Paper 2026/1976

Properties of the Me Operation and Me-Scalar Multiplication on Elliptic Curves over Finite Fields

Abstract

The M operation was introduced by Yura as an alternative to the max operation appearing in the box-ball system (BBS) to construct a BBS over finite fields. The Me operation is a version of the M operation for an elliptic curve $E$ over a finite field ${\mathbb F}_p$. As with the M operation, the Me operation satisfies the idempotent law and does not satisfy the associative law. Nevertheless, for $P,Z \in E({\mathbb F}_p)$ and $n \in {\mathbb N}$, the 1st Me-scalar multiplication $P_{n,Z}^{\,I}$ with auxiliary element $Z$ can be defined. Moreover, for $P,Z \in E({\mathbb F}_p)$ and $n \in {\mathbb Q}_+$, the 2nd Me-scalar multiplication $P_{n,Z}^{II}$ with auxiliary element $Z$ can be defined. This paper shows the following properties that may be useful to construct cryptographic protocols: $(P_{n_0,Z}^{\,I})_{n_1,Z}^{\,I}=(P_{n_1,Z}^{\,I})_{n_0,Z}^{\,I}$, $(P_{n_0,Z}^{II})_{n_1,Z}^{II}=(P_{n_1,Z}^{II})_{n_0,Z}^{II}=P_{n_0n_1,Z}^{II}$; the 1st MeDLP and the 2nd MeDLP, which are Me versions of the ECDLP, are difficult to solve on classical computers under certain conditions; the 1st MeCDH and the 2nd MeCDH, which are Me versions of the ECCDH, are NOT difficult to solve; and the sequence $\{ P_{n,Z}^{II}:n=1,2,3,\ldots \}$ is nonperiodic unless it is constant.