105 lines
3.1 KiB
Python
105 lines
3.1 KiB
Python
from consts import *
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from components import *
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import numpy as np
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from manim import *
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class Introduction(TitledScene):
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def construct(self):
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self.add_title("Goldreich--Goldwasser--Halevi")
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self.wait()
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text_1 = Tex(
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r"""
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\begin{itemize}
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\item Lattice-based cryptosystem.
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\item Devised in 1997 by Goldreich, Goldwasser, and Halevi.
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\item Broken in 1999 by Nguyen.
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\end{itemize}
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""",
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font_size=MEDIUM_FONT,
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)
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self.add(text_1)
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self.play(Write(text_1, run_time=4.0))
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self.wait()
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class Premise(TitledScene):
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def construct(self):
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self.add_title("Lattices")
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# A lattice is a subspace of a vector space that is constructed by taking integer multiples of some basis
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# vectors.
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# For example, take the real plane R2.
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plane = NumberPlane(axis_config={"stroke_width": 0.0})
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plane.set_z_index(-10)
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plane.set_opacity(0.75)
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dot = Dot(ORIGIN)
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self.add(dot, plane)
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self.play(Create(dot), Create(plane))
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self.wait()
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# We can construct the 2D grid of integers with the elementary basis
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lattice_1 = VGroup()
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arrow_1 = Arrow(ORIGIN, [1, 0, 0], buff=0)
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arrow_2 = Arrow(ORIGIN, [0, 1, 0], buff=0)
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self.play(Create(arrow_1), Create(arrow_2))
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self.wait()
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for i in range(-7, 8):
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for j in range(-6, 7):
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lattice_1.add(Dot([i, j, 0]))
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self.play(Create(lattice_1))
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# By moving these basis vectors but maintaining their linear independency, other lattices can be formed
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self.play(
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Transform(arrow_1, Arrow(ORIGIN, [1.5, 0, 0], buff=0)),
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*[
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Transform(
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dot,
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Dot(
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dot.get_center()
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* np.matrix([[1.5, 0, 0], [0, 1, 0], [0, 0, 1]])
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),
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)
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for dot in lattice_1
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]
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)
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self.wait()
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self.play(
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Transform(arrow_1, Arrow(ORIGIN, [1, -0.5, 0], buff=0)),
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*[
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Transform(
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dot,
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Dot(
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dot.get_center()
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* np.matrix([[2 / 3, 0, 0], [0, 1, 0], [0, 0, 1]])
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* np.matrix([[1, -0.5, 0], [0, 1, 0], [0, 0, 1]])
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),
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)
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for dot in lattice_1
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]
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)
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self.wait()
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self.play(
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Transform(arrow_1, Arrow(ORIGIN, [1, -1, 0], buff=0)),
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Transform(arrow_2, Arrow(ORIGIN, [1, 1, 0], buff=0)),
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*[
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Transform(
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dot,
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Dot(
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dot.get_center()
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* np.matrix([[1, 0.5, 0], [0, 1, 0], [0, 0, 1]])
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* np.matrix([[1, -1, 0], [1, 1, 0], [0, 0, 1]])
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),
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)
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for dot in lattice_1
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]
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)
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self.wait()
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