cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences

In this article, we introduce new notions $cc$-differential uniformity, $cc$-differential spectrum, PccN functions and APccN functions, and investigate their properties. We also introduce $c$-CCZ equivalence, $c$-EA equivalence, and $c1$-equivalence. We show that $c$-differential uniformity is invariant under $c1$-equivalence, and $cc$-differential uniformity and $cc$-differential spectrum are preserved under $c$-CCZ equivalence. We characterize $cc$-differential uniformity of vectorial Boolean functions in terms of the Walsh transformation. We investigate $cc$-differential uniformity of power functions $F(x)=x^d$. We also illustrate examples to prove that $c$-CCZ equivalence is strictly more general than $c$-EA equivalence.Comment: 18 pages. Comments welcom

On r-th Root Extraction Algorithm in F_q For q=lr^s+1 (mod r^(s+1)) with 0 < l < r and Small s

We present an r-th root extraction algorithm over a finite field F_q. Our algorithm precomputes a primitive r^s-th root of unity where s is the largest positive integer satisfying r^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms

Square Root Algorithm in F_q for q=2^s+1 (mod 2^(s+1))

We present a square root algorithm in F_q which generalizes Atkins\u27s square root algorithm for q=5(mod 8) and Kong et al.\u27s algorithm for q=9(mod 16) Our algorithm precomputes a primitive 2^s-th root of unity where s is the largest positive integer satisfying 2^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, Muller and Kong et al

Trace Expression of r-th Root over Finite Field

Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root formula which generalizes Müller\u27s result on square root, and which provides a possible improvement of the Cipolla-Lehmer algorithm for general case. More precisely, for given $r$-th power $c\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial of degree $r$ over $\mathbb F_q$

Anisotropic Thermal Conductivity of Nickel-Based Superalloy CM247LC Fabricated via Selective Laser Melting

Efforts to enhance thermal efficiency of turbines by increasing the turbine inlet temperature have been further accelerated by the introduction of 3D printing to turbine components as complex cooling geometry can be implemented using this technique. However, as opposed to the properties of materials fabricated by conventional methods, the properties of materials manufactured by 3D printing are not isotropic. In this study, we analyzed the anisotropic thermal conductivity of nickel-based superalloy CM247LC manufactured by selective laser melting (SLM). We found that as the density decreases, so does the thermal conductivity. In addition, the anisotropy in thermal conductivity is more pronounced at lower densities. It was confirmed that the samples manufactured with low energy density have the same electron thermal conductivity with respect to the orientation, but the lattice thermal conductivity was about 16.5% higher in the in-plane direction than in the cross-plane direction. This difference in anisotropic lattice thermal conductivity is proportional to the difference in square root of elastic modulus. We found that ellipsoidal pores contributed to a direction-dependent elastic modulus, resulting in anisotropy in thermal conductivity. The results of this study should be beneficial not only for designing next-generation gas turbines, but also for any system produced by 3D printing

New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field

In this paper, we present a new cube root algorithm in finite field $\mathbb{F}_{q}$ with $q$ a power of prime, which extends the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root method is inspired by the work of Müller \cite{Muller} on quadratic case. For given cubic residue $c \in \mathbb{F}_{q}$ with $q \equiv 1 \pmod{9}$, we show that there is an irreducible polynomial $f(x)=x^{3}-ax^{2}+bx-1$ with root $\alpha \in \mathbb{F}_{q^{3}}$ such that $Tr(\alpha^{\frac{q^{2}+q-2}{9}})$ is a cube root of $c$. Consequently we find an efficient cube root algorithm based on third order linear recurrence sequence arising from $f(x)$. Complexity estimation shows that our algorithm is better than previously proposed Cipolla-Lehmer type algorithms

Forecasting Warping Deformation Using Multivariate Thermal Time Series and K-Nearest Neighbors in Fused Deposition Modeling

Over the past decades, additive manufacturing has rapidly advanced due to its advantages in enabling diverse material usage and complex design production. Nevertheless, the technology has limitations in terms of quality, as printed products are sometimes different from their desired designs or are inconsistent due to defects. Warping deformation, a defect involving layer shrinkage induced by the thermal residual stress generated during manufacturing processes, is a major factor in lowering the quality and raising the cost of printed products. This study utilized a variety of thermal time series data and the K-nearest neighbors (KNN) algorithm with dynamic time warping (DTW) to detect and predict the warping deformation in the printed parts using fused deposition modeling (FDM) printers. Multivariate thermal time series data extracted from thermocouples were trained using DTW-based KNN to classify warping deformation. The results showed that the proposed approach can predict warping deformation with an accuracy of over 80% by only using thermal time series data corresponding to 20% of the whole printing process. Additionally, the classification accuracy exhibited the promising potential of the proposed approach in warping prediction and in actual manufacturing processes, so the additional time and cost resulting from defective processes can be reduced