Pairings for cryptographers

Many research papers in pairing-based cryptography treat pairings as a “black box”. These papers build cryptographic schemes making use of various properties of pairings. If this approach is taken, then it is easy for authors to make invalid assumptions concerning the properties of pairings. The cryptographic schemes developed may not be realizable in practice, or may not be as efficient as the authors assume.The aim of this paper is to outline, in as simple a fashion as possible, the basic choices that are available when using pairings in cryptography. For each choice, the main properties and efficiency issues are summarized. The paper is intended to be of use to non-specialists who are interested in using pairings to design cryptographic schemes.     

Regularity of Optimal Controls for State Constrained Problems

Conditions are given under which optimal controls are Lipschitz continuous, for dynamic optimization problems with functional inequality constraints. The linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters. Smoothness conditions on the data are also relaxed. A key part of the proof involves an analysis of the implications of first order optimality conditions in the form of a nonsmooth Maximum Principle.     

Utility of the tandem Pauson-Khand reaction in the construction of tetracycles

The scope of the tandem Pauson-Khand reaction has been explored for the regiospecific construction of [5.5.5.5]- and [5.6.6.5]tetracyclic systems via the photolytic method of Livinghouse. The rapid regiospecific entry into the two dicyclopentapentanoid systems 17 and 29 was accomplished from the key diene-diynes 11 and 19b. A photochemically mediated catalytic tandem Pauson-Khand cyclization was employed to prepare the parent ring systems of dicyclopenta[a,e]pentalene (from 19b) and dicyclopenta[a,f]pentalene (from 11) in regiospecific fashion in a one-pot process. Under these conditions, conversion of acyclic diene-diyne 16 into tetracyclic system 17 was achieved in 74% yield, while a similar process was employed to convert 28 into tetracycle 29 in 90% yield. This is much improved over the previous conditions that employed NMO. Six carbon-carbon bonds were generated in this process constituting up to 98% yield for each carbon-carbon bond so formed. Furthermore, tetracyclic [5.6.6.5] systems such as dicyclopenta[b,g]decalins 37, 38, and 40 were prepared from similar diene-diyne precursors via the tandem Pauson-Khand cyclization. Importantly, acetal 36 provided the desired cis-fused [5.6.6.5] system 38a (via 40a/b) in stereospecific fashion. This reaction is unique in that it provides a cis-decalin ring system; moreover, the yield of each of the six carbon-carbon bonds formed in this process was at least 89%. The structure of cis diol 38a was confirmed by X-ray crystallography.     

A non-uniform birthday problem with applications to discrete logarithms

We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends on the colour of the ball. We determine the expected number of trials until two balls of different colours are placed in the same urn. As an aside we present an amusing “paradox” about birthdays.     

Students,mathematics, and technology: assessing the present – challenging the future

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships.     

On the distribution of the coefficients of normal forms for Frobenius expansions

Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions).