Finite-field groups and scalar operations
In a common prime-field model, coordinates satisfy a curve equation under arithmetic modulo p, together with an identity element. Defined point addition forms a group, and public Q=dG denotes scalar multiplication through group operations, not coordinate-wise multiplication.
Parameters include a base point and subgroup order as well as the curve. Recovering d from G and Q is the relevant discrete-log problem, whose difficulty depends on suitable choices rather than merely long-looking coordinates.
References: [1]
Agreement and signatures are different schemes
ECDH combines a private scalar with a peer’s public value to establish material, whereas curve-based signatures use a different generation and verification relation. Shared mathematics does not license arbitrary reuse of one scheme’s key in every other interface.
Point encodings differ too. X25519 and X448 in RFC 7748 have specified coordinate representations and input handling, not interchangeable formats for any curve or signature with a similar bit count. Mathematical objects, serialization and interfaces must agree.
Validation, authentication and quantum limits
Protocols apply their own rules for valid points, subgroups and exceptional outputs, and basic ECDH also needs authenticated exchange values. Input conventions vary, so neither copying unrelated validation steps nor accepting a nonempty output establishes correct peer authentication.
ECC is not inherently post-quantum: a sufficiently capable general-purpose quantum computer threatens its discrete-log basis. Key-size, performance and storage comparisons must identify schemes and threat models rather than treating a compact key as assumption-free security.