Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field F. The authors of this paper revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi [math.NT/0409209, to appear in Mathematics of Computation]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g2.376) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. They show show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an n × n matrix over a finite field that requires O(n3) field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worst-case complexity of O(n4). Finally, we report the results of practical experiments with an implementation of our algorithms in comparison with the current algorithms in the GAP library.

We construct two embeddings of finite groups into groups of Lie type. These embeddings have the interesting property that the finite subgroup acts irreducibly on a minimal module for the group of Lie type. We present our constructions as examples of a general method that obtains embeddings into groups of Lie type.

In this paper, the authors examine a number of ways of implementing characteristic three arithmetic for use in cryptosystems based on the Tate pairing. Three alternative representations of the field elements are examined, and the resulting algorithms for the field addition, multiplication and cubing are compared. Issues related to the arithmetic of supersingular elliptic curves over fields of characteristic three are also examined. Details of how to compute the Tate pairing itself are not covered, since these are well documented elsewhere.

We classify the simple Lie algebras of dimension at most 9 over GF(2). There is one of dimension 3 and one of dimension 6, there are two of dimension 7 and two of dimension 8, and there is one of dimension 9. The two simple Lie algebras of dimension 8 are restricted Lie algebras. If we extend the ground field to GF(4), then the six-dimensional algebra is no longer simple, and if we extend the ground field to GF(8) then the nine-dimensional algebra is no longer simple. But the other algebras are all central simple.

Corrigendum to Proposition 14, on page 87 of the paper ‘Reduction of binary cubic and quartic forms’, LMS Journal of Computation and Mathematics, Volume 2, pp. 62–92.

In this paper, the authors analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP) for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field F2N, where N is in [100,600], elliptic curve parameters are identified such that: (i) there should exist a cryptographically interesting elliptic curve E over F2N with these parameters; and (ii) the GHS attack is more efficient for solving the ECDLP in E(F2N) than for solving the ECDLP on any other cryptographically interesting elliptic curve over F2N. The feasibility of the GHS attack on the specific elliptic curves is examined over F2176, F2208, F2272, F2304 and F2368, which are provided as examples in the ANSI X9.62 standard for the elliptic curve signature scheme ECDSA. Finally, several concrete instances are provided of the ECDLP over F2N, N composite, of increasing difficulty; these resist all previously known attacks, but are within reach of the GHS attack.

We present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite held, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds ‘by induction on the dimension’. The ‘inductive step’ combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of Christol and Dwork on p-adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane.

A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.

We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.

Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.

Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 + v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

In this paper we display an explicit polynomial having Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunter Malle. Furthermore, the polynomial has small Galois root discriminant; this fact answers a question of John Jones and David Roberts. The computation of this polynomial uses modular forms and their Galois representations.

Let A be a finite dimensional algebra over a finite field F. Condensing an A-module V with two different idempotents e and e′ leads to the problem that to compare the composition series of V e and V e′, we need to match the composition factors of both modules. In other words, given a composition factor S of V e, we have to find a composition factor S′ of V e′ such that there exists a composition factor Ŝ of V with Ŝ e ≅ S and Ŝ e′ ≅ S′, or prove that no such S′ exists. In this note, we present a computationally tractable solution to this problem.

We apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

This paper presents a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. The new procedure is a Monte Carlo algorithm, and it is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. The authors' major objective was that the procedures could be proved to be Monte Carlo algorithms, and the costs computed. In addition, they explicitly determined suitable subset chains for six of the sporadic groups, and then implemented the algorithms involving these chains in the GAP computational algebra system. It turns out that sample imple-mentations perform well in practice. The implementations will be made available publicly in the form of a GAP package.

The author considers the development of algorithms for deciding whether a finitely generated matrix group over a field of positive characteristic is finite. A deterministic algorithm for deciding the finiteness is presented for the case of a field of transcendence degree one over a finite field.

The authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.

Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

Given a Markov chain, a stochastic flow that simultaneously constructs sample paths started at each possible initial value can be constructed as a composition of random fields. Here, a method is described for coupling flows by modifying an arbitrary field (consistent with the Markov chain of interest) by an independence Metropolis-Hastings iteration. The resulting stochastic flow is shown to have many desirable coalescence properties, regardless of the form of the original flow.

In previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra 28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz for q even and n = 6,8,12,16. In the range 11 < n < 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3), PΩ±(8, 3k), PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) for n = 9,11,13, PSp(8, 7k) are not Hurwitz.