Counting Points on the Jacobian Variety of a
نویسندگان
چکیده
Counting the order of the Jacobian group of a hyperelliptic curve over a nite eld is very important for constructing a hyperelliptic curve cryptosystem (HECC), but known algorithms to compute the order of a Jacobian group over a given large prime eld need very long running times. In this note, we propose a practical polynomial-time algorithm to compute the order of the Jacobian group for a hyperelliptic curve of type y 2 = x 5 + ax over a given large prime eld F p , e.g. an 80-bit eld. We also investigate the order of the Jacobian group for such curve and determine the necessary condition to be suitable for HECC, that is, to satisfy that the order of the Jacobian group is of the form l 1 c where l is a prime number greater than about 2 160 and c is a very small integer. Moreover we show some examples of a suitable curve for HECC obtained by using our algorithm.
منابع مشابه
Counting points on varieties over finite fields of small characteristic
We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic. ...
متن کاملA p-adic quasi-quadratic point counting algorithm
In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field Fq of cardinality q with time complexity O(n) and space complexity O(n), where n = log(q). In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalizat...
متن کاملCounting Points on Genus 2 Curves with Real Multiplication
We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field Fq of large characteristic from Õ(log q) to Õ(log q). Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applica...
متن کاملCounting Points for Hyperelliptic Curves of Type y2= x5 + ax over Finite Prime Fields
Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y = x + ax over given large...
متن کاملPoints Counting Algorithm for One-dimensional Family of Genus 3 Nonhyperelliptic Curves over Finite Fields
In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of one-dimensional family of genus 3 nonhyperelliptic curves over finite fields. We also provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of C : y3 = x4 + a over a finite field Fp with p ≡ 1 (mod 3) and p 6≡ 1 (mod 4). Moreover, we give s...
متن کامل