Differential addition on Jacobi quartic Curves

2026-06-18

AI Summary

This work addresses the inefficiency of differential addition and doubling operations on Jacobi quartic curves by proposing three novel, highly efficient formulae. Assuming input points are given in affine coordinates with their difference known, the proposed methods optimize field operations—specifically multiplications (M), squarings (S), and multiplications by constants (D)—achieving computational costs of 5M+4S+1D, 3M+7S+1D, and 3M+6S+3D, respectively. These formulae significantly reduce arithmetic complexity while preserving correctness, thereby enhancing the performance of relevant operations in elliptic curve cryptography. The contributions offer both practical utility and theoretical innovation, advancing the state of the art in efficient elliptic curve arithmetic.

Abstract

In this paper we present new differential addition and doubling formulas for Jacobi quartic curves. Several differential addition and doubling formulas are presented with cost of 5M+4S+1D,3M+7S+1D,3M+6S+3D when the given difference point is in affine form. Here M,S,D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively.

Problem

• differential addition
• doubling
• Jacobi quartic curves
• elliptic curve arithmetic
• efficient formulas

Innovation

• differential addition
• Jacobi quartic curves
• point doubling
• elliptic curve cryptography
• efficient formulas