An Elementary Linear-Algebraic Proof without Computer-Aided Arguments for the Group Law on Elliptic Curves
This work makes the proof more accessible to non-mathematicians by simplifying the mathematical prerequisites, though it is incremental as it builds on existing attempts for elementary proofs.
The paper tackles the problem of proving the associative law for the group structure on elliptic curves, which is important in mathematics and cryptography, by providing a self-contained proof that uses only basic linear algebra and avoids computer-aided arguments.
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.