Efficient Implementation of Karatsuba Algorithm Based Three-Operand Multiplication Over Binary Extension Field

Three-operation multiplication (TOM) over binary extension field is frequently encountered in cryptosystems such as elliptic curve cryptography. Though digit-serial polynomial basis multipliers are usually preferred for the realization of TOM due to their efficient tradeoff in implementation complexity, the Karatsuba algorithm (KA)-based strategy is rarely employed to reduce the complexity further. Based on this reason, in this paper, we derive a novel low-complexity implementation of TOM based on a new KA-based digit-serial multiplier. The proposed TOM is obtained through two novel coherent interdependent efforts: 1) mapping an efficient KA-based algorithm into a novel digit-serial multiplier and 2) obtaining a new TOM structure through the novel derivation of the TOM algorithm. From the estimated results, it is shown that the proposed structure has significant lower area-time-complexities when compared with the existing competing TOMs. The proposed TOM is highly regular with low-complexity, and hence can be employed in many cryptographic applications.