saymrwulf/NTT-learning
1
0
Fork 0
mirror of https://github.com/saymrwulf/NTT-learning.git synced 2026-09-07 19:40:49 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions
NTT-learning/notebooks/foundations/02_negative_wrapped_ntt/lecture.ipynb

165 lines
8.5 KiB
Text
Raw Blame History

{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "meta",
"difficulty": 1,
"kind": "orientation",
"title": "Objectives"
}
},
"source": "## META | difficulty 1 | Objectives\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 | difficulty 2 | Why \u03c8 Shows Up\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 | difficulty 2 | Inspect \u03c9, \u03c8, And The Direct Transform Matrix\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 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(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 | difficulty 2 | Direct NTT\u03c8 Is Mechanically Clear But Still Quadratic\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 | difficulty 2 | Run A Direct NTT\u03c8 / INTT\u03c8 Round Trip\n\nfrom ntt_learning.toy_ntt import find_psi, forward_ntt_psi, inverse_ntt_psi\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))\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 | difficulty 3 | Use Direct NTT\u03c8 For Negacyclic Multiplication\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 | difficulty 2 | Retrieval Check\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 | difficulty 4 | Optional: Compare Positive And Negative Wrapped Transforms\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 | difficulty 1 | Next Notebook\n\nNext notebook: `lab.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
}
Reference in a new issue View git blame Copy permalink