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The basic functionality unit that to executes modular arithmetic in different solving codes and uses a Pseudo-random bit generator (PRBG) algorithm. The Carry-save-adder (CS3A) is the most used a technique to execute the three-operand-adder. Whenever a high propagated delay O(n) occurs in the Ripple-carry stage of CS3A. In addition, Han-carlson adder is undergoing a parallel-prefix two-operand-adder. It can be used for three-operand-additions that reduce the criticalpath delay at the cost of an additional hardware. To implement the proposed architecture using Xilinx tool for a simulation and synthesis. The proposed adder post-synthesis results were observed 3.12, 5.31 and 9.28 times rapid than the CS3A for 32-bits, 64-bits, and 128-bits architectures, respectively. In addition, compared to CS3A, HC3A has a lower power consumption, a smaller delays and a less area. The lowest PDP and ADP in the proposed adder than in the current three-operand adder methods. [ABSTRACT FROM AUTHOR]

In most studies concerning Galois fields the class of self-complementary basis have found considerable interest due to applications in coding theory and cryptology. Moreover they are useful in the construction of devices for the arithmetic in finite fields such as multiplication, exponentiation, discrete logarithms. Among the class of self-complementary bases the abundance of self-dual bases of finite fields is less as compared to that of almost self-dual bases. Almost self-dual bases exist for almost all extensions of finite fields. Thus, to study the intrinsic relationship of a self-dual basis and an almost self-dual basis of 픽pn, present work presents construction of almost self-dual basis from a self-dual basis of 픽pn with odd characteristics along with supporting demonstrations. [ABSTRACT FROM AUTHOR]

Residue Multiplication operations are extensively used in Residue Number System (RNS) based cryptosystem architecture. Pointing to increase the speed performance of RNS crypto processors, the new parallel unsigned 2n+1 residue multiplier is designed in this work. Mathematical model, Algorithm, Architecture and FPGA implementation is done in this work. The proposed residue multiplier is described in Verilog HDL and synthesized in Application-Specific Integrated Circuits (ASIC) environment. Cadence RTL Compiler estimates the Area, Power and Delay performance parameters using various CMOS libraries. The proposed 2n+1 residue multiplication scheme saves 13% of the area, improves the speed by 19% and PDP by 23% compared to the recent 2n+1 residue multipliers. [ABSTRACT FROM AUTHOR]

Cadets majoring in Cybersecurity at the United States Coast Guard Academy are typically enrolled in the department's Introductory Cryptography course during the Fall of their Junior year. Many of these students struggled to understand the need to use modular operations to solve quadratic equations modulo N. This was because homework assignments requiring the solution of these equations typically employed equations for which regular real operations were perfectly satisfactory. Quadratic equations modulo N are assigned to help the students recognize the similarities between modular and real arithmetic while also helping them see the differences. These equations are also introduced to prepare the students for the study of elliptic curves. Three specific differences between modular quadratic equations and real arithmetic quadratic equations were identified. Modular arithmetic equations can fail to have any solutions if the modulus for an otherwise acceptable equation does not permit a multiplicative inverse required for solution. Modular quadratic equations can also have solutions where real quadratic equations would fail to have any solutions (complex solutions). And finally, modular quadratic equations with composite moduli can have many more than two solutions. All three of these differences can be exploited to prevent the students from solving a quadratic equation using real arithmetic methods and thus compel them to use modular techniques. It was observed on the homework assignments that a significant fraction of the students failed to use modular techniques to solve the quadratic equations that were assigned and it was suspected that many of those that did were unsure of why they needed to employ them. After constructing quadratic equations that took advantage of the peculiar properties of modular arithmetic the students were unable to use real arithmetic techniques and had to resort to modular methods thus reinforcing the need to use those methods to solve the modular equations. [ABSTRACT FROM AUTHOR]

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Hurwitz groups are the nontrivial finite quotients of the (2, 3, 7) triangle group of 7). In this paper we study the action of the smallest Hurwitz group is the simple group PS L(2, 7) acts via linear fractional transformation on the projective line P¹(7) over the field by using directed suborbital graphs. We obtain edge and circuit conditions. [ABSTRACT FROM AUTHOR]

Modular multiplication finds the major role in RSA Cryptography and Elliptical Curve Cryptography. Timing analysis measures the delay along the various timing paths and verifies the performance and operation of the design. We have implemented a 256-bit Modular multiplier using Montgomery Reduction Algorithm in VHDL. The output of the Montgomery multiplier is Z=X∗Y R−1 mod M. Both RSA key generation component and data encryption component are too big to fit into a single Altera Cyclone II Device on Field Programmable platform, so that we are unable to test them in real hardware. However, each sub-component was simulated in Model-Sim SE 6.0 and Altera Quartus II 8.0 and proved functionally correct. [ABSTRACT FROM PUBLISHER]

In the last years, many modulus sets in the Residue Number System (RNS) for increasing Dynamic Range (DR) are presented. Hence, for achieve this goal a new 4-Moduli Set22n+1, 22n + l, 2n + 1, 2n−1, and its efficient reverse converter design are introduced in this paper. This modulus set contains pair-wise relatively prime, and it offers the maximum feasible DR. New CRT-I is used to obtain a high-performance memory-less reverse converter for this modulus set. Also our modulus set can speed up internal RNS arithmetic processing, compared to the modulus set 2n − 1, 2n + 1, 22n − 2, 22n+1 −3. Finally, we review improving the modular multiplication with this modulus set, and its dynamic range compared with other modulus sets. [ABSTRACT FROM PUBLISHER]

This paper describes a modification to the conventional Hill Cipher system. The purpose of this paper is to explore the adeptness of Hill Cipher for binary data. The plaintext is any binary data such as images, audio, video, etc. The plaintext is subjected to scrambling by dividing it into 8 planes. Each of these planes is encrypted using a different key. From the study conducted on Bitwise Hill Cipher, it is found that it has enough security required for commercial applications. [ABSTRACT FROM PUBLISHER]

Education is increasingly important nowadays. Researches in education are increasing year by year. The enhancement of the researches is the important aspect in the development process of new educational methods and techniques. Nowadays, the information society in our region needs wider educational spaces supported by the internet connection that provides a good tool to make use of educational technologies and online applications. Indeed, interactive tools give us privileges to have access to a huge amount of information in a multi-sequential way by using specifically suitable tools for the learning specific forms of knowledge. These interactive tools have many advantages for learning Mathematics. In our case, we present an interactive tool to show the main applications of Modular Arithmetic in a brief theoretical environment. [ABSTRACT FROM PUBLISHER]

This paper presents two designs of residue-to-binary converters, namely reverse converters. These designs are based on two well known algorithms for reverse conversion; Mixed-radix conversion (MRC) and the new Chinese remainder theorem 1 (new CRT-I). A comparison is done in order to highlight the differences between the two algorithms when implemented on field programmable gate array FPGA, in terms of time and area consumption. Both converters are dedicated for the moduli set (2n-1, 2n, 2n+1). [ABSTRACT FROM AUTHOR]

To ensure the security to the applications of business, the business sectors use Public Key Cryptographic Systems (PKCS). An RSA system generally belongs to the category of PKCS for both encryption and authentication. This paper describes an introduction to RSA through encryption and decryption schemes, mathematical background which includes theorems to combine modular equations and correctness of RSA. In short, this paper explains some of the maths concepts that RSA is based on, and then provides a complete proof that RSA works correctly. We can proof the correctness of RSA through combined process of encryption and decryption based on the Chinese Remainder Theorem (CRT) and Euler theorem. However, there is no mathematical proof that RSA is secure, everyone takes that on trust!. [ABSTRACT FROM AUTHOR]

An efficient construction of lattice points FM of any density and any admissible symmetry is described in a finite region F⊃FM of a real n-dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n underlying the construction. The density of FM is fixed by our choice of a positive integer M, where 1<=M<∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM together with the possibility of expanding digital data on FM into finite series of such special functions. [ABSTRACT FROM AUTHOR]

We introduce the modular class of a twisted Jacobi structure on a Lie algebroid and establish a relation between this modular class and the modular class of the underlying Lie algebroid. [ABSTRACT FROM AUTHOR]