Uniform Extensions Research Articles

Uniform instruction set extensions for multiplications in contemporary and post-quantum cryptography

Hybrid key encapsulation is in the process of becoming the de-facto standard for integration of post-quantum cryptography (PQC). Supporting two cryptographic primitives is a challenging task for constrained embedded systems. Both contemporary cryptography based on elliptic curves or RSA and PQC based on lattices require costly multiplications. Recent works have shown how to implement lattice-based cryptography on big-integer coprocessors. We propose a novel hardware design that natively supports the multiplication of polynomials and big integers, integrate it into a RISC-V core, and extend the RISC-V ISA accordingly. We provide an implementation of Saber and X25519 to demonstrate that both lattice- and elliptic-curve-based cryptography benefits from our extension. Our implementation requires only intermediate logic overhead, while significantly outperforming optimized ARM Cortex M4 implementations, other hardware/software codesigns, and designs that rely on contemporary accelerators.

Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

Extensions of uniform algebras

The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole’s method for extending uniform algebras by adding square roots of functions to a given uniform algebra has been used to solve many problems within uniform algebra theory. However, there are numerous other examples in the theory of uniform algebras that can be realized as extensions of a uniform algebra, and these more general extensions have received little attention in the literature. In this paper, we investigate more general classes of uniform algebra extensions. We introduce a new class of extensions of uniform algebras, and show that several important properties of the original uniform algebra are preserved in these extensions. We also show that several well-known examples from the theory of uniform algebras belong to these more general classes of uniform algebra extensions.

Toward a Principled Sampling Theory for Quasi-Orders.

Quasi-orders, that is, reflexive and transitive binary relations, have numerous applications. In educational theories, the dependencies of mastery among the problems of a test can be modeled by quasi-orders. Methods such as item tree or Boolean analysis that mine for quasi-orders in empirical data are sensitive to the underlying quasi-order structure. These data mining techniques have to be compared based on extensive simulation studies, with unbiased samples of randomly generated quasi-orders at their basis. In this paper, we develop techniques that can provide the required quasi-order samples. We introduce a discrete doubly inductive procedure for incrementally constructing the set of all quasi-orders on a finite item set. A randomization of this deterministic procedure allows us to generate representative samples of random quasi-orders. With an outer level inductive algorithm, we consider the uniform random extensions of the trace quasi-orders to higher dimension. This is combined with an inner level inductive algorithm to correct the extensions that violate the transitivity property. The inner level correction step entails sampling biases. We propose three algorithms for bias correction and investigate them in simulation. It is evident that, on even up to 50 items, the new algorithms create close to representative quasi-order samples within acceptable computing time. Hence, the principled approach is a significant improvement to existing methods that are used to draw quasi-orders uniformly at random but cannot cope with reasonably large item sets.

Strongly uniform extensions of dual 2-designs

A strongly α-uniform partial line space of order (s, t) is called an α-partial geometry. If α = t+1, then the geometry is a dual 2-design. Locally triangular and locally Grassman graphs correspond to triangular extensions of certain dual 2-designs, and the class of strongly uniform quasi-biplanes coincides with the class of strongly uniform extensions of dual 2-designs. We study strongly uniform extensions of dual 2-designs.

(Uniform) convergence of twisted ergodic averages

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p>1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

Sub-Exponential Decay and Uniform Holomorphic Extensions for Semilinear Pseudodifferential Equations

The goal of the present paper is to derive a simultaneous description of the decay and the regularity properties for elliptic equations in ℝ n with coefficients admitting irregular decay at infinity of the type O(|x|−σ), σ > 0, filling the gap between the case of Cordes globally elliptic operators and the case of regular/Fuchs behavior at infinity. Representative examples in ℝ n are the equations where 0 < σ <2, ⟨x⟩ = (1 + |x|2)1/2, ω(x) a bounded smooth function, f given and F[u] a polynomial in u, and similar Schrödinger equations at the endpoint of the spectrum. Other relevant examples are given by linear and nonlinear ordinary differential equations with irregular type of singularity for x → ∞, admitting solutions y(x) with holomorphic extension in a strip and sub-exponential decay of type |y(x)| ≤Ce −ϵ|x| r ; 0 < r < 1. Sobolev estimates for the linear case are proved in the frame of a suitable pseudodifferential calculus; decay and uniform holomorphic extensions are then obtained in terms of Gelfand–Shilov spaces by an inductive technique. The same technique allows to extend the results to the semilinear case.

Uniform extensions of partial geometries

A geometry of rank 2 is an incidence system (P, \(\mathcal{B}\)), where P is a set of points and \(\mathcal{B}\) is a set of subsets from P, called blocks. Two points are called collinear if they lie in a common block. A pair (a, B) from (P, \(\mathcal{B}\)) is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called φ-uniform (φ is a natural number) if for any antiflag (a, B) the number of points in the block B collinear to the point a equals 0 or φ, and strongly φ-uniform if this number equals φ. In this paper, we study φ-uniform extensions of partial geometries pGα(s, t) with φ = s and strongly φ-uniform geometries with φ = s − 1. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.

On graphs the neighbourhoods of whose vertices are strongly regular with

It is proved that an amply regular graph of diameter greater than 2 the neighbourhoods of whose vertices are strongly regular with is a Taylor graph. A description of the locally Paley graphs is obtained. Uniform extensions of the partial geometries are found.

Natural convection below heated horizontal rectangular plates

The boundary layer equations describing the high Grashof number laminar natural convection flow beneath a heated horizontal surface are solved numerically for an infinitely long strip with a uniform temperature central region and horizontal adiabatic extensions on the sides, and for rectangular plates of two different aspect ratios kept at constant temperature. The boundary condition required by these equations at the edge of the plate is discussed. The results are compared with existing approximate theoretical results and with experiments, as well as with the solutions obtained previously using the same method for a strip and a circular disk.

Higher bruhat orders and cyclic hyperplane arrangements

We study the higher Bruhat orders $B(n,k)$ of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets $U(C^{n+1,r},n+1)$ of uniform extensions of the alternating oriented matroids $C^{n,r}$ for $r:=n-k$ (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), - show that $B(n,k)$ is a lattice for $k =1$ and for $r\le 3$, but not in general, - show that $B(n,k)$ is ordered by inclusion of inversion sets for $k=1$ and for $r\le 4$. However, $B(8,3)$ is not ordered by inclusion. This implies that the partial order $B_\subseteq (n,k)$ defined by inclusion of inversion sets differs from $B(n,k)$ in general. We show that the proper part of $B_\subseteq (n,k)$ is homotopy equivalent to $S^{r-2}$. Consequently, - $B(n,k)\simeq S^{r-2}$ for $k=1$ and for $r\le 4$. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for $r=3$, and that the proper part is not always homotopy equivalent to $S^{r(M)-2}$.

Qualitative Robustness of Rank Tests

An asymptotic notion of robust tests is studied which is based on the requirement of equicontinuous error probabilities. If the test statistics are consistent, their robustness in Hampel's sense and robustness of the associated tests turn out to be equivalent. Uniform extensions are considered. Moreover, test breakdown points are defined. The main applications are on rank statistics: they are generally robust, under a slight condition even uniformly so; their points of final breakdown coincide with the breakdown points of the corresponding $R$ - estimators.

Some Methods of Experimental Studies into the Extension of Polymeric Liquids

Abstract Experimental studies into the extension of polymeric liquids have revealed an envelope in the stress versus recoverable strain relationships, flow retardation under considerable elastic strains, and a correlation between uniform and non-uniform extensions.

Small deformation superposed on large deformation of an elastic dielectric

General theory dealing with problems of small deformation superposed on large deformation of an elastic polarized dielectric is formulated on the lines of Green and Zerna [1]. Small twist superposed on large uniform extensions of a hollow cylindrical dielectric is considered as a particular example. The new stress field required to keep the body thus deformed in equilibrium is found to be affected by the polarization vector. For zero polarization, the results reduce to those of classical finite strain theory [1].

Small uniform extensions superposed on the finite torsion of a homogeneous cylinder

Using linear stress-strain law and finite components of strain tensor, a general theory dealing with problems of different types of small deformations superposed on finite deformations has been developed in this paper. The case of small uniform extensions along three perpendicular directions superposed on a homogeneous isotropic solid right circular cylinder of compressible elastic material already subjected to finite torsion and extension has been discussed as an illustration. It has been found that in addition to equal and opposite forces applied at The plane ends and a couple about the axis, a surface traction on the curved surface is also needed to keep the final state of the body in equilibrium. The stresses in the final state are found to depend upon the previous strain history of the body.

Thermo-elastic Rayleigh waves in a prestressed medium

ABSTRACTWe consider the propagation of thermo-elastic Rayleigh waves in a half space, which has been subjected to large uniform extensions, at constant temperature, in three mutually perpendicular directions, one being perpendicular to the surface, and the extension ratios in the other two directions being equal. Rayleigh waves, assuming (a) isentropic conditions and (b) isothermal conditions, are studied, and the relation of the approximation (a) to the exact thermo-elastic theory is considered.

Plane thermo-elastic waves in an initially stressed medium

The propagation of plane thermo-elastic waves of small amplitude in an infinite body is studied, the body having been subjected to large uniform extensions, at constant temperature, in three perpendicular directions, two of the extension ratios being equal.

A Simple Constant Stress Apparatus for Creep Testing

A simple device is described which can maintain the stress on a creep specimen constant to within 9.8% during uniform extensions up to 100% The apparatus is suitable for slow rates of strain. Examples are given of creep curves obtained with this device from tests on lead and a lead-tin alloy extending to 1,000 and 400 hours respectively; good agreement with the Andrade creep equation is observed in both cases

The theory of aplanatic surfaces

The necessary and sufficient condition that an optical system should have a pair of aplanatic surfaces is that the eikonal of the system can be expressed as a homogeneous function of the first order of three variables, each of which is a linear function of the direction cosines of the ray in the object and image spaces. Methods are given for finding the equations of these surfaces when the eikonal is given, and for finding the eikonal when the surfaces are given. Conjugate aplanatic surfaces are similar apart from arbitrary uniform unidirectional extensions or compressions. In general only one pair of aplanatic surfaces is possible, but in spherically symmetrical systems two pairs are found. In addition there may be aplanatic imagery at isolated points.