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

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

A reduced τ-adic Naf (RTNAF) representation for an efficient scalar multiplication on Anomalous Binary Curves (ABC)

Elliptic curve cryptosystems (ECC) provides better security for each bit key utilized compared to the RSA cryptosystem. For this reason, it is projected to have more practical usage than the RSA. In ECC, scalar multiplication (or point multiplication) is the dominant operation, namely, computing nP from a point P on an elliptic curve, where n is an integer defined as the point resulting from adding P + P + ... + P , n times. However, for practical uses, it is very important to improve the efficiency of the scalar multiplication. Solinas (1997) proposes that the τ-adic Non-Adjacent Form (τ-NAF) is one of the most efficient algorithms used to compute scalar multiplications on Anomalous Binary curves. In this paper, we give a new property (i.e., Theorem 1.2) of τ-NAF(n) representation for every length, l. This is useful for evaluating the maximum and minimum norms occurring among all length-l elements of Z(τ). We also propose a new cryptographic method by using randomization of a multiplier n to ñ an element of Z(τ). It is based on τ-NAF. We focused on estimating the length of RTNAF(ñ) expansion by using a new method

Pseudo τ - adic non adjacent form for scalar multiplication on Koblitz curves

In ECC, scalar multiplication is the dominant operation, namely computing nP from a point P on an elliptic curve where the multiplier n is an integer, defined as the point resulting from adding P+P+⋯+P, n times. The τ-NAF proposed by Solinas, is one of the most efficient algorithms to compute scalar multiplications on Koblitz curves. In this paper, we introduced an equivalent multiplier to τ-NAF namely pseudo TNAF. It is based on the idea of transforming the τ-NAF expression to a reduced τ-NAF that has been done by some researchers. It can eliminate the elliptic doublings in scalar multiplication method, and double the number of elliptic additions. We provide the formula for obtaining a total of lattice points in Voronoi region of modulo r+st where r+sτ an element of ring Z(τ). This helps us to find all the multipliers n that based on τ-NAF. We also discuss the estimation of operational costs when using pseudo TNAF as a multiplier of scalar multiplication

Noise induced synchronization of time-delayed semiconductor lasers and authentication based asymmetric encryption.

In this work, we propose to enable security mechanisms on a chaotic communication system based upon common noise induced synchronization between two time-delayed semiconductor laser systems. The cryptosystem subjected to the common additive Gaussian colored noise undergoes a transition to follow identical trajectories. An investigation of the system together with a novel scheme for authentication based message encryption process are presented. The encrypted message is also sent over a public channel, while the key is never transmitted at all. The advantage of the scheme is its security, based on the authentication and asymmetric encryption. Extended statistical tests with the proposed two phase cryptography scheme demonstrate the efficiency of the system being robust and tolerant to different types of statistical attacks

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