This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to " realize " a matrix product via the regular algebra of a (fin...

An algebra is a vector space (over C) with a multiplication such that A is a ring with identity, i.e. there is a map A × A → A, (a, b) 7→ ab, which is bilinear and satisfies the associative and distributive laws. The following are examples of algebras: (1) The group algebra of a group G is the vector space CG with basis G and with multiplication forced by the multiplication in G (and the biline...

We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak∗-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals C0...

For a large integer m, we obtain an asymptotic formula for the number of solutions of a certain congruence modulo m with four variables, where the variables belong to special sets of residue classes modulo m. This formula are applied to obtain a new bound for a double trigonometric sum with an exponential function and new information on the exceptional set of the multiplication table problem in...

This paper presents the comparison and optimization of Galois field algorithm and IDEA(International data encrypted algorithm) using VHDL(xc2v40-6cs144)with the help of Xilinx-Ise (9.2) .In terms of security IDEA algorithm is very much superior where as in terms of timings Galois field is much better and is already patented of Ascom. These are cryptographic algorithm used for security purpose. ...

This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to " realize " a matrix product via the regular algebra of a (fin...

We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Schützenberger graphs; having a context-free multiplication table; or having word hyperbolic maximal subgroups. Our main resul...

A new elliptic curve scalar multiplication algorithm is proposed. The algorithm o ers about twice the troughput of some conventional OEF-base algorithms because it combines the Frobenius map with the table reference method based on baseexpansion. Furthermore, since this algorithm suits conventional computational units such as 16, 32 and 64 bits, its base eld F p m is expected to enhance ellipti...

For n > 3, every n×n partial Cayley matrix with at most n−1 holes can be reconstructed by quadrangle criterion. Moreover, the holes can be filled in given order. Without additional assumptions, this is the best possible result. Reconstruction of other types of multiplication tables is discussed.

In abstract algebra, a field is a structure with total operations of addition, subtraction and multiplication. Moreover, every element has a multiplicative inverse—except 0. In a field, the rules hold which are familiar from the arithmetic of ordinary numbers. The prototypical example is the field of rational numbers. Fields can be specified by the axioms for commutative rings with identity ele...