The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle⁄ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.

This paper introduces ‘hyper-and-elliptic-curve cryptography’, in which a single high-security group supports fast genus-2-hyperelliptic-curve formulas for variable-base-point single-scalar multiplication (for example, Diffie–Hellman shared-secret computation) and at the same time supports fast elliptic-curve formulas for fixed-base-point scalar multiplication (for example, key generation) and multi-scalar multiplication (for example, signature verification).

We compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2F4(22n+1) for all n ≥ 1 and prove that these Borel subgroups are M-groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2F4(22n+1) and show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for 2F4(22n+1) in characteristic 2. For most of the calculations we use CHEVIE.

Given a sextic CM field $K$, we give an explicit method for finding all genus-$3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.

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.

This paper presents a new geometric algorithm to construct a Ck-smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the De Casteljau procedure, this new algorithm is implemented in only three steps, independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function, which is defined as soon as the degree of smoothness is fixed.

Constructing numerical models of noisy partial differential equations is a very delicate task. Our long-term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step, we consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscale time-scales resolved by the model.

We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of the Mathieu sporadic group M22. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.

We present a domain-theoretic version of Picard's theorem for solving classical initial value problems in ℝn. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, from below and above respectively, exponentially fast, to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus we obtain a sound implementation framework to solve initial value problems. In particular, the use of rational arithmetic guarantees that implementations of our technique will adhere to the bounds on convergence speed and algebraic complexity.

In this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr–Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr–Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire’s problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.

The discriminant of a trinomial of the form $x^{n}\pm \,x^{m}\pm \,1$ has the form $\pm n^{n}\pm (n-m)^{n-m}m^{m}$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have that $((n^{2}-n+1)/3)^{2}$ always divides $n^{n}-(n-1)^{n-1}$. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as $n$ varies seems to be independent of $m$, and this set can be seen as a generalization of the Wieferich primes, those primes $p$ such that $2^{p}$ is congruent to 2 (mod $p^{2}$). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.

A generalisation of Milner's ‘LCF approach’ is described. This allows algorithms based on binary decision diagrams (BDDs) to be programmed as derived proof rules in a calculus of representation judgements. The derivation of representation judgements becomes an LCF-style proof by defining an abstract type for judgements analogous to the LCF type of theorems. The primitive inference rules for representation judgements correspond to the operations provided by an efficient BDD package coded in C (BuDDy). Proof can combine traditional inference with steps inferring representation judgements. The resulting system provides a platform to support a tight and principled integration of theorem proving and model checking. The methods are illustrated by using them to solve all instances of a generalised Missionaries and Cannibals problem.

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.

Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the index ℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ≤16 is obtained.

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$-module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.

Supplementary materials are available with this article.

The maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).

Supplementary materials are available with this article.

We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.