumulate\n   B. the modulus changed mid-computation\n   C. bit-reversal requires scaling\n\n3. In the standard pairing, GS expects:\n   A. NO input and BO output\n   B. BO input and NO output\n   C. random order on both ends\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "quiz",
          "title": "Answer Key"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 2 | Answer Key\n\nanswers = {1: \"B\", 2: \"A\", 3: \"B\"}\nprint(answers)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "Paper Example Check"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Paper Example Check\n\nVerify that the GS trace on the paper\u2019s BO input scales back to `[1,2,3,4]`.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "Check The Final Scaling"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 2 | Check The Final Scaling\n\nfrom ntt_learning.toy_ntt import fast_intt_psi_gs_trace\n\ntrace = fast_intt_psi_gs_trace([1467, 3471, 2807, 7621], 7681, 1925)\nprint(trace.scaled_output)\nassert list(trace.scaled_output) == [1, 2, 3, 4]\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "reflection",
          "title": "Written Reflection"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Written Reflection\n\nIn one paragraph, explain why \u201csame structure, opposite direction\u201d is a better intuition for GS than \u201ctotally different algorithm\u201d.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "facultative",
          "difficulty": 4,
          "kind": "exploration",
          "title": "Optional Challenge"
        }
      },
      "outputs": [],
      "source": "# FACULTATIVE | difficulty 4 | Optional Challenge\n\nfrom ntt_learning.toy_ntt import fast_intt_psi_gs_trace\n\ntrace = fast_intt_psi_gs_trace([2489, 6478, 7489, 6607], 7681, 1925)\nfor stage in trace.stages:\n    print(stage.stage_index, stage.output_values)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "meta",
          "difficulty": 1,
          "kind": "handoff",
          "title": "Next Notebook"
        }
      },
      "source": "## META | difficulty 1 | Next Notebook\n\nNext notebook: `studio.ipynb`\n"
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    },
    "ntt_learning": {
      "title": "Problems: Fast Inverse GS",
      "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
}