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Differential analysis is an important cryptanalytic technique on block ciphers. In one form, this measures the probability of occurrence of the differences between certain input vectors and the corresponding output vectors. For this analysis, the constituent S-boxes of a block cipher need to be studied carefully. In this direction, we derive further cryptographic properties of the multiplicative inverse function, especially the ones related to higher order differentials. This improves some theoretical results of Boukerrou et al. [ToSC 2020(1)]. Further, we prove that the multiplicative inverse function defined over F 2 n has an error (bias) in its second order differential spectrum with probability 1 2 n − 2 , and that error occurs in more than one place. Next, we analyze the Gowers uniformity norm of S-boxes, which is also a measure connected to higher order approximations. Finally, the bounds related to the nonlinearity profile of the multiplicative inverse function are derived using both Gowers U 3 norm and Walsh–Hadamard spectrum. Some of our findings here provide slightly improved bounds over the work of Carlet [IEEE-IT, 2008]. These theoretical insights might have implications towards non-randomness of a block cipher, where the multiplicative inverse function is used as a primitive.

Aiming at the security and efficiency problems in the process of image transmission, an image compression–encryption scheme based on 2D compressive sensing and hyperchaotic system is proposed in this paper. First, we construct a hyperchaotic system with more complex chaotic behavior, which is used to construct the measurement matrix of compressive sensing. Then, two-dimensional compressive sensing is used to compress the image. Compared with one-dimensional sensing, it achieves faster execution efficiency and better image reconstruction quality. Finally, to improve the encryption security, we use the multiplicative inverse operation in the finite domain to diffuse the cipher image after compressive sensing. The experimental simulation results show that the algorithm in this paper has higher execution efficiency, better image reconstruction quality, great security and robustness.

In this paper, authors investigate the stability results of multiplicative inverse quartic functional (MIQF) equation associating a constant, sum of powers of norms, general control function, mixed...

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

A classical problem of D. H. Lehmer suggests the study of distributions of elements of Z/pZ of opposite parity to the multiplicative inverse mod p. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved, from which one may see that Zhang’s error term is optimal up to the logarithm factor. The method also applies to the case of arbitrary positive integral moments.

For the numerical solution of the Cauchy problem with multiple poles, we propose a reciprocal function method. In the case of first-order poles, it makes it possible to continue the solution through the poles and to determine the solution and the pole positions with good accuracy. The method allows one to employ conventional explicit and implicit schemes, for example, explicit Runge–Kutta schemes. A test problem with multiple poles is computed as an example. The proposed method is useful for construction of software for direct computation of special functions.

The aim of this study is to introduce a new multiplicative inverse cubic functional equation, to accomplish its general solution, to employ Hyers’ method for solving its stability problems in Felbin’s type fuzzy normed linear spaces, to present an apt example for justifying its stability result is invalid for singular case and to elucidate its interpretation through an application in electromagnetism.

We obtain a new version of the Hardy theorem about power series reciprocal to the power series with positive coefficients in this work. We prove that, if the sequence {an}, n ≥ K is logarithmically convex, then the reciprocal power series has only negative coefficients bn, n > 0 for any K if the first coefficient a0 is sufficiently large. The classical Hardy theorem corresponds to the case K = 0. These results are useful in the Nevanlinna–Pick theory. For instance, if the function k(x, y) can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{a}_{n}}{{{(x\bar {y})}}^{n}}} $$, an > 0, and the reciprocal function $$\frac{1}{{k(x,y)}}$$ can be represented as the power series $$\sum\nolimits_{n \geqslant 0} {{{b}_{n}}{{{(x\bar {y})}}^{n}}} $$ such that bn 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc $$\mathbb{D}$$ with the Nevanlinna–Pick property. The reproducing kernel $$\frac{1}{{1 - x\bar {y}}}$$ of the classical Hardy space H2($$\mathbb{D}$$) is a prime example for our theorems. We also obtained new estimates on the growth of analytic functions reciprocal to analytic functions with positive Taylor coefficients.

Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ F n , for every positive integer n not divisible by 3, where $$F_n$$ F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo $$F_n$$ F n , for every fixed integer $$a \ge 3$$ a ≥ 3 and for all positive integers n with $$\gcd (a, F_n) = 1$$ gcd ( a , F n ) = 1 . Our proof makes use of the so-called base-$$\varphi $$ φ expansion of real numbers.

In this paper, we consider the congruence equation [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the multiplicative inverse of [Formula: see text]. We prove that its number of solutions is asymptotic to [Formula: see text] when [Formula: see text] by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when [Formula: see text], the number of solutions has order of magnitude at least [Formula: see text]. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet [Formula: see text]-functions at [Formula: see text].

The paper deals with a new type of intuitionistic fuzzy number, in the complex plane called as complex horizontal relative trapezoidal intuitionistic fuzzy number (CHRTrIFN). Arithmetic operations like addition, multiplicative inverse and division on these numbers are defined and discussed. Complex horizontal relative trapezoidal intuitionistic fuzzy number(CHRTrIFN) and geometric representation of CHRTrIFN and complex trapezoidal intuitionistic fuzzy number(CTrIFN) are presented. CHRTrIFNs are used to solve complex intuitionistic fuzzy linear system of equations and is applied in a RLC intuitionistic fuzzy circuit to find the current flow in the circuit.

Modular inversion, the multiplicative inverse of an integer in the ring of integers modulo a prime number, is widely used in public-key cryptography. However, it is one of the most computationally intensive operations, thus, it remains the main performance bottleneck for many cryptographic algorithms.This paper presents to the best of the author’s knowledge, the first FPGA-based hardware design for computing the multiplicative inverse using the recently proposed fast constant-time Greatest Common Divisor (GCD) algorithm. This paper introduces two distinct design architectures targeting different applications: (a) a full-width design and (b) a sequential design. The presented designs are compact, parameterizable, and scalable in terms of area and speed. The evaluation shows the proposed designs, which are constant-time and protect against timing-based attacks, outperform existing software and hardware implementations that use other modular inversion techniques. As a specific example, this work presents an evaluation focusing on the use of the multiplicative inverse hardware module to accelerate the ElGamal cryptosystem. The proposed design achieves a speed-up of 90% in the modular inverse calculation and a speed-up of 45% in the overall ElGamal decryption algorithm using our sequential hardware design of fast constant-time GCD algorithm.In addition to developing the fast hardware implementation, this work potentially opens up a new direction for designing cryptosystems: the inverse operation is often avoided when designing algorithms, due to its complexity. With the new hardware module, using the inverse becomes more tractable, making it more appealing to use in the design of new cryptosystems.

A quantum search algorithm with phase shift of π/18 is proposed to find a multiplicative inverse of an equation in an unsorted database. The unitary operator $$Q{\text{ }}$$ ( $$Q{\text{ = }}I_{t} U^{{ - 1}} I_{0} U$$ ) is operated repeatedly on the initial state with $$\left\lfloor {9\sqrt {{N \mathord{\left/ {\vphantom {N M}} \right. \kern-\nulldelimiterspace} M}} } \right\rfloor$$ times to find a marked item successfully. At least 99.81% of the probability of success could be achieved by this proposed quantum search algorithm with time complexity $$O\left( {\sqrt {{N \mathord{\left/ {\vphantom {N M}} \right. \kern-\nulldelimiterspace} M}} } \right)$$ . Compared with four typical algorithms, the proposed search algorithm has better performance to find out a match in terms of success probability with the same time complexity $$O\left( {\sqrt {{N \mathord{\left/ {\vphantom {N M}} \right. \kern-\nulldelimiterspace} M}} } \right)$$ .

In literature, there are a number of cryptographic algorithms (RSA, ElGamal, NTRU, etc.) that require multiple computations of modulo multiplicative inverses. In this paper, we describe the modulo operation and we recollect the main approaches to computing the modulus. Then, given a and n positive integers, we present the sequence (zj)j≥0, where zj=zj−1+aβj−n, a&lt, n and GCD(a,n)=1. Regarding the above sequence, we show that it is bounded and admits a simple explicit, periodic solution. The main result is that the inverse of a modulo n is given by a−1=⌊im⌋+1 with m=n/a. The computational cost of such an index i is O(a), which is less than O(nlnn) of the Euler’s phi function. Furthermore, we suggest an algorithm for the computation of a−1 using plain multiplications instead of modular multiplications. The latter, still, has complexity O(a) versus complexity O(n) (naive algorithm) or complexity O(lnn) (extended Euclidean algorithm). Therefore, the above procedure is more convenient when a&lt, &lt, n (e.g., a&lt, lnn).

The recent boom of decision-making under uncertain information has attracted many researchers to the field of integrating various types of sets with decision-making methods. In this paper, a combined decision-making trial and evaluation laboratory (DEMATEL) with single-valued neutrosophic sets is proposed to solve the decision problem. This new model combines the advantages of multiplicative inverse of decision matrix in DEMATEL and neutrosophic numbers in linguistic variables, which can find the interrelationship among factors of decision problem. Differently from the typical multiplicative inverse of DEMATEL, which directly used inverse of matrix using real numbers, this method introduces the concept of inverse of matrix using the proposed left–right neutrosophic numbers. This step will enhance the validity of multiplicative inverse of decision matrix in the DEMATEL with neutrosophic numbers. The proposed neutrosophic DEMATEL is also be compared with the DEMATEL and fuzzy DEMATEL. This paper includes a case study that demonstrates the applicability of the neutrosophic DEMATEL in establishing the relationship among influential factors of coastal erosion. Extensive empirical studies using 12 factors of coastal erosion were presented to study the benefits of the proposed method. The result unveils the cause-and-effect relationships among the factors, where seven factors are identified as cause factors and five factors are grouped as effect factors. It is discovered that the factor ‘imbalance sediment supply’ gives a significant influence to coastal erosion. It is also shown that the degree of importance of the factors is almost consistent with the other two methods despite differences in type of numbers used in defining linguistic variables.

The analysis of stability results of different multiplicative inverse functional equations is an emerging trend in the research field of fundamental stability of equations emanated through the quer...

The multiplicative inverse over binary fields is one of main arithmetic operations used in cryptography. This paper presents two new inversion architectures. First, an improved architecture for classic inversion scheme using single multiplier is presented. The new improved inverter achieves lower latency through loading input registers during last multiplication cycle, at the expense of higher propagation delay. After this, a novel inversion architecture which uses half the latency to process the classic-based addition chains (or improved ones) is presented. The latter architecture, named Classical-Interleaved, is constructed based on a novel fully-serial-in square-multiply processor (FSISM). The FSISM, squares one operand, and multiply it to the second one, concurrently while the two inputs are absorbed serially digit-by-digit. The new classical inverter and the new classical-interleaved inverter reduce the latency compared to other schemes. In addition, the proposed classical and interleaved inverters outperform the original Itoh-Tsujii algorithm (ITA) and Ternary Itoh-Tsujii / optimal 3-chain algorithms in terms of its higher throughput and improved hardware efficiency for a number of digit sizes. The efficiency of the proposed field inverters are demonstrated by comparisons based on application specific integrated circuits (ASIC) implementations results using the standard 65 nm CMOS technology libraries.

This study reveals an organised algebraic view on the Advanced Encryption Standard (AES), and in particular the S-box. A clear visualisation of algebraic expressions is granted for both the cipher part and the key schedule part of AES. Additionally, by introducing the combined representation of the S-box, alternative methods can be constructed to serve algebraic cryptanalysis on AES. The process to obtain a combined representation of the S-box, as a combination of four bits, the input value's multiplicative inverse and bytes in hexadecimal notion, may be of high importance for other proposed ciphers too built with layers of S-boxes.

This paper presents how the steps that used to determine a multiplicative inverse by method based on the Euclidean algorithm, can be used to find a greatest common divisor for polynomials in the Galois field (2^n ).

We identify two faults in a published algorithm for fast computation of multiplicative inverses modulo prime powers. We patch the algorithm and present machine-assisted proofs of correctness of the repair. Our formal proofs also reveal that being prime is an unnecessary demand for the power base, thus attributing a wider scope of applications to the repaired algorithm.

This paper gives some techniques to compute the set of multiplicative inverses, which uses in the Advanced Encryption Standard (AES).

This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties of Q , the system Q of rational numbers is constructed and its properties verified (being those of Q , plus the existence of additive inverse). Taking the basic operative properties of N for the axioms, we define the algebraic structure {S, , ×

The substitution table (S-Box) of Advanced Encryption Standard (AES) and its properties are key elements in cryptanalysis ciphering. We aim here to propose a straightforward method for the non-linear transformation of AES S-Box construction. The method reduces the steps needed to compute the multiplicative inverse, and computes the matrices multiplication used in this transformation, without a need to use the characteristic matrix, and the result is a modern method constructing the S-Box.

This paper makes two major contributions. First, we propose an algorithm to compute a multiplicative inverse modulo $2^{k}$ without using integer division by refining the Arazi–Qi algorithm. We then generalize this algorithm into a scalable algorithm to compute an inverse modulo $2^{kl}$ by iteratively exploiting a $k$ -bit-based process when the inverse modulo $2^{k}$ is given. Because of its scalability, this generalized algorithm can be cost-effectively implemented on hardware and applied to larger modulus operations by iteratively applying a processing unit. Furthermore, we highlight the possibility of the derivation of a binary representation of an inverse $P^{-1}$ in terms of a representation of $P$ and demonstrate this operation for up to eight bits. Second, using this direct relationship, we propose a multi-bit Montgomery inverse algorithm that is at least three times faster than the original version. Finally, we derive various properties of this algorithm and compare it with previous algorithms. This fast calculation of a Montgomery inverse is important for public key cryptographic applications, such as elliptic curve cryptosystems, because the relative speed of calculating a modular inverse for modular multiplication is a critical parameter for deciding which coordinate systems to adopt.

The first step in converting a plaintext to ciphertext by the famous Advanced Encryption Standard (AES), which is called Rijndael ByteSub Transformation, involves some operations: computing a multiplicative inverse, multiplying this multiplicative inverse by a specific matrix, and adding the result to a specific vector. The purpose of this research is to simplify these operations. This paper gives elegant techniques and presents the matrices multiplication as simple XOR operations, and the result is a simple, straightforward way finding the transformation.

This paper describes an attack on the Rivest, Shamir and Adleman (RSA) cryptosystem utilizing the modulus N=p2q where p and q are two large balanced primes. Let e1,e2&lt, Nγ be the integers such that d1,d2&lt, Nδ be their multiplicative inverses. Based on the two key equations e1d1−k1ϕ(N)=1 and e2d2−k2ϕ(N)=1 where ϕ(N)=p(p−1)(q−1), our attack works when the primes share a known amount of least significant bits (LSBs) and the private exponents share an amount of most significant bits (MSBs). We apply the extended strategy of Jochemsz–May to find the small roots of an integer polynomial and show that N can be factored if δ&lt, 1110+94α−12β−12γ−130180γ+990α−180β+64. Our attack improves the bounds of some previously proposed attacks that makes the RSA variant vulnerable.

In this investigation, various approximate multiplicative inverse Lie \(\star \)-derivations are determined in the framework of normed \(\star \)-algebras pertinent to Ulam–Hyers stability theory. These results are applied to achieve other classical stabilities by taking different upper bounds. The results obtained are compared with concluding remarks.

In [9], the notion of c-differentials was introduced as a potential expansion of differential cryptanalysis against block ciphers utilizing substitution boxes. Drawing inspiration from the technique of higher order differential cryptanalysis, in this paper we propose the notion of higher order c-derivatives and differentials and investigate their properties. Additionally, we consider how several classes of functions, namely the multiplicative inverse function and the Gold function, perform under higher order c-differential uniformity.

Summary Probability forecasts for binary events play a central role in many applications. Their quality is commonly assessed with proper scoring rules, which assign forecasts numerical scores such that a correct forecast achieves a minimal expected score. In this paper, we construct e-values for testing the statistical significance of score differences of competing forecasts in sequential settings. E-values have been proposed as an alternative to $p$-values for hypothesis testing, and they can easily be transformed into conservative $p$-values by taking the multiplicative inverse. The e-values proposed in this article are valid in finite samples without any assumptions on the data-generating processes. They also allow optional stopping, so a forecast user may decide to interrupt evaluation, taking into account the available data at any time, and still draw statistically valid inference, which is generally not true for classical $p$-value-based tests. In a case study on post-processing of precipitation forecasts, state-of-the-art forecast dominance tests and e-values lead to the same conclusions.

Advanced Encryption Standard (AES) algorithm is one of the most widely used symmetric block cipher that is utilized in data protection through symmetric cryptography algorithm, as it offers high efficiency and ability in securing information. In AES, the SubByte/InvSubByte transformation costs an amount of memories for the lookup tables (LUT) that leads to area usage in hardware. As per mathematical equations, the SubByte/InvSubByte is able to share the same resource which is the multiplicative inverse value table. Instead of using LUTs, the multiplicative inverse values can be generated on-the-fly as needed. Therefore, this paper presents the custom hardware design and implementation of multiplicative inverse value generator using logarithm method specifically for AES algorithm. The logarithm method benefits the usage of antilog and log values in obtaining the multiplicative inverse value. The EDA tools as Intel Quartus Prime, Synopsys Design Compiler and IC Compiler are used to perform functional simulation, on synthesis analysis and block-level implementation. Detailed comparison is made between this work and previous implementation. The newly designed multiplicative inverse value generator has achieved the improvement on speed and area as compared to previous implementation. The area of the design is reduced around 8.5% with current improvement of 63.091 μm2, while the speed has increased to 2.54 ns with a positive slack of 0.11 ns, which is shorter than the previous implementation. The new design also outreached the performance of the common LUT-based design.

A multiplicative inverse (MI) algorithm is used in several fields, like cryptography algorithms. There are many MIs, however, these algorithms suffer from using several multiplication and division operations, which take more execution time than additions and subtractions. We created a new algorithm called Clockwise-Displacement (CDMI) by using addition and subtraction operations in the iterative steps instead of multiplications and divisions. Additionally, numerous MIs face the undecidable problem because of the floating-point issue. Whereas, CDMI tackles this issue by converting the domain from the floatingpoint space to integer space. Therefore, CDMI declines the time consuming to calculate the multiplicative inverse by applying fewer divisions and multiplications (expensive operations) and addresses the rounding error issue in some MI Algorithms.

We present a simple criterion to determine if the graph of a rational function of the form $ f(x)=\frac{ax+b}{cx+d} $ includes a reflection of the reciprocal function $ r(x)=\frac{1}{x} $ over the $ x $-axis. In addition, we determine an algebraic expression for the factor by which the graph of $ y=r(x) $ is stretched vertically to produce the graph of $ y=f(x) $.

Advanced Encryption Standard (AES) Algorithm has been extensively applied in the present financial applications. Sub-channel attacks are one of the main problems occurred n the AES Algorithm. Asynchronous AES Architecture is one of the leading solutions of the sub-channel attacks due to its natural properties. The AES architecture with the enhanced mix column to be proposed with reduced number of transistor counts.. Then, the Verilog A modeling is used to evaluate the performance of the proposed AES Architecture. Finally, the VLSI Implementations of the AES Processor is implemented with CMOS technology 0.25 µm. By using the net list generations, the proposed AES Architecture is analyzed regarding the VLSI design environment. The simulation results of the proposed structure are performed with the minimum number of transistor counts as well as power utilizations. Moreover, the proposed CMOS technology based AES Algorithm is integrated into the backend based chip technology. &nbsp

The Advanced Encryption Standard (AES) is the most popular algorithm used in symmetric key cryptography. The efficient computation of AES is essential for many computing platforms. The S-box is the only nonlinear transformation step of the AES algorithm. Efficient implementation of the AES S-box is very crucial for AES hardware. The AES S-box could be implemented by using look-up table method or by using finite field arithmetic. The finite field arithmetic design approach to implement the AES S-box is superior in saving the hardware resources. The main objective of this thesis is to evaluate, compare and improve the hardware implementations of the forward, inverse and combined AES S-box in terms of area and/or delay. Both the composite field GF((2^4)^2) and the tower field GF(((2^2)^2)^2) are considered. Our first improvement is the optimization of the input and output linear mappings of the S- box in order to design a more compact circuit. Our second improvement aims at modifying the architecture of the S-box to achieve a higher speed. We used multiplication of the S-box input by an 8-bit binary field element to optimize the input and output transformation matrices of the S-box. A Matlab® search is then conducted to find more compact linear mappings for the S-box. We also modified the fast S-box architecture, in addition to optimizing and searching the extended linear input mappings to improve the speed of Reyhani et al. fast S-box. The improved fast S-box, Fast 3, is the fastest and most efficient (measured by area × delay) AES S-box available in the literature, up to our knowledge. We also improved the area and delay of the inversion circuit of the lightweight and fast S-boxes in [69], by slightly modifying the exponentiation block and designing a new subfield inverter block. The improved inversion circuit leads to a more compact and a faster lightweight S-box and it yields a lower area fast S-box. Moreover, we show that the “tech. XORs” concept proposed by Maximov et al. [54] to estimate the delay of the S-box is not accurate. We show how to use the logical effort method [74] instead to estimate and compare the delay of previous and improved S-boxes, regardless of the CMOS technology library used for the implementation. We verified all the codes at the RTL level using Mentor Graphics Modelsim®, by comparing against the legitimate S-box outputs. We synthesized the designs using STM 65nm CMOS standard cell library and we used VHDL coding as the design entry method to Synopsys Design Compiler®. The synthesis results confirm the lower area and / or delay of the improved S-box designs and match our space and timing analyses.

While most software automation research concentrates on programs' code, we have started investigating if Genetic Improvement (GI) of data can assist developers by automating aspects of the maintenance of parameters embedded in source code. We extend recent GI work on optimising compile time constants to give new functionality and describe the transformation of a GNU C library square root function into the double precision reciprocal function, drcp. Multiplying by 1/x (drcp) allows division free division without requiring the hardware to support division. The evolution (6 seconds) and indeed the GI dp division (7.14 ± 0.012 nS) are both surprisingly fast.

The aim of this present article is to investigate various classical stability results of the multiplicative inverse difference and adjoint functional equations $$ m_{d} \biggl(\frac{rs}{r+s} \biggr)-m_{d} \biggl( \frac{2rs}{r+s} \biggr)= \frac{1}{2} \bigl[m_{d}(r)+m_{d}(s) \bigr] $$md(rsr+s)−md(2rsr+s)=12[md(r)+md(s)] and $$ m_{a} \biggl(\frac{rs}{r+s} \biggr)+m_{a} \biggl( \frac{2rs}{r+s} \biggr)= \frac{3}{2} \bigl[m_{a}(r)+m_{a}(s) \bigr] $$ma(rsr+s)+ma(2rsr+s)=32[ma(r)+ma(s)] in the framework of non-zero real numbers. A proper counter-example is illustrated to prove the failure of the stability results for control cases. The relevance of these functional equations in optics is also discussed.

In this paper, we study the four-dimensional real algebra of bihyperbolic numbers. Under consideration of the spectral representation of the bihyperbolic numbers, we give a partial order of bihyperbolic numbers which allows us to obtain some relations in the ordered vector space of bihyperbolic numbers. Moreover, we state that the set of bihyperbolic numbers form a real Banach algebra with a new defined norm. We introduce conjugates, three hyperbolic valued moduli, real moduli, and multiplicative inverse of the bihyperbolic numbers. We give the concept of the absolute value of a bihyperbolic number which generalizes that of real numbers. Also, we represent the polar form of invertible bihyperbolic numbers.

In this chapter, an inexact solution near to the exact solution of a multiplicative inverse trevigintic and quottuorvigintic functional equations are achieved in the sense of Ulam stability hypothesis in matrix normed spaces. Proper examples are also illustrated to prove the instabilities for control cases.

In this chapter, the generalized Hyers-Ulam stability of multiplicative inverse unvigintic and duovigintic functional equations in paranormed spaces using direct and fixed point methods are presented. Counter-examples to invalidate the stability results for critical cases are also discussed.

This chapter is devoted to study various classical stability results of multiplicative inverse novemdecic and vigintic functional equations in intuitionistic fuzzy normed spaces and also counter-examples to disprove the validity of stability results for singular cases.

This chapter deals with the investigation of validity of various fundamental stabilities of multiplicative inverse type tredecic and quottuordecic functional equations relevant to Ulam stability theory in non-Archimedean fields via fixed point method. Two suitable counter-examples are also included to prove that the stability results are not valid for singular cases.

This chapter contains the classical investigation of various fundamental stability results of multiplicative inverse septendecic and octadecic functional equations in quasi-\(\beta \)-normed spaces using fixed point technique and also includes two proper examples to disprove stability results for control cases.

Designing S-box with good cryptographic properties still remains as one of the most important areas of research in symmetric key cryptography. For quite sometime, inverse function (\(x\mapsto x^{-1}\), i.e., \(x^{2^n-2}\)) over \(\mathbb {F}_{2^n}\) has been the most popular choice for S-boxes due to good resistance against differential and linear cryptanalysis. Very recently Tang et. al. (2020) proved that inverse function admits a bias (error) of \(\frac{1}{2^{n-2}}\) when considered in its second-order differential spectrum. In this paper we present experimental results related to higher-order differential spectrum of multiplicative inverse functions for \(n=6\) and 8 and compare the result with APN permutation for \(n=6\). In particular, we observe that APN permutation over \(\mathbb {F}_{2^6}\) has larger bias in its second-order differential spectrum with probability \(\frac{1}{8}\) ( \(\frac{1}{2^{n-2}}\)). This fact admits the possibility of higher-order differential attacks against block ciphers which employ APN permutations as a nonlinear layer.

This chapter is devoted to demonstrate the validation of various stabilities of multiplicative inverse quindecic and multiplicative inverse sexdecic functional equations via fixed point technique in the framework of Felbin’s type fuzzy normed spaces. Proper illustrations are presented to disprove the stability results for singular cases.

This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.