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 .
Read more about this topic: Hyperelliptic Curve Cryptography
Famous quotes containing the words order of the and/or order:
“The world men inhabit ... is rather bleak. It is a world full of doubt and confusion, where vulnerability must be hidden, not shared; where competition, not co-operation, is the order of the day; where men sacrifice the possibility of knowing their own children and sharing in their upbringing, for the sake of a job they may have chosen by chance, which may not suit them and which in many cases dominates their lives to the exclusion of much else.”
—Anna Ford (b. 1943)
“To me the “female principle” is, or at least historically has been, basically anarchic. It values order without constraint, rule by custom not by force. It has been the male who enforces order, who constructs power structures, who makes, enforces, and breaks laws.”
—Ursula K. Le Guin (b. 1929)