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NTT-learning/notebooks/butterfly_mechanics/03_fast_forward_ct/lecture.ipynb

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{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "meta",
"difficulty": 1,
"kind": "orientation",
"title": "Objectives"
}
},
"source": "## META | difficulty 1 | Objectives\n\nThis bundle is where the transform stops being a full matrix multiplication and becomes a staged butterfly network.\n\nFocus:\n\n- CT as the fast forward NTT strategy\n- visible stage arrays\n- explicit zeta values per pair\n- BO output vs NO output\n"
},
{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "explanation",
"title": "CT Is A Schedule For Reusing Work"
}
},
"source": "## MANDATORY | difficulty 3 | CT Is A Schedule For Reusing Work\n\nThe point of the CT butterfly is not to invent a new transform.\nThe point is to compute the same transform by reusing shared bracket terms instead of recomputing everything from scratch.\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "demo",
"title": "See The Schedule Geometry Before The Numbers"
}
},
"outputs": [],
"source": "# MANDATORY | difficulty 3 | See The Schedule Geometry Before The Numbers\n\nfrom IPython.display import display\n\nfrom ntt_learning.visuals import plot_stage_pairing_map, plot_stage_schedule\n\ndisplay(plot_stage_schedule(8, title=\"CT schedule skeleton for n=8\"))\ndisplay(plot_stage_pairing_map(8, 2, title=\"Stage 1 pairings for n=8\"))\ndisplay(plot_stage_pairing_map(8, 4, title=\"Stage 2 pairings for n=8\"))\ndisplay(plot_stage_pairing_map(8, 8, title=\"Stage 3 pairings for n=8\"))\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "demo",
"title": "Trace The Exact n=4 Paper Example"
}
},
"outputs": [],
"source": "# MANDATORY | difficulty 3 | Trace The Exact n=4 Paper Example\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_ntt_psi_ct_trace, forward_ntt_psi\nfrom ntt_learning.visuals import interactive_trace, plot_butterfly_network, plot_trace_overview\n\nsignal = [1, 2, 3, 4]\nmodulus = 7681\npsi = 1925\ntrace = fast_ntt_psi_ct_trace(signal, modulus, psi)\n\nprint(\"raw CT output (BO):\", trace.raw_output)\nprint(\"bit-reversed back to NO:\", trace.normal_order_output)\nprint(\"direct NTT_psi:\", forward_ntt_psi(signal, modulus, psi))\ndisplay(plot_trace_overview(trace, title=\"CT overview for [1,2,3,4]\"))\ndisplay(plot_butterfly_network(trace, title=\"Full CT network for [1,2,3,4]\"))\ndisplay(interactive_trace(trace, title=\"CT forward trace\"))\n"
},
{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "explanation",
"title": "What You Should Notice In The Stage Viewer"
}
},
"source": "## MANDATORY | difficulty 3 | What You Should Notice In The Stage Viewer\n\nDo not just read the final answer.\nNotice:\n\n- which pairs talk to each other in each stage\n- which `zeta` each pair uses\n- how the array order changes before the final bit-reversal correction\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "demo",
"title": "Run The Second n=4 Paper Example"
}
},
"outputs": [],
"source": "# MANDATORY | difficulty 3 | Run The Second n=4 Paper Example\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_ntt_psi_ct_trace\nfrom ntt_learning.visuals import interactive_trace\n\nsignal = [5, 6, 7, 8]\ntrace = fast_ntt_psi_ct_trace(signal, 7681, 1925)\nprint(\"BO output:\", trace.raw_output)\nprint(\"NO output:\", trace.normal_order_output)\ndisplay(interactive_trace(trace, title=\"Second CT trace\"))\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "demo",
"title": "Go One Stage Deeper With n=8"
}
},
"outputs": [],
"source": "# MANDATORY | difficulty 3 | Go One Stage Deeper With n=8\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_ntt_psi_ct_trace, find_psi\nfrom ntt_learning.visuals import interactive_trace, plot_butterfly_network, plot_trace_overview\n\nsignal = [0, 1, 2, 3, 4, 5, 6, 7]\nmodulus = 97\npsi = find_psi(8, modulus)\ntrace = fast_ntt_psi_ct_trace(signal, modulus, psi)\n\nprint(\"psi:\", psi)\nprint(\"BO output:\", trace.raw_output)\nprint(\"NO output:\", trace.normal_order_output)\ndisplay(plot_trace_overview(trace, title=\"Three CT stages for n=8\"))\ndisplay(plot_butterfly_network(trace, title=\"Full CT network for n=8\"))\ndisplay(interactive_trace(trace, title=\"n=8 CT stage explorer\"))\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 3,
"kind": "demo",
"title": "See BO Output And NO Output Side By Side"
}
},
"outputs": [],
"source": "# MANDATORY | difficulty 3 | See BO Output And NO Output Side By Side\n\nfrom IPython.display import display\n\nfrom ntt_learning.toy_ntt import fast_ntt_psi_ct_trace\nfrom ntt_learning.visuals import plot_bit_reversal_mapping, plot_vector_comparison\n\ntrace = fast_ntt_psi_ct_trace([1, 2, 3, 4], 7681, 1925)\ndisplay(\n plot_vector_comparison(\n trace.raw_output,\n trace.normal_order_output,\n left_label=\"BO\",\n right_label=\"NO\",\n title=\"Same CT values, different ordering\",\n )\n)\ndisplay(plot_bit_reversal_mapping(4, title=\"Why BO becomes NO after bit-reversal\"))\n"
},
{
"cell_type": "markdown",
"metadata": {
"pedagogy": {
"role": "mandatory",
"difficulty": 2,
"kind": "quiz",
"title": "Retrieval Check"
}
},
"source": "## MANDATORY | difficulty 2 | Retrieval Check\n\n1. What does CT change: the transform definition or the computation schedule?\n2. Why is the output naturally in BO rather than NO?\n3. In a stage diagram, what are the first three things you should inspect before any formula?\n"
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"pedagogy": {
"role": "facultative",
"difficulty": 4,
"kind": "exploration",
"title": "Optional: Inspect Stage Rows As Data"
}
},
"outputs": [],
"source": "# FACULTATIVE | difficulty 4 | Optional: Inspect Stage Rows As Data\n\nfrom ntt_learning.toy_ntt import fast_ntt_psi_ct_trace, stage_rows\n\ntrace = fast_ntt_psi_ct_trace([1, 2, 3, 4], 7681, 1925)\nfor stage in trace.stages:\n print(\"stage\", stage.stage_index)\n for row in stage_rows(stage):\n print(row)\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: Fast Forward CT",
"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
}
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