Multiplier Design For Even-type Gaussian Normal Basis

Elliptic curve encryption(ECC)is widely used in lots of applications,its security is proven in practice.The encryption and decryption of elliptic curve encryption involves the basic arithmetic operations over GF(2~m).In these arithmetic operations,multiplication is one of the most important operations.To accomplish these arithmetic operations,efficient multipliers are required.The time and space complexities of multiplication in finite field GF(2~m) are strongly related to the representation of basis in the finite field.In other words,the base determines efficiency.Polynomial basis(PB),normal basis(NB)and dual basis(DB)are three kinds of commonly used basis representations.They are used to design different types of multipliers,due to each of them has its own advantages.In NB multiplication square operation is very efficient,and it can be achieved by cyclically shifting its binary form.Gaussian normal basis(GNB)is a special class of NB,which inherits the advantages of the NB.Even-type GNB multiplier is very efficient to achieve the exponent arithmetic and the inverse of multiplication operation.Even-type GNB received widespread attention because of its low complexity.Based on the consideration of space complexity,an even-type Gaussian normal basis is selected in this paper.The purpose of this paper is to design a multiplier which can make the space complexity as small as possible to improve the efficiency of elliptic curve encryption applications.In this paper,three even-type Gaussian normal basis multipliers are proposed.The first is the symmetric matrix and vector multiplication structure,it can reduce 50% multiplications than pure matrix-vector product;the second is the block symmetric matrix and vector multiplication structure,which is extend from the first structure;The third is the Hankel and Teoplitz systolic array structure.Through the analysis of the complexities of the three multipliers,the three multiplier structures have a good result in the reduction of space complexity.The three multiplier structures can improve the efficiency of elliptic curve encryption applications.Compared to other existing even-type GNB multipliers,the proposed Hankel and Teoplitz systolic array multiplier use one core multiplier instead of two TMVPs.According to theoretical analysis,the proposed multiplier has low space complexity for even-type GNBmultipliers.