Open AccessAn elementary linear-algebraic proof without computer-aided arguments for the group law on elliptic curves
Author(s) -
Koji Nuida
Publication year - 2021
Publication title -
international journal of mathematics for industry
Language(s) - English
Resource type - Journals
eISSN - 2661-3352
pISSN - 2661-3344
DOI - 10.1142/s2661335221500015
Subject(s) - mathematical proof , associative property , algebra over a field , symbolic computation , group (periodic table) , elliptic curve , cryptography , mathematics , elementary algebra , property (philosophy) , algebraic number , computer science , pure mathematics , algorithm , geometry , mathematical analysis , epistemology , philosophy , chemistry , organic chemistry
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.