Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

In CRYPTO 2008, one year earlier than Gentry\u27s pioneering \lq\lq bootstrapping\u27\u27 technique on constructing the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. Namely, they showed that the $\mathsf{NAND}$ operator can be realized in some \emph{non-commutative} groups; consequently, in combination with the $\mathsf{NAND}$ operator realized in such a group, homomorphically encrypting the elements of the group will yield an FHE scheme. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature based on their approach. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism $\pi \colon \widetilde{G} \to G$ between finite groups for which bit operators are realized in $G$ and the elements of the kernel of $\pi$ are indistinguishable from the general elements of $\widetilde{G}$. Secondly, we propose new methodologies to realize bit operators in some groups, which enlarges the possibility of the group $G$ to be used in our framework. Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the \lq\lq linearity\u27\u27 of the construction. Then we propose an idea to avoid such \lq\lq linearity\u27\u27 by using combinatorial group theory, and give a prototypical but still \emph{incomplete} construction in the sense that it is \lq\lq non-compact\u27\u27 FHE, i.e., the ciphertext size is unbounded (though the ciphertexts are noise-free as opposed to the existing FHE schemes). Completely realizing FHE schemes based on our proposed framework is left as a future research topic

Efficient Finite Groups Arising in the Study of Relative Asphericity

We study a class of two-generator two-relator groups, denoted Jn(m, k), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups Fa,b,c with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to ?bicyclically presented groups? that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups

Locating Pleistocene Refugia: Comparing Phylogeographic and Ecological Niche Model Predictions

Ecological niche models (ENMs) provide a means of characterizing the spatial distribution of suitable conditions for species, and have recently been applied to the challenge of locating potential distributional areas at the Last Glacial Maximum (LGM) when unfavorable climate conditions led to range contractions and fragmentation. Here, we compare and contrast ENM-based reconstructions of LGM refugial locations with those resulting from the more traditional molecular genetic and phylogeographic predictions. We examined 20 North American terrestrial vertebrate species from different regions and with different range sizes for which refugia have been identified based on phylogeographic analyses, using ENM tools to make parallel predictions. We then assessed the correspondence between the two approaches based on spatial overlap and areal extent of the predicted refugia. In 14 of the 20 species, the predictions from ENM and predictions based on phylogeographic studies were significantly spatially correlated, suggesting that the two approaches to development of refugial maps are converging on a similar result. Our results confirm that ENM scenario exploration can provide a useful complement to molecular studies, offering a less subjective, spatially explicit hypothesis of past geographic patterns of distribution

Verifiable Delay Functions

We study the problem of building a verifiable delay function (VDF). A VDF requires a specified number of sequential steps to evaluate, yet produces a unique output that can be efficiently and publicly verified. VDFs have many applications in decentralized systems, including public randomness beacons, leader election in consensus protocols, and proofs of replication. We formalize the requirements for VDFs and present new candidate constructions that are the first to achieve an exponential gap between evaluation and verification time

Elevational Distribution and Extinction Risk in Birds

Mountainous regions are hotspots of terrestrial biodiversity. Unlike islands, which have been the focus of extensive research on extinction dynamics, fewer studies have examined mountain ranges even though they face increasing threats from human pressures – notably habitat conversion and climate change. Limits to the taxonomic and geographical extent and resolution of previously available information have precluded an explicit assessment of the relative role of elevational distribution in determining extinction risk. We use a new global species-level avian database to quantify the influence of elevational distribution (range, maximum and midpoint) on extinction risk in birds at the global scale. We also tested this relationship within biogeographic realms, higher taxonomic levels, and across phylogenetic contrasts. Potential confounding variables (i.e. phylogenetic, distributional, morphological, life history and niche breadth) were also tested and controlled for. We show that the three measures of elevational distribution are strong negative predictors of avian extinction risk, with elevational range comparable and complementary to that of geographical range size. Extinction risk was also found to be positively associated with body weight, development and adult survival, but negatively associated with reproduction and niche breadth. The robust and consistent findings from this study demonstrate the importance of elevational distribution as a key driver of variation in extinction dynamics in birds. Our results also highlight elevational distribution as a missing criterion in current schemes for quantifying extinction risk and setting species conservation priorities in birds. Further research is recommended to test for generality across non-avian taxa, which will require an advance in our knowledge of species’ current elevational ranges and increased efforts to digitise and centralise such data

Cleavage modification did not alter blastomere fates during bryozoan evolution

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.The study was funded by the core budget of the Sars Centre and by The European Research Council Community’s Framework Program Horizon 2020 (2014–2020) ERC grant agreement 648861 to A

A REMARK ON INFINITE TORSION GROUPS WITH PERIODIC COHOMOLOGY

In this paper we obtain the structure of locally finite groups which have periodic cohomology after some steps