{
  "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 direct negative-wrapped transform is now mechanically understandable.\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. In the negacyclic transform, `\u03c8` matters because:\n   A. it makes the modulus disappear\n   B. it encodes the `x^n + 1` sign structure\n   C. it avoids all inverses\n\n2. The inverse transform differs from the forward transform by:\n   A. an inverse root and an `n^-1` scaling\n   B. a larger modulus\n   C. removing all twiddle factors\n\n3. The direct matrix transform is still pedagogically useful because:\n   A. it keeps every coefficient contribution visible\n   B. it is how Kyber is implemented directly at full size\n   C. it removes the need for butterflies\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: \"A\"}\nprint(answers)\n"
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "Round-Trip Check"
        }
      },
      "source": "## MANDATORY | difficulty 2 | Round-Trip Check\n\nVerify by code that `INTT\u03c8(NTT\u03c8(a)) = a` for a nontrivial vector.\n"
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "pedagogy": {
          "role": "mandatory",
          "difficulty": 2,
          "kind": "exercise",
          "title": "Check The Round Trip"
        }
      },
      "outputs": [],
      "source": "# MANDATORY | difficulty 2 | Check The Round Trip\n\nfrom ntt_learning.toy_ntt import find_psi, forward_ntt_psi, inverse_ntt_psi\n\nsignal = [6, 0, 5, 2]\npsi = find_psi(4, 17)\nrecovered = inverse_ntt_psi(forward_ntt_psi(signal, 17, psi), 17, psi)\nprint(recovered)\nassert recovered == signal\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 the direct transform is the right place to understand the algebra, but not the right place to stop if you care about algorithmic speed.\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 find_psi, forward_ntt_psi\n\npsi = find_psi(4, 17)\nfor signal in ([1, 1, 1, 1], [0, 1, 0, 1], [3, 5, 7, 9]):\n    print(signal, \"->\", forward_ntt_psi(signal, 17, psi))\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: 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
}