Modular multiplication is the most dominant arithmetic operation in elliptic curve cryptography (ECC), which is a type of public-key cryptography. Montgomery multiplication is commonly used as a technique for the modular multiplication and required scalability since the bit length of operands varies depending on the security levels. Also, ECC is performed in  GF(P)  or  GF(2 n ) , and unified architectures for  GF(P)  and  GF(2 n )  multiplier are needed. However, in previous works, changing frequency or dual-radix architecture is necessary to deal with delay-time difference between  GF(P)  and  GF(2 n )  circuits of the multiplier because the critical path of  GF(P)  circuit is longer. This paper proposes a scalable unified dual-radix architecture for Montgomery multiplication in  GF(P)  and  GF(2 n ) . The proposed architecture unifies 4 parallel radix-2 16  multipliers in  GF(P)  and a radix-2 64  multiplier in  GF(2 n )  into a single unit. Applying lower radix to  GF(P)  multiplier shortens its critical path and makes it possible to compute the operands in the two fields using the same multiplier at the same frequency so that clock dividers to deal with the delay-time difference are not required. Moreover, parallel architecture in  GF(P)  reduces the clock cycles increased by dual-radix approach. Consequently, the proposed architecture achieves to compute  GF(P)  256-bit Montgomery multiplication in 0.23μs.
['Kazuyuki Tanimura', 'Ryuta Nara', 'Shunitsu Kohara', 'Kazunori Shimizu', 'Youhua Shi', 'Nozomu Togawa', 'Masao Yanagisawa', 'Tatsuo Ohtsuki']
Scalable unified dual-radix architecture for montgomery multiplication in GF ( P ) and GF (2 n )