Paper 2024/1767

ECPM Cryptanalysis Resource Estimation

Abstract

Elliptic Curve Point Multiplication (ECPM) is a key component of the Elliptic Curve Cryptography (ECC) hierarchy protocol. However, the specific estimation of resources required for this process remains underexplored despite its significance in the cryptanalysis of ECC algorithms, particularly binary ECC in GF (2𝑚). Given the extensive use of ECC algorithms in various security protocols and devices, it is essential to conduct this examination to gain valuable insights into its cryptanalysis, specifically in terms of providing precise resource estimations, which serve as a solid basis for further investigation in solving the Elliptic Curve Discrete Logarithm Problem. Expanding on several significant prior research, in this work, we refer to as ECPM cryptanalysis, we estimate quantum resources, including qubits, gates, and circuit depth, by integrating point addition (PA) and point-doubling (PD) into the ECPM scheme, culminating in a Shor’s algorithm-based binary ECC cryptanalysis circuit. Focusing on optimizing depth, we elaborate on and implement the most efficient PD circuit and incorporate optimized Karatsuba multiplication and FLT-based inversion algorithms for PA and PD operations. Compared to the latest PA-only circuits, our preliminary results showcase significant resource optimization for various ECPM implementations, including single-step ECPM, ECPM with combined or selective PA/PD utilization, and total−step ECPM (2𝑛 PD+2 PA).