A Novel and High-Performance Modular Square Scheme for Elliptic Curve Cryptography Over GF( ${p}$ )

In this brief, we present a novel and high-performance modular squaring scheme with low complexity and a small hardware area for elliptic curve cryptography over GF( ${p}$ ). First, we develop a method to reduce half of partial products in a squaring operation by using the proposed same items merging and logic combination. Second, we propose the modular squaring scheme that can compress the partial products based on the method above and accomplish accumulation and reduction simultaneously. Third, we devise the implementation circuits for the proposed modular squaring scheme and then simplify the circuits by using the property of the prime number. Finally, we implement the circuits on different platform and the 0.13- ${\mu }\text{m}$ CMOS ASIC implementation demonstrates that our design can perform a 256-bit modular squaring in 0.36- $ {\mu }\text{s}$ with 17200 gates, which achieves a desirable balance between hardware resource and performance.