Lattices with transforms

.idea/watcherTasks.xml

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+<?xml version="1.0" encoding="UTF-8"?>
+<project version="4">
+  <component name="ProjectTasksOptions">
+    <TaskOptions isEnabled="true">
+      <option name="arguments" value="$FilePath$" />
+      <option name="checkSyntaxErrors" value="true" />
+      <option name="description" />
+      <option name="exitCodeBehavior" value="ERROR" />
+      <option name="fileExtension" value="py" />
+      <option name="immediateSync" value="false" />
+      <option name="name" value="Black" />
+      <option name="output" value="" />
+      <option name="outputFilters">
+        <array />
+      </option>
+      <option name="outputFromStdout" value="false" />
+      <option name="program" value="$PROJECT_DIR$/venv/bin/black" />
+      <option name="runOnExternalChanges" value="false" />
+      <option name="scopeName" value="Project Files" />
+      <option name="trackOnlyRoot" value="false" />
+      <option name="workingDir" value="" />
+      <envs />
+    </TaskOptions>
+  </component>
+</project>

components.py

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+from manim import *
+from consts import *
+
+
+class TitledScene(Scene):
+    def add_title(self, title_text):
+        title = Tex(title_text, font_size=LARGE_FONT)
+        title.set_z_index(100)
+        title.to_edge(LEFT + UP)
+
+        self.add(title)
+        self.play(Write(title))
+
+        return title

consts.py

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+LARGE_FONT = 48
+MEDIUM_FONT = 40
+SMALL_FONT = 32

main.py

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