TY - JOUR

T1 - An elementary linear-algebraic proof without computer-aided arguments for the group law on elliptic curves

AU - Nuida, Koji

N1 - Funding Information:
The author thanks Go Yamashita and Tsuyoshi Takagi for their valuable comments. The author also thanks the anonymous reviewer for the careful review, especially for pointing out the related work in Ref. 10. This work is supported by JST CREST Grant Number JPMJCR14D6 and JSPS KAKENHI Grant Number JP19H01804.
Publisher Copyright:
© 2021 The Author(s).

PY - 2021/12/1

Y1 - 2021/12/1

N2 - 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.

AB - 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.

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U2 - 10.1142/S2661335221500015

DO - 10.1142/S2661335221500015

M3 - Article

AN - SCOPUS:85144971892

VL - 13

JO - International Journal of Mathematics for Industry

JF - International Journal of Mathematics for Industry

SN - 2661-3352

IS - 1

M1 - 2150001

ER -