Point counting on Picard curves in large characteristic

Point counting on Picard curves in large characteristic

We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O( √ p).

Point Counting in Families of Hyperelliptic Curves in Characteristic 2

Let ĒΓ be a family of hyperelliptic curves over F cl 2 with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ēγ̄, with γ̄ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and me...

Point Counting on Elliptic and Hyperelliptic Curves

Counting points on elliptic curves in medium characteristic

In this paper, we revisit the problem of computing the kernel of a separable isogeny of degree ` between two elliptic curves defined over a finite field Fq of characteristic p. We describe an algorithm the asymptotic time complexity of which is equal to e O(`(1 + `/p) log q) bit operations. This algorithm is particularly useful when ` > p and as a consequence, we obtain an improvement of the co...

Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits...