Cryptanalysis of elliptic curve scalar multiplication algorithms

Leong, Zhen Hong (2025) Cryptanalysis of elliptic curve scalar multiplication algorithms. Final Year Project, UTAR.

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Abstract

This project explores scalar multiplication algorithms in Elliptic Curve Cryptography, focusing on the binary method and elliptic net method applied in Elliptic Curve Diffie-Hellman and Elliptic Curve Digital Signature Algorithm. Scalar multiplication is the most computationally intensive operation in Elliptic Curve Cryptography and directly impacts both cryptographic strength and performance. There is lack of standardized scalar multiplication algorithm or parameter set to ensure compatibility and interoperability in cryptographic implementations. This creates challenges in developing secure Elliptic Curve Cryptography systems and performing cryptanalysis for scalar multiplication algorithms. This research implemented both methods on secure Twisted Edwards curves (numsp384t1 and numsp512t1) using the affine coordinate system for clearer point representation. The binary method uses a double-and-add approach, which introduces conditional branches that increase execution variability, making it more vulnerable to timing-based side-channel attacks. In contrast, the elliptic net method structures point operations more uniformly, reducing observable patterns and improving leakage resistance despite its higher complexity. Simulated attack scenarios, including timing and power analysis, revealed that the elliptic net method maintained more consistent behavior and offered better protection against information leakage. Overall, the findings highlighted the performance of Elliptic Curve Cryptography Scalar Multiplications over side-channel attacks in the implementations. Area of Study: Cryptography Keywords: Binary, Diffie-Hellman, Digital Signature, Elliptic Net, Power Analysis, Timing

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