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In this work we investigated the resistance of different kP implementations based on the Montgomery ladder against horizontal, i.e. single trace, attacks. Applying statistical methods for the analysis we were able to reveal the secret value k completely. The reason causing the success of our attacks is the key-dependent addressing of the registers and other design blocks, which is an inherent feature of binary kP algorithms. This dependency was successfully exploited in the past by Itoh et al. analyzing many hundreds of kP traces, i.e. this attack is a vertical address-bit differential power analysis attack against Montgomery ladder. The vulnerability of the Montgomery ladder against horizontal address-bit attacks was detected and demonstrated during our investigations. We were able to reveal the scalar k exploiting the address-bit vulnerability in single trace attacks using not only statistical methods, but also Fourier transform, selected clustering methods as well as one of the simplest methods – the automatized simple SCA. We performed successful horizontal address-bit SCA attacks against both types of ECs, i.e. against highly regular Montgomery ladder and against a binary kP algorithm implementing atomic patterns. The success of our attacks shows that the regularity and atomicity principles are not effective against horizontal address-bit attacks. As a means for reducing the attack success, we investigated the hiding ability of the field multiplier which is usually the largest block of kP designs. We implemented our field multiplier for ECs over prime fields corresponding to the 4-segment Karatsuba multiplication formula that reduces the execution time and the energy consumption for a kP operation by about 40 % in comparison to multipliers exploiting the classical multiplication formula. However, the energy consumption per clock remained in our multiplier without significant changes, i.e. the protective hiding properties of the multiplier as a noise source were not decreased. Another advantage of our field multiplier is its inherent resistance to horizontal collision attacks, in contrast to multipliers based on the classic multiplication formula. Additionally, we proposed regular scheduling for the block addressing as an effective strategy for reducing the success of horizontal address-bit attacks. Combining this approach with the hiding features of the field multipliers can increase the resistance of the kP designs for both types of ECs against a broad spectrum of SCA attacks. The mentioned analysis methods can be successfully applied for determining SCA leakage sources in the early design phase.