Efficient and Constant Time Modular Inversions Over Prime Fields

Abstract

As an important operation, modular inversion is crucial for high-performance public key cryptosystems (PKC), especially in Elliptic curve-based schemes over prime fields. Both security and efficiency must be considered in a specific implementation. Straightforward implementation leaks side channel information which can be used for breaking Elliptic curve signature algorithm (ECDSA) through a combination attack. Modular inversion is also the most time-consuming operation which has important impact on the performance. Therefore, efficient and constant time modular inversion is an optimal option to ensure both security and efficiency. In this paper, we describe a general principle on how to construct efficient constant time modular inversion based on Fermat's little theorem (FLT) over prime fields. We give the tight upper bounder of multiplications needed in our schemes. Improvements are obtained from both algorithm architecture and Montgomery trick. We extended our scheme to NIST and Chinese Elliptic curve standard, which can save 90% multiplications. The total improvement is a factor of 2 by comparing the straightforward implementation.