Secure and Efficient RNS Software Implementation for Elliptic Curve Cryptography

Elliptic Curve Cryptography operations rely heavily on the strong security of scalar multiplication. However, this operation is vulnerable to side channel (SCA) and fault injection (FA) attacks. The use of alternative arithmetic systems like Residue Number System (RNS) for all scalar multiplication underline operations has been proposed as an efficient countermeasure approach for the above attacks. In RNS, a number is represented as a set of smaller numbers, where each one is the result of the modular reduction with a given moduli basis. Under certain requirements, a number can be uniquely transformed from the integers to the RNS domain (and vice versa) and all arithmetic operations can be performed in RNS. This representation provides an inherent SCA and FA resistance to many attacks and can be further enhanced by additional RNS arithmetic manipulations or more traditional algorithmic countermeasures. In this paper, extending our previous work, we explore the potentials of RNS as an SCA and FA countermeasure. A description of RNS based SCA and FA resistance means is provided through appropriate scalar multiplication algorithmic variations, traces of the proposed algorithm are collected and the results are analyzed regarding the RNS countermeasure strength. More specifically, in this paper, a secure RNS based Montgomery Power Ladder based scalar multiplication algorithm is provided and is implemented on an ARM Cortex A7 processor. The implementation SCA-FA resistance is evaluated by collecting preliminary leakage trace results that validate our initial assumptions.