Horizontal DPA Attacks against ECC: Impact of Implemented Field Multiplication Formula

Due to the nature of applications such as critical infrastructure and the Internet of Things etc. side channel analysis attacks are becoming a serious threat. Side channel analysis attacks take advantage from the fact that the behavior of crypto implementations can be observed and provides hints that simplify revealing keys. A new type of SCA are the so called horizontal SCAs. Well known randomization based countermeasures are effective means against vertical DPA attacks but they are not effective against horizontal DPA attacks. In this paper we investigate how the formula used to implement the multiplication of $GF(2^n)$-elements influences the results of horizontal DPA attacks against a Montgomery kP implementation. We implemented 5 designs with different partial multipliers, i.e. based on different multiplication formulae. We used two different technologies, i.e. a 130 and a 250 nm technology, to simulate power traces for our analysis. We show that the implemented multiplication formula influences the success of horizontal attacks significantly, but we also learned that its impact differs from technology to technology. Our analysis also reveals that the use of different multiplication formulae as the single countermeasure is not sufficient to protect cryptographic designs against horizontal DPA attacks.