Scalable Unified Dual-Radix Architecture for Montgomery Multiplication in GF (P) and GF (2n)

Abstract — Modular multiplication is the most dominant arith-metic 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 depend-ing on the security levels. Also, ECC is performed in GF (P) or GF (2n), and unified architectures for GF (P) and GF (2n) multi-plier are needed. However, in previous works, changing frequency or dual-radix architecture is necessary to deal with delay-time dif-ference between GF (P) and GF (2n) circuits of the multiplier be-cause the critical path of GF (P) circuit is longer. This paper pro-poses a scalable unified dual-radix architecture for Montgomery multiplication in GF (P) and GF (2n). The proposed architecture unifies 4 parallel radix-216 multipliers in GF (P) and a radix-264 multiplier in GF (2n) 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 multi-plier at the same frequency so that clock dividers to deal with the delay-time difference are not required. Moreover, parallel archi-tecture 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. I