Formulas for the arithmetic geometric mean of curves of genus 3

The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of genus 3.Comment: 26 pages, amslatex, xypic +2 eps figure

Higher dimensional 3-adic CM construction

We find equations for the higher dimensional analogue of the modular curve X_0(3) using Mumford's algebraic formalism of algebraic theta functions. As a consequence, we derive a method for the construction of genus 2 hyperelliptic curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime 3. We prove the existence of a quasi-quadratic time algorithm for computing a canonical lift in characteristic 3 based on these equations, with a detailed description of our method in genus 1 and 2.Comment: 23 pages; major revie

Counting hyperelliptic curves that admit a Koblitz model

Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)

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

Some heuristics about elliptic curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict $cX^{5/6}$ total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions

Elliptic curve cryptography: Generation and validation of domain parameters in binary Galois Fields

Elliptic curve cryptography (ECC) is an increasingly popular method for securing many forms of data and communication via public key encryption. The algorithm utilizes key parameters, referred to as the domain parameters. These parameters must adhere to specific characteristics in order to be valid for use in the algorithm. The American National Standards Institute (ANSI), in ANSI X9.62, provides the process for generating and validating these parameters. The National Institute of Standards and Technology (NIST) has identified fifteen sets of parameters; five for prime fields, five for binary fields, and five for Koblitz curves. The parameter generation and validation processes have several key issues. The first is the fast reduction within the proper modulus. The modulus chosen is an irreducible polynomial having degree greater than 160. Choosing irreducible polynomials of a particular order is less critical since they have isomorphic properties, mathematically. However, since there are differences in performance, there are standards that determine the specific polynomials chosen. The NIST standards are also based on word lengths of 32 bits. Processor architecture, primality, and validation of irreducibility are other important characteristics. The area of ECC that is researched is the generation and validation processes, as they are specified for binary Galois Fields F (2m). The rationale for the parameters, as computed for 32 bit and 64 bit computer architectures, and the algorithms used for implementation, as specified by ANSI, NIST and others, are examined. The methods for fast reduction are also examined as a baseline for understanding these parameters. Another aspect of the research is to determine a set of parameters beyond the 571-bit length that meet the necessary criteria as determined by the standards

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics included modular forms, varieties over finite fields, rational and integral points on varieties, class groups, and integer factorization