Revisiting Atomic Patterns for Elliptic Curve Scalar Multiplication Revealing Inherent Vulnerability to Simple SCA

Elliptic Curve Scalar Multiplication, denoted as $k P$ operation, is the basic operation in all Elliptic Curve-based cryptographic protocols. The atomicity principle and different atomic patterns for $\boldsymbol{k P}$ algorithms were proposed in the past as countermeasures against simple side-channel analysis. In this work, we investigated the resistance of a $k P$ algorithm implemented in hardware using Longa’s atomic patterns. We analysed its simulated power trace. We show in the example of our $k P$ implementation for the NIST EC P-256 that the field squaring operations are distinguishable from the field multiplications even if they are performed by the same field multiplier, due to the addressing of the second multiplicand. This inherent vulnerability of atomic patterns can be successfully exploited to reveal the scalar k.