Multiplication of three elements over finite fields is used extensively in multivariate public key cryptography and solving system of linear equations over finite fields. This contribution shows the enhancements of multiplication of three elements over finite fields by using specific architecture. We firstly propose a versatile multi-input multiplier over finite fields. The parameters of this m...

In this paper we present two algorithms for the multiplication of sparse Laurent polynomials and Poisson series (the latter being algebraic structures commonly arising in Celestial Mechanics from the application of perturbation theories). Both algorithms first employ the Kronecker substitution technique to reduce multivariate multiplication to univariate multiplication, and then use the schoolb...

The current paper presents a fast implementation of finite field multiplication, squaring, and exponentiation operations using two look-up tables, thereby reducing the number of operations, saving time through precomputation, and avoiding the complicated computations required in the Montgomery method. Furthermore, look-up table-based algorithms have the advantage of speed and are particularly s...

Before discussing multiplication, let us speak about addition. The number A(k) of distinct sums i+ j ≤ k such that 1 ≤ i ≤ k/2, 1 ≤ j ≤ k/2 is clearly 2 bk/2c − 1. Hence the number A(2n) of distinct elements in the n × n addition table involving {1, 2, . . . , n} satisfies limn→∞A(2n)/n = 2, as expected. We turn to multiplication. Let M(k) be the number of distinct products ij ≤ k such that 1 ≤...

Group theory is a branch of mathematics that studies, of course, the structure and behavior of "groups." A mathematical group is composed of a set of elements and a group operation (call it "multiplication" and denote it as *); the set is closed under multiplication--combining elements using multiplication never leads outside the group. It is a closed system. In addition to being closed, a grou...

let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

In this paper we give some general results on the non-splitextension group $overline{G}_{n} = 2^{2n}{^{cdot}}Sp(2n,2), ngeq2.$ We then focus on the group $overline{G}_{4} =2^{8}{^{cdot}}Sp(8,2).$ We construct $overline{G}_{4}$ as apermutation group acting on 512 points. The conjugacy classes aredetermined using the coset analysis technique. Then we determine theinertia factor groups and Fischer...

For computing Padé approximants, we present presumably stable recursive algorithms that follow two adjacent rows of the Padé table and generalize the well-known classical Levinson and Schur recurrences to the case of a nonnormal Padé table. Singular blocks in the table are crossed by look-ahead steps. Ill-conditioned Padé approximants are skipped also. If the size of these lookahead steps is bo...