MEMORY EFFICIENT HYPERELLIPTIC CURVE POINT COUNTING

Memory efficient hyperelliptic curve point counting

Let E be a hyperelliptic curve of genus g over a finite field of degree n and small characteristic. Using deformation theory we present an algorithm that computes the zeta function of E in time essentially cubic in n and quadratic memory. This improves substantially upon Kedlaya’s result which has the same time asymptotic, but requires cubic memory size. AMS (MOS) Subject Classification Codes: ...

Elliptic and Hyperelliptic Curve Point Counting through Deformation

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A Quasi Quadratic Time Algorithm for Hyperelliptic Curve Point Counting

We describe an algorithm to compute the cardinality of Jacobians of ordinary hyperelliptic curves of small genus over finite fields F2n with cost O(n ). This algorithm is derived from ideas due to Mestre. More precisely, we state the mathematical background behind Mestre’s algorithm and develop from it a variant with quasiquadratic time complexity. Among others, we present an algorithm to find ...

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...

Point Counting on Elliptic and Hyperelliptic Curves