Animation Specification: secp256k1 Public Key Generation

Animation Description and Purpose

This animation visually demonstrates how a public key is generated on the secp256k1 elliptic curve using a private key. The goal is to illustrate the core concept of elliptic curve cryptography: scalar multiplication of a generator point by a private key to produce a public key.

Mathematical Elements and Formulas

• Elliptic Curve Equation: $y^2 = x^3 + 7$ over the finite field $\mathbb{F}_p$, where $p = 2^{256} - 2^{32} - 977$.
• Public Key Generation Formula: $Q = d \cdot G$, where:
  • $Q$ is the public key (a point on the curve).
  • $d$ is the private key (a scalar).
  • $G$ is the generator point (a fixed point on the curve).
• Scalar Multiplication: Repeated point addition on the elliptic curve.

Visual Elements

Shapes and Objects

1. Elliptic Curve: A 2D representation of the secp256k1 curve, simplified for visualization.
2. Generator Point $G$: A distinct point on the curve, highlighted in green.
3. Private Key $d$: Displayed as a small integer (e.g., $d = 3$) for simplicity, with an opaque background.
4. Public Key $Q$: The resulting point after scalar multiplication, highlighted in blue.
5. Point Addition Lines: Dashed lines connecting points during scalar multiplication.
6. Tangent Lines: For visualizing point doubling (part of scalar multiplication).

Colors

• Curve: Light gray.
• Generator Point $G$: Green.
• Private Key $d$: White text with a dark gray opaque background.
• Public Key $Q$: Blue.
• Point Addition Lines: Light blue dashed lines.
• Tangent Lines: Orange dashed lines.

Animation Sequence

1. Introduction (0s - 3s):

   • Display the elliptic curve $y^2 = x^3 + 7$ with the title "secp256k1 Public Key Generation".
   • Highlight the generator point $G$ on the curve.
   • Show the private key $d = 3$ appearing near the bottom of the screen.

2. Scalar Multiplication (3s - 10s):

   • Animate the process of $Q = d \cdot G$ as repeated point addition:
     • Start with $G$ and show $G + G = 2G$ (point doubling).
     • Add $G$ again to show $2G + G = 3G$.
   • Use dashed lines to connect points during addition.
   • Highlight the tangent line for point doubling.

3. Result (10s - 12s):

   • The final point $Q = 3G$ is highlighted in blue.
   • Display the text "Public Key $Q$