SCHEME FOR ARITHMETIC OPERATIONS IN FINITE FIELD AND GROUP OPERATIONS OVER ELLIPTIC CURVES REALIZING IMPROVED COMPUTATIONAL SPEED

A scheme for arithmetic operations in finite field and group operations over elliptic curves capable of realizing a very fast implementation. According to this scheme, by using a normal basis [.alpha. .alpha.+1], the multiplicative inverse calculation and the multiplication in the finite field GF(2.sup.2n) can be realized as combinations of multiplications, additions and a multiplicative inverse calculation in the subfield GF(2.sup.n). Also, by using a standard basis [1.alpha.], the multiplication, the square calculation, and the multiplicative inverse calculation in the finite field GF(2.sup.2n) can be realized as combinations of multiplications, additions and a multiplicative inverse calculation in the subfield GF(2.sup.n). These arithmetic operations can be utilized for calculating rational expressions expressing group operations over elliptic curves that are used in information security techniques such as elliptic curve cryptosystems.