Workshop on Cryptographic Hardware and Embedded Systems CHES’2002, pages 485-500, San Francisco Bay (Redwood City), USA, August 13-15, 2002



Scalable and Unified Hardware to Compute Montgomery Inverse in GF(p) and GF(2n


Adnan Abdul-Aziz Gutub*, Alexandre F. Tenca, Erkay Savaş**, and Çetin K. Koç

Department of Electrical and Computer Engineering
Oregon State University, Corvallis, Oregon 97331, USA


*Now with King Fahd University, Dhahran, Saudi Arabia,


**Now with Sabanci University, Istanbul, Turkey, 


 Computing the inverse of a number in finite fields GF(p) or GF(2n) is equally important for cryptographic applications. This paper proposes a novel
scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2n) fields. We adjust and modify a GF(2n)
Montgomery inverse algorithm to accommodate multi-bit shifting hardware, making it very similar to a previously proposed GF(p) algorithm. The
architecture is intended to be scalable, which allows the hardware to compute the inverse of long precision numbers in a repetitive way. After implementing
this unified design it was compared with other designs. The unified hardware was found to be eight times smaller than another reconfigurable design, with
comparable performance. Even though the unified design consumes slightly more area and it is slightly slower than the scalable inverter implementations for
GF(p) only, it is a practical solution whenever arithmetic in the two finite fields is needed.