Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Uniform Cyclic Group Factorizations of Finite Groups

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group $G$ is said to admit a uniform group factorization if there exist subgroups $H_1, H_2, \ldots, H_k$ such that $G = H_1 H_2 \cdots H_k$ and the number of ways to represent any element $g \in G$ as $g = h_1 h_2 \cdots h_k$ ($h_i \in H_i$) does not depend on the choice of $g$. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations.Comment: 10 pages. To appear in Communications in Algebr

Applications of Finite non-Abelian Simple Groups to Cryptography in the Quantum Era

The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context

Above the noise : the search for periodicities in the inner heliosphere

Remote sensing of coronal and heliospheric periodicities can provide vital insight into the local conditions and dynamics of the solar atmosphere. We seek to trace long (one hour or longer) periodic oscillatory signatures (previously identified above the limb in the corona by, e.g., Telloni et al., 2013, Astrophys. J., 767, 138) from their origin at the solar surface out into the heliosphere. To do this, we combine on-disk measurements taken by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) and concurrent extreme ultra-violet (EUV) and coronagraph data from one of the Solar Terrestrial Relations Observatory (STEREO) spacecraft to study the evolution of two active regions in the vicinity of an equatorial coronal hole over several days in early 2011. Fourier and wavelet analysis of signals are performed. Applying white-noise-based confidence levels to the power spectra associated with detrended intensity time series yields detections of oscillatory signatures with periods from 6 − 13 hours in both AIA and STEREO data. As was found by Telloni et al. (2013), these signatures are aligned with local magnetic structures. However, typical spectral power densities all vary substantially as a function of period, indicating spectra dominated by red (rather than white) noise. Contrary to the white-noise-based results, applying global confidence levels based on a generic background noise model (allowing a combination of white noise, red noise, and transients following Auch`ere et al., 2016, Astrophys. J., 825, 110) without detrending the time series, uncovers only sporadic, spatially uncorrelated evidence of periodic signatures in either instrument. Automating this method to individual pixels in the STEREO/COR coronagraph field of view is non-trivial.Publisher PDFPeer reviewe

Lidar cloud studies for FIRE and ECLIPS

Optical remote sensing measurements of cirrus cloud properties were collected by one airborne and four ground-based lidar systems over a 32 h period during this case study from the First ISCCP (International Satellite Cloud Climatology Program) Regional Experiment (FIRE) Intensive Field Observation (IFO) program. The lidar systems were variously equipped to collect linear depolarization, intrinsically calibrated backscatter, and Doppler velocity information. Data presented describe the temporal evolution and spatial distribution of cirrus clouds over an area encompassing southern and central Wisconsin. The cirrus cloud types include: dissipating subvisual and thin fibrous cirrus cloud bands, an isolated mesoscale uncinus complex (MUC), a large-scale deep cloud that developed into an organized cirrus structure within the lidar array, and a series of intensifying mesoscale cirrus cloud masses. Although the cirrus frequently developed in the vertical from particle fall-streaks emanating from generating regions at or near cloud tops, glaciating supercooled (-30 to -35 C) altocumulus clouds contributed to the production of ice mass at the base of the deep cirrus cloud, apparently even through riming, and other mechanisms involving evaporation, wave motions, and radiative effects are indicated. The generating regions ranged in scale from approximately 1.0 km cirrus uncinus cells, to organized MUC structures up to approximately 120 km across

Cryptanalysis of the MST_3 Public Key Cryptosystem

In this paper we describe a cryptanalysis of MST_3, a public key cryptosystem based on non-commutative groups recently proposed by Lempken, Magliveras, van Trung and Wei