Introduction 2 1. Preliminaries on conformal algebras and modules 4 2. Basic definitions 9 3. Extensions and deformations 13 4. Homology 17 5. Exterior multiplication, contraction, and module structure 18 6. Cohomology of conformal algebras and their annihilation Lie algebras 19 6.1. Cohomology of the basic complex 19 6.2. Cohomology of the reduced complex 21 6.3. Cohomology of conformal algebr...

Securing the data stored on E-passport is a very important issue. RSA encryption algorithm is suitable for such application with low data size. In this paper the design and implementation of 1024 bit-key RSA encryption and decryption module on an FPGA is presented. The module is verified through comparing the result with that obtained from MATLAB tools. The design runs at a frequency of 36.3 MH...

Galois/Counter Mode of Operation (GCM) is a block cipher mode operation used to provide encryption and authentication using universal Hashing based on multiplication over binary Galois/Finite Field.GCM can be implemented on both hardware and software effectively and efficiently. GCM supports pipelined and parallelized implementations to have minimal computational latency in order to be useful a...

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R (M), the idealization of M . Homogeneous ideals of R (M) have the form I (+)N where I is an ideal of R, N a submodule of M and IM ⊆ N . The purpose of this paper is to investigate how properties of a homogeneous ideal I (+)N of R (M) are related to those of I and N . We show that if M is a m...

We investigate interconnected aspects of hyperderivatives polynomials over finite fields, q -th powers polynomials, and specializations Vandermonde matrices. construct formulas for Carlitz multiplication coefficients using symmetric we prove identities hyperderivative power sums in terms the inverse matrix. As an application these results give a new proof theorem Thakur on explicit Anderson's s...

A Q-curve is an elliptic curve defined over Q that is isogenous to all its Galois conjugates. The term Q-curve was first used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication, the c...

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calder\'on-Zygmund operators, characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes mixed-norm integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a bi-parameter version recent approximate weak factorization method.

Finally, we proved that the hardness of factoring implies that the inverse function (that is to say the multiplication operation or fmult(x, y)) is a weak one way function. This class will combine these definitions together. Particularly, today’s lesson will focus on proving an important theorem which states that if we are able to build a weak one-way function, then we are able to build a stron...

Let T be a Noetherian ring and f a nonzerodivisor on T . We study concrete necessary and sufficient conditions for a module over R = T/(f) to be weakly liftable to T , in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftab...

This research is motivated by the need to provide teaching materials in form of electronic-based modules (e-modules) integrated with numeracy literacy learning. Where math books are less able delivery related literacy. Given lack a learning process during covid 19 period improve students' abilities field numeracy. The objective develop an e-module for multiplication and division fractions grade...