Newton Polyhedra and p-Adic Estimates of Zeros of Polynomials in np[x, y]

Newton polyhedron associated with a polynomial in n pix, is introduced. Existence of a relationship between a Newton polyhedron and zeros of its associated polynomial is proved. This relationship is used to arrive at the p-adic estimates of the zeros. An upper bound to the p-adic orders of these zeros is found using the Newton polyhedron method

Newton Polyhedral Method of Determining p-adic Orders of Zeros Common to Two Polynomials in Qp[x, y]

To obtain p-adic orders of zeros common to two polynomials in Q [x,y], the combination of P . Indicator diagrams assodated with both polynomials are examined. It is proved that the p-adic orders of zeros common to both polynomials give the coordinates of certain intersection points of segments of the Indicator diagrams assodated with both polynomials. We make a conjecture that if ( A, IJ. ) is a point of intersection of non-coinddent segments in the combination of Indicator diagrams associated with two polynomials in Q [ x,y l then there exists a zero (L Tl) common to both polynomials such that ord ~. = A , ord Tl::: IJ. . A special case of this conjecture is proved

A Method for Determining the Cardinality of the Set of Solutions to Congruence Equations

The cardinality of the set of solutions to a system of congruence equations module a prime power is estimated by applying the Newton polyhedral method. Estimates to this value are obtained for an n-tuple of polynomials f = _ (f1, ... ,fn) in coordinates -f = (xl' ... ,xn) with coefficients in Zp. The discussion is 011 the estimates corresponding to the polynomials f that are linear in x and a specific pair of quadratics in Zp(x,y

On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form

The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton polyhedral technique. The method involves reduction of the partial derivatives of f to single-variable polynomials and finding 8 the determinant factor in the estimation. f x and f yare reduced to one-variable polynomials by the employment of suitable parameters. The Newton polyhedrons associated with the polynomials so obtained are then considered. There exist common zeros of the single-variable polynomials whose p-adic orders correspond to the intersection points in the combination of the indicator diagrams associated with the respective Newton polyhedrons. This leads to sizes of common zeros of the partial derivatives of f. This information is then used to arrive at the above estimate

An average density of τ-adic Naf (τ-NAF) representation: an alternative proof

In order to improve the efficiency of scalar multiplications on elliptic Koblitz curves, expansions of the scalar to a complex base associated with the Frobenius endomorphism are commonly used. One such expansion is the τ-adic Non Adjacent Form (τ-NAF), introduced by Solinas (1997). Some properties of this expansion, such as the average density, are well known. However in the literature there is no description on the same sequences occuring as length- NAF's and length-l τ-NAF's to proof that the average density is approximately 1/3. In this paper we provide an alternative proof of this fact

On the integral solutions of the diophantine equation x4 + y4 = z3

This paper is concerned with the existence, types and the cardinality of the integral solutions for diophantine equation x4y4z3+ = where x , y and z are integers. The aim of this paper was to develop methods to be used in finding all solutions to this equation. Results of the study show the existence of infinitely many solutions to this type of diophantine equation in the ring of integers for both cases, x=y and x y. For the case when x=y, the form of solutions is given by (x,y,z)=(4n3,4n3,8n4), while for the case when x y, the form of solutions is given by (x,y,z)=(un3k-1,vn3k-1,n4k-1). The main result obtained is a formulation of a generalized method to find all the solutions for both types of diophantine equations

Number theoretical foundations in cryptography

In recent times the hazards in relationships among entities in different establishments worldwide have generated exciting developments in cryptography. Central to this is the theory of numbers. This area of mathematics provides very rich source of fundamental materials for constructing secret codes. Some number theoretical concepts that have been very actively used in designing crypto systems will be highlighted in this presentation. This paper will begin with introduction to basic number theoretical concepts which for many years have been thought to have no practical applications. This will include several theoretical assertions that were discovered much earlier in the historical development of number theory. This will be followed by discussion on the “hidden” properties of these assertions that were later exploited by designers of cryptosystems in their quest for developing secret codes. This paper also highlights some earlier and existing cryptosystems and the role played by number theoretical concepts in their constructions. The role played by cryptanalysts in detecting weaknesses in the systems developed by cryptographers concludes this presentation

Improvement to scalar multiplication on Koblitz curves by using pseudo τ-adic non-adjacent form

Pseudo τ-adic non-adjacent form (pseudoTNAF) for elliptic scalar multiplication on Koblitz Curve was developed by Faridah et al. since 2012. This is analog to binary method and alternative to τ-adic non-adjacent form (TNAF) and reduced τ-adic non-adjacent form (RTNAF) methods that was produced by Solinas at the year 1997 and 2000 respectively. The objective of this paper is to improve the scalar multiplication algorithm with pseudoTNAF that was published earlier. Consequently, to prove that the density of the pseudoTNAF Hamming weights (HW) is less four percents than the HW of both TNAF and RTNAF

The fascinating numbers

We see numbers almost everywhere and every day in our lives. Numbers such as telephone numbers, car plate numbers and IC numbers serve to organize our daily activities into some order. Numbers are money-spinners for TNB, Telekom and other business concerns and more recently IWK despite the ignorance of its activities by some quarters. Numbers, especially big ones seem to be the nightmares of politicians and economists although the empty one (zero) is an elusive dram for consumers in this age of concern over the threat of inflation. Numbers are conveniently employed to determine the status of salary increments under the SSB

On contractions and invariants of Leibniz algebras.

In this paper, contractions of complex Leibniz algebras are considered. A short summary of the history, relationships of different definitions and comparisons of them are given. We focus on the contractions of three-dimensional case of complex Leibniz algebras. A several contraction invariants that are useful in determining whether one algebra can be obtained as an contraction of another algebra are given