241 lines
7.2 KiB
Python
241 lines
7.2 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(-50, 50):
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for j in range(-25, 25):
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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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self.play(
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Transform(arrow_1, Arrow(ORIGIN, [2, 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 / 2, 1 / 2, 0], [-1 / 2, 1 / 2, 0], [0, 0, 1]])
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* np.matrix([[2, 0, 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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class MatrixRep(TitledScene):
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def construct(self):
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self.add_title("Lattices")
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matrix_comp = MathTex(
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r"\begin{bmatrix}1 & 1\\ 1 & -1\end{bmatrix} \sim \begin{bmatrix}1 & 2\\ 1 & 0\end{bmatrix}"
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)
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lattice_comp = MathTex(
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r"L\left(\begin{bmatrix}1 & 1\\ 1 & -1\end{bmatrix}\right) = L\left(\begin{bmatrix}1 & 2\\ 1 & 0\end{bmatrix}\right)"
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)
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self.play(Create(matrix_comp))
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self.wait()
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self.play(Transform(matrix_comp, lattice_comp))
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self.wait()
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self.play(ApplyMethod(matrix_comp.shift, 2 * UP))
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self.wait()
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l_def = MathTex(
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r"L(\begin{bmatrix}\mathbf{b_1} & \mathbf{b_2}\end{bmatrix}) = \{ k_1 \mathbf{b_1} + k_2 \mathbf{b_2} \mid k_i \in \mathbb{Z} \}"
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)
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l_def.shift(DOWN)
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self.play(Create(l_def))
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self.wait()
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class LatticeProblems(Scene):
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def construct(self):
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text = Tex("How does this give us a trapdoor?", font_size=LARGE_FONT)
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self.play(Create(text))
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self.wait()
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self.play(Transform(text, Tex("Lattice problems", font_size=LARGE_FONT)))
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self.wait()
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self.play(ApplyMethod(text.to_corner, UP + LEFT))
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self.wait()
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diagram_1 = VGroup()
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diagram_1.add(Dot(ORIGIN))
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diagram_1.add(Arrow(ORIGIN, [0.25, 1, 0], buff=0))
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diagram_1.add(Arrow(ORIGIN, [0.5, 1, 0], buff=0))
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arrow = Arrow(ORIGIN, [1, 0, 0], buff=0)
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arrow.set_color(BLUE)
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arrow.set_z_index(-1)
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diagram_1.add(arrow)
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diagram_1.shift(3 * LEFT)
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label_1 = Tex("Closest Vector Problem", font_size=MEDIUM_FONT)
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label_1.next_to(diagram_1, DOWN)
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diagram_2 = VGroup()
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diagram_2.add(Dot(ORIGIN))
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diagram_2.add(Arrow(ORIGIN, [1, 1, 0], buff=0))
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diagram_2.add(Arrow(ORIGIN, [-1, 0, 0], buff=0))
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arrow_1 = Arrow(ORIGIN, [1, 0, 0], buff=0)
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arrow_1.set_color(BLUE)
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arrow_1.set_z_index(-1)
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diagram_2.add(arrow_1)
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arrow_2 = Arrow(ORIGIN, [0, 1, 0], buff=0)
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arrow_2.set_color(BLUE)
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arrow_2.set_z_index(-1)
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diagram_2.add(arrow_2)
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diagram_2.shift(3 * RIGHT)
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label_2 = Tex("Shortest Basis Problem", font_size=MEDIUM_FONT)
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label_2.next_to(diagram_2, DOWN)
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self.play(Create(diagram_1), Create(diagram_2))
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self.wait()
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self.play(Create(label_1), Create(label_2))
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self.wait()
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class OrthoDefect(Scene):
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def construct(self):
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text = Tex("How do we decide which bases are ``good''?", font_size=LARGE_FONT)
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self.play(Create(text))
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self.wait()
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self.play(Transform(text, Tex("Orthogonal defect", font_size=LARGE_FONT)))
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self.wait()
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self.play(ApplyMethod(text.to_corner, UP + LEFT))
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self.wait()
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diagram_1 = VGroup()
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diagram_1.add(Dot(ORIGIN))
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diagram_1.add(Arrow(ORIGIN, [1, 0, 0], buff=0))
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diagram_1.add(Arrow(ORIGIN, [0, 1, 0], buff=0))
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diagram_1.shift(3 * LEFT)
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diagram_1.set_color(GREEN)
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label_1 = Tex("Defect 1", font_size=MEDIUM_FONT)
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label_1.next_to(diagram_1, DOWN)
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diagram_2 = VGroup()
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diagram_2.add(Dot(ORIGIN))
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diagram_2.add(Arrow(ORIGIN, [0.25, 1, 0], buff=0))
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diagram_2.add(Arrow(ORIGIN, [0.5, 1, 0], buff=0))
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diagram_2.shift(3 * RIGHT)
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diagram_2.set_color(RED)
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label_2 = Tex(r"Defect $\approx$ 4.6", font_size=MEDIUM_FONT)
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label_2.next_to(diagram_2, DOWN)
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self.play(Create(diagram_1), Create(diagram_2))
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self.wait()
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self.play(Create(label_1), Create(label_2))
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self.wait()
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