Atomic Patterns: Field Operation Distinguishability on Cryptographic ASICs

Scalar multiplication $k P$ is the most frequently attacked operation in Elliptic Curve Cryptosystems (ECC) protocols. To counter simple Side-Channel Analysis (SCA), the atomicity principle and various atomic block constructions have been proposed in the past. In this paper, we demonstrate that binary $k P$ algorithms based on atomic patterns remain vulnerable to simple SCA, due to different energy consumption of the field multiplier while obtaining the second multiplicand for field multiplication or squaring operations. The reasons for this difference are the data-bit as well as the address-bit effects. Please note, this leakage does not depend on the multiplication formula used. We validate this vulnerability through experiments using Longa’s atomic patterns, revealing that such leakage significantly compromises the SCA resistance of numerous atomic implementations and facilitates potential key extraction. We conclude by providing some potential solutions to mitigate the revealed vulnerability.