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{
"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"
},
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{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "meta",
"difficulty": 1,
"kind": "route_nav",
"title": "Route Guardrails"
}
},
"source": "## META | difficulty 1 | Route Guardrails\n\nYou are at **step 9 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 / Studio](studio.ipynb)\n- Previous notebook: [Foundations / 02 Negative Wrapped NTT / Lab](lab.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](lecture.ipynb)\n8. [Foundations / 02 Negative Wrapped NTT / Lab](lab.ipynb)\n9. **Foundations / 02 Negative Wrapped NTT / Problems** <- you are here\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"
},
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{
"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"
}
},
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"source": "## META | difficulty 1 | Next Notebook\n\nYou finished **Foundations / 02 Negative Wrapped NTT / Problems**.\n\n**Primary next action**\n- Next notebook: [Step 10 of 27 - Foundations / 02 Negative Wrapped NTT / Studio](studio.ipynb)\n\n**Recovery links if you get lost**\n- Previous notebook: [Foundations / 02 Negative Wrapped NTT / Lab](lab.ipynb)\n- Restart route: [Start Here](../../START_HERE.ipynb)\n"
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}
],
"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"
]
}
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