Fast modular inversion algorithm to match any operation unit
Speeding up modular inversion is one of the most important subjects in the field of information security. Over the elliptic curve - on the prime finite field in particular goals - public-key cryptosystems and digital signature schemes frequently use modular inversion if affine coordinates are selected. In the regular computer environment, most data transmission via networks and data storage on memories as well as the operation set of processors are performed in multiples of eight bits or bytes. A fast modular multiplication algorithm that matches these operation units for DSP was proposed to accelerate the Montgomery method by Dusse and Kaliski [2]. However, modular inversion algorithms were developed using bit by bit operation and so do not match the operation unit. This paper proposes two techniques for modular inversion that suits any arbitrary processing unit. The first technique proposes a new extended GCD procedure with out any division. It can be constructed by the shifting, adding and multiplying operations, all of which a Montgomery modular arithmetic algorithm employs. The second technique can reduce the delay time of post processing in the modular inversion algorithm. In particular, it is of great use for the modular inversion defined in the Montgomery representation. These proposed techniques make modular inversion about 5.5 times faster.
modular inversion montgomery method binary gcd euclidean algorithm elliptic-curve cryptosystem