Fast arithmetic in unramified p-adic fields

Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmuller lifting and computing minimal polynomials.Comment: 6 page

Quasi-quadratic elliptic curve point counting using rigid cohomology

We present a deterministic algorithm that computes the zeta function of a nonsupersingular elliptic curve E over a finite field with p^n elements in time quasi-quadratic in n. An older algorithm having the same time complexity uses the canonical lift of E, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.Comment: 14 page

The probability that the number of points on the Jacobian of a genus 2 curve is prime

In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus 2 and study various related issues, such as the probability of cyclicity and the probability of primality of the number of points on the curve itself. Finally, we discuss the asymptotic behavior as the genus tends to infinity.Comment: Minor edits, 37 pages. To appear in Proceedings of the London Mathematical Societ

Elliptic and hyperelliptic curve point counting through deformation

The use in cryptography of the group structure on elliptic curves or the jacobians of hyperelliptic curves over finite fields has been suggested already a few decades ago. In order to exploit the difficulty of the discrete logarithm problem on these groups, their size is a parameter of central importance. In this thesis we develop a certain p-adic analytic cohomology theory which gives rise to algorithms that can compute those sizes in a very efficient way. In 2001 Kedlaya came up with an algorithm for computing the zeta function of a hyperelliptic curve of genus g using Monsky-Washnitzer cohomology. This is a certain 2g-dimensional p-adic vector space on which a Frobenius operator lives. The characteristic polynomial of this operator yields then the zeta function. A few years later Lauder used in the context of determining the size of hypersurfaces an old result of Dwork concerning deformation. Deformation is a technique that looks at families of curves, where a specific fiber is easy to treat and the variation of Frobenius throughout the family satisfies a p-adic differential equation. Our work combines these two approaches and consists mainly of constructing a relative Monsky-Washnitzer cohomology (i.e. for one-dimensional families) for hyperelliptic curves and finding interesting forms of the corresponding differential equation. By taking well-chosen families E(G), so that E(0) is defined over a very small finite field, and solving the corresponding differential equation, we can construct an algorithm for a certain type of curves that uses far less memory than Kedlaya's and is also faster. An implementation in odd characteristic gives some very nice results. By reconsidering the solution technique of the differential equation and choosing other types of families we developed also an algorithm that works for any hyperelliptic curve and still requires far less memory than Kedlaya's algorithm. A final result concerns elliptic curves, where we showed that the Frobenius operator can be 'semi-diagonalized'. This allows a running time that is quadratic in the extension degree of the finite field and e.g. lets us compute the zeta function of an elliptic curve over a field of degree 100 and characteristic 3 in half a second.status: publishe