Super singular isogeny-Based Cryptography (Key Concepts for Beginners)

1. Elliptic Curves:

Definition: Elliptic curves are geometric objects that can be represented by equations of the form y^2 = x^3 + ax + b, where a and b are constants.
Points on an Elliptic Curve: Each point (x, y) on the curve represents an element of a group, where the group operation (addition) is defined using a specific formula.
Finite Fields: To work with elliptic curves, we often use finite fields, which are sets of numbers with a finite number of possible values.

Definition: Supersingular elliptic curves are a special type of elliptic curve with specific properties, such as a small number of rational points.
Example: An example of a supersingular elliptic curve is E: y^2 = x^3 + x.

Definition: Isogenies are homomorphisms (functions) between elliptic curves that preserve the group structure.
Example: If we have two elliptic curves E1 and E2, an isogeny can be represented as f: E1 → E2.

Overview: Supersingular isogeny-based cryptography is a type of public-key cryptography that uses the mathematical structure of supersingular elliptic curves and their isogenies to create cryptographic primitives.
Security: The security of supersingular isogeny-based cryptography is based on the difficulty of solving specific mathematical problems related to elliptic curves and isogenies.

Overview: SIDH is a key-exchange protocol that uses supersingular elliptic curves and isogenies to allow two parties to agree on a shared secret over an insecure channel.
Security: The security of SIDH is based on the difficulty of computing an isogeny between two supersingular elliptic curves.

Definition: An isogeny graph is a directed graph where each node represents an elliptic curve, and each edge represents an isogeny between two curves.
Example: The isogeny graph for a set of supersingular elliptic curves can be visualized as a directed graph with nodes and edges representing isogenies.

Overview: The computational complexity of supersingular isogeny-based cryptography is based on the difficulty of solving mathematical problems related to elliptic curves and isogenies.
Example: The SIDH key-exchange protocol relies on the difficulty of computing an isogeny between two supersingular elliptic curves.

Overview: Supersingular isogeny-based cryptography is resistant to quantum attacks, making it a promising candidate for post-quantum cryptography.
Example: The SIDH protocol is believed to be secure against quantum attacks, making it a viable option for quantum-resistant cryptography.

Overview: Implementing supersingular isogeny-based cryptography can be challenging due to the complexity of the underlying mathematics.
Example: Implementing the SIDH protocol requires careful handling of elliptic curves, isogenies, and finite fields.

Overview: Research and development in supersingular isogeny-based cryptography is ongoing to improve the efficiency, security, and usability of the cryptographic primitives.
Example: Researchers are working on optimizing the implementation of the SIDH protocol and exploring its applications in real-world scenarios.