An Elementary Linear-Algebraic Proof without Computer-Aided Arguments for the Group Law on Elliptic Curves

TL;DR

This paper presents a straightforward, elementary proof of the group law on elliptic curves that avoids complex mathematics and computer-aided calculations, making the proof more accessible.

Contribution

It provides a self-contained, linear algebra-based proof of the elliptic curve group law, specifically the associativity, without advanced mathematics or computer assistance.

Findings

Proof is accessible to those with basic linear algebra knowledge.

The proof confirms the associativity of the elliptic curve group law.

No computer-aided calculations are needed in this proof.

Abstract

The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this operation, assuming mathematical knowledge only at the level of basic linear algebra and not requiring computer-aided arguments.