Research Of Modular Multiplier Based On Normal Basis In Elliptic Curve Cryptography

Elliptic Curve Cryptography (ECC) has gradually become the new generation of publickey cryptography standards with its advantages in security, processing speed and circuit area.Finite field modular multiplication is the key operation in ECC and its speed is also the mainfactor influencing the processing performance. Owing to its lowest operation complexity inhardware circuit, there are more and more researches and applications of the modularmultiplication on Type-Ⅱ Optimal Normal Bases.Modular multiplication on Type-Ⅱ ONB is essentially the operation of random high-ordersparse matrix, so multiplication on normal bases, which is length-variable and reconfigurable,has low speed and large area. To solve these problems, this paper designs three hardwarearchitectures of serial, parallel and combination of the both two in different ECC applications.The paper first analyses the characteristics of elements expressed by Type-Ⅱ ONB andprovides a base conversion algorithm from Type-Ⅱ ONB to Reordered Normal Bases. Wedesign a fast reconfigurable conversion unit to make modular multiplication briefer and easilyperformed in hardware. We optimize the multiplication algorithm on Type-Ⅱ ONB with thetheory of base conversion and serial input¶llel output. A serial structured multiplicationwith reconfiguration is designed, which has the least resources. Based on the word theory, weoptimize multiplication on Type-Ⅱ ONB deeply and provide reconfigurable unit by combinationof both serial and parallel, which is balanced in processing speed and circuit area. After deepstudy of Toeplitz Matrix-Vector Product algorithm, we improve the algorithm and design alength-fixed parallel multiplication, which is of subquadrate space complexity and high speed.At last, we carry out simulations and verification of the three multiplications and useSynopsys Design Compiler to comply the circuit in CMOS0.18um Process Standard CellLibrary. The results show that the three designs can meet the different demands of ECCapplications and has an obvious advantage in resource access, performance and flexibility,compared with other similar designs.