Hyperelliptic Curve Cryptography - Order of The Jacobian

Order of The Jacobian

Hence, in order to choose a good curve and a good underlying finite field, it is important to know the order of the Jacobian. Consider a hyperelliptic curve of genus over the field where is the power of a prime number and define as but now over the field . It can be shown that the order of the Jacobian of lies in the interval, called the Hasse-Weil interval. But there is more, we can compute the order using the zeta-function on hyperelliptic curves. Let be the number of points on . Then we define the zeta-function of as . For this zeta-function it can be shown that where is a polynomial of degree with coefficients in . Furthermore factors as where for all . Here denotes the complex conjugate of . Finally we have that the order of equals . Hence orders of Jacobians can be found by computing the roots of .