HERatio: Homomorphic Encryption of Rationals using Laurent Polynomials

In this work we present $\mathsf{HERatio}$, a homomorphic encryption scheme that builds on the scheme of Brakerski, and Fan and Vercauteren. Our scheme naturally accepts Laurent polynomials as inputs, allowing it to work with rationals via their bounded base-$b$ expansions. This eliminates the need for a specialized encoder and streamlines encryption, while maintaining comparable efficiency to BFV. To achieve this, we introduce a new variant of the Polynomial Learning With Errors (PLWE) problem which employs Laurent polynomials instead of the usual ``classic\u27\u27 polynomials, and provide a reduction to the PLWE problem

Surface effects associated with dislocations in layer crystals

Dislocation configurations in thin foils cannot be accurately interpreted unless the effects of anisotropy and surfaces on the stresses and energies of edge and screw dislocations are known. Expressions for these effects are derived here for a semi-infinite hexagonal crystal with dislocations in the basal plane. lt is then shown that in plate-like crystals, as used in electron-microscopic investigations, the finite thickness of the specimen leads to observable effects on the dislocation patterns. In particular, the width of a ribbon decreases as it approaches the surface, due to the reduced repulsion between the partials, so that care is needed in deducing stacking fault energies from ribbon widths. Also the energy of a dislocation is a function of its distance from a surface, so that if it is crossed by a surface step it suffers a "refraction" which, in simple cases, follows Snell's law. lt is further shown that dislocations will tend tobe aligned with surface steps, artd the interaction energy between a step and a parallel dislocation line can thus be derived from experimental data. Finally, a method is suggested for obtaining information on the elastic constants from electron microscopic data

Die direkte Messung von Stapelfehlerenergien

Different methods for determining stacking fault energies from dislocation configurations observed in the electron microscope are discussed. Configurations discussed are simple, threefold, and fourfold ribbons, arrays of many parallel ribbons, and dislocation nodes. The latter are treated taking the mutual interaction of the partials approximately into account. Results are given for measurementsin graphite, MoS$_{2}$, AIN, and talc

The buckling of a thin plate due to the presence of an edge dislocation

lt is shown that an edge dislocation parallel to the surface of a thin foil causes buckling of this foil by an angle of about $\theta$ = b/t. (b = Burgers vector; t = thickness of the foil). The angle $\theta$ depends on the position of the dislocation. lt is maximum for a dislocation in the middle of the foil and it tends to zero as the dislocation approaches to the surface. lt is shown that the buckling is responsible for the discontinuous change in contrast along a dislocation as observed in transmission electron microscopy. The sense of buckling which can be determined by means of Kikuchi lines depends on the sign of the dislocation. The effect therefore provides an easy means to determine the sign of edge dislocations

PIE: pp-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption

A large part of current research in homomorphic encryption (HE) aims towards making HE practical for real-world applications. In any practical HE, an important issue is to convert the application data (type) to the data type suitable for the HE. The main purpose of this work is to investigate an efficient HE-compatible encoding method that is generic, and can be easily adapted to apply to the HE schemes over integers or polynomials. $p$-adic number theory provides a way to transform rationals to integers, which makes it a natural candidate for encoding rationals. Although one may use naive number-theoretic techniques to perform rational-to-integer transformations without reference to $p$-adic numbers, we contend that the theory of $p$-adic numbers is the proper lens to view such transformations. In this work we identify mathematical techniques (supported by $p$-adic number theory) as appropriate tools to construct a generic rational encoder which is compatible with HE. Based on these techniques, we propose a new encoding scheme PIE, that can be easily combined with both AGCD-based and RLWE-based HE to perform high precision arithmetic. After presenting an abstract version of PIE, we show how it can be attached to two well-known HE schemes: the AGCD-based IDGHV scheme and the RLWE-based (modified) Fan-Vercauteren scheme. We also discuss the advantages of our encoding scheme in comparison with previous works

Leveled Fully Homomorphic Encryption Schemes with Hensel Codes

We propose the use of Hensel codes (a mathematical tool lifted from the theory of $p$-adic numbers) as an alternative way to construct fully homomorphic encryption (FHE) schemes that rely on the hardness of some instance of the approximate common divisor (AGCD) problem. We provide a self-contained introduction to Hensel codes which covers all the properties of interest for this work. Two constructions are presented: a private-key leveled FHE scheme and a public-key leveled FHE scheme. The public-key scheme is obtained via minor modifications to the private-key scheme in which we explore asymmetric properties of Hensel codes. The efficiency and security (under an AGCD variant) of the public-key scheme are discussed in detail. Our constructions take messages from large specialized subsets of the rational numbers that admit fractional numerical inputs and associated computations for virtually any real-world application. Further, our results can be seen as a natural unification of error-free computation (computation free of rounding errors over rational numbers) and homomorphic encryption. Experimental results indicate the scheme is practical for a large variety of applications