Lattices with transforms

main.py

@@ -0,0 +1,104 @@
+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()