orbital in cryptography

 Written for me by Bing Chat. 

That's an interesting question. Here is one possible analogy between elliptic curve encryption and orbital mechanics:

Elliptic curve encryption is a technique that uses the mathematical properties of elliptic curves to generate public and private keys for secure communication. Elliptic curves are curves that have the form y^2 = x^3 + ax + b, where a and b are constants. They have some special features that make them useful for cryptography, such as the fact that it is easy to perform a certain operation called point addition on them, but hard to reverse it. This operation is analogous to adding two vectors in a plane.

Orbital mechanics is the study of the motion of objects in space under the influence of gravity and other forces. One of the basic concepts in orbital mechanics is the orbital velocity, which is the speed and direction that an object needs to have in order to maintain a stable orbit around another object. The orbital velocity depends on the mass of the objects, the distance between them, and the shape of the orbit. The shape of the orbit can be circular, elliptical, parabolic, or hyperbolic, depending on the energy and angular momentum of the system.

One way to think of elliptic curve encryption and orbital mechanics is to imagine that each point on an elliptic curve represents a possible orbit around a central body, such as the Earth or the Sun. The point at infinity, which is not on the curve but is considered part of it, represents an escape orbit, where an object has enough energy to leave the gravitational field of the central body. The point addition operation on the elliptic curve corresponds to changing the orbit of an object by applying a thrust or a gravity assist from another object. The result is a new point on the curve, which represents a new orbit.

The analogy can be extended to explain how public and private keys are generated and used in elliptic curve encryption. Suppose Alice and Bob want to communicate securely using this technique. They agree on a common elliptic curve and a common point on it, which we can call G. This point G is like their home planet, where they start their journey. Alice chooses a random number a as her private key, and calculates her public key A by performing point addition a times on G. This means she changes her orbit a times by applying some thrust or gravity assist. Her public key A is like her destination planet, where she wants to go. She sends A to Bob over an insecure channel.

Bob does the same thing: he chooses a random number b as his private key, and calculates his public key B by performing point addition b times on G. He sends B to Alice over an insecure channel.

Now Alice and Bob have each other's public keys, but not their private keys. They can use them to calculate a shared secret key S by performing point addition again. Alice takes Bob's public key B and performs point addition a times on it, using her private key a. This means she changes her orbit from B to S by applying some thrust or gravity assist a times. Bob does the same thing: he takes Alice's public key A and performs point addition b times on it, using his private key b. He changes his orbit from A to S by applying some thrust or gravity assist b times.

The result is that Alice and Bob end up at the same point S on the elliptic curve, which represents their shared secret key. They can use this key to encrypt and decrypt their messages using a symmetric encryption scheme. No one else can find out their secret key S, because they would have to know either Alice's private key a or Bob's private key b, which are not transmitted over the channel. Finding these private keys from the public keys A and B is very hard, because it would require reversing the point addition operation on the elliptic curve, which is like finding out how much thrust or gravity assist was applied at each step of changing orbits.

This analogy is not perfect, of course, but it may help you understand some of the basic ideas behind elliptic curve encryption and orbital mechanics.

If you want to learn more about these topics, you can check out these links:

- [Elliptic-curve cryptography - Wikipedia](https://en.wikipedia.org/wiki/Elliptic-curve_cryptography)

- [Orbital mechanics - Wikipedia](https://en.wikipedia.org/wiki/Orbital_mechanics)

- [Elliptic Curve Cryptography | CSRC](https://csrc.nist.gov/Projects/Elliptic-Curve-Cryptography)

- [Orbital Mechanics I – The Physics Hypertextbook](https://physics.info/orbital-mechanics-1/)

Source: Conversation with Bing, 6/21/2023

(1) Elliptic-curve cryptography - Wikipedia. https://en.wikipedia.org/wiki/Elliptic-curve_cryptography.

(2) Elliptic Curve Cryptography | CSRC. https://csrc.nist.gov/Projects/Elliptic-Curve-Cryptography.

(3) What is Elliptical Curve Cryptography (ECC)? - TechTarget. https://www.techtarget.com/searchsecurity/definition/elliptical-curve-cryptography.

(4) Orbital mechanics - Wikipedia. https://en.wikipedia.org/wiki/Orbital_mechanics.

(5) Orbital Mechanics I – The Physics Hypertextbook. https://physics.info/orbital-mechanics-1/.

(6) Basics of Space Flight: Orbital Mechanics - braeunig.us. http://www.braeunig.us/space/orbmech.htm.

(7) 3.1: Orbital Mechanics - Geosciences LibreTexts. https://geo.libretexts.org/Courses/University_of_California_Davis/GEL_056%3A_Introduction_to_Geophysics/Geophysics_is_everywhere_in_geology.../03%3A_Planetary_Geophysics/3.01%3A_Orbital_Mechanics.