71 research outputs found

### Variational Formulations for Electromagnetic Field and Charged-Particle Stream Configurations and Their Linearization

Two variational formulations for electromagnetic field and charged-particle stream configurations, in which both field and particles are described by the field-like variables suited for the problems of electrodynamics, are presented. One of them is directly obtained through slight modifications of Sturrock's original procedure but has a complicated form. The other is obtained through linearization of the preceding one and has a compact form. Both formulations lend themselves to straightforward derivation of the well-known energy-momentum tensor and/or its conservation law. Specifically the latter one is of academic interest because of its compact form. Moreover, as a proof of its practical usefulness the variational principle under the small-amplitude approximation is derived from it, which is known to provide a basis for the study of certain types of instability in plasmas. It is, however, hoped that it will find main applications in the electrodynamics problems concerned with large-amplitude behavior

### Determining Minimal Polynomial of Proper Element by Using Higher Degree Traces

Modern communication engineerings, such as elliptic curve cryptographies, often requires algebra on finite extension field defined by modulus arithmetic with an irreducible polynomial. This paper provides a new method to detemine the minimal (irreducible) polynomial of a given proper element in finite extension field. In the conventional determination method, as we have to solve the simultaneous equations, the computation is very involved. In this paper, the well known "trace" is extended to higher degree traces. Using the new traces, we yield the coefficient formula of the desired minimal polynomial. The new method becomes very simple without solving the simultaneous equations, and about twice faster than the conventional method in computation speed

### The Number of the Irreducible Cubic Polynomials in the Form of x(3) + ax + b with a Certain Fixed Element a

In this paper, we first show the number of x's such that x(2) +u, u âˆˆ F(*)(p) , becomes a quadratic residue in F(p), and then this number is proven to be equal to (p+1)/2 if âˆ’u is a quadratic residue in Fp, which is a necessary fact for the following. With respect to the irreducible cubic polynomials over Fp in the form of x(3)+ax+b, we give a classification based on the trace of an element in F(p3) and based on whether or not the coefficient of x, i.e. the parameter a, is a quadratic residue in Fp. According
to this classification, we can know the minimal set of the irreducible cubic polynomials, from which all the irreducible cubic polynomials can be generated by using the following two variable transformations: x=x + i, x=jâˆ’1x, i, j âˆˆ Fp, j â‰ 0. Based on the classification and that necessary fact, we show the number of the irreducible cubic polynomials in the form of x(3)+ax+b, b âˆˆ F(p), where a is a certain fixed element in F(p)

### Ordinary Pairing Friendly Curve of Embedding Degree 3 Whose Order Has Two Large Prime Factors

This paper proposes a method for generating a certain composite order ordinary pairingâ€“friendly elliptic curve of embedding degree 3. In detail, the order has two large prime factors such as the modulus of RSA cryptography. The method is based on the property that the order of the target pairingâ€“friendly curve is given by a polynomial as r(X) of degree 2 with respect to the integer variable X. When the bit size of the prime factors is about 500 bits, the proposed method averagely takes about 15 minutes on Core 2 Quad (2.66Hz) for generating one

### The Orders of Elliptic Curves y(2) = x(3) + b, b âˆˆ F(* q)

This paper particularly deals with elliptic curves in the form of E(x, y) = y(2) âˆ’ x(3) âˆ’b = 0, b âˆˆ F(* q) , where 3 divides qâˆ’1. In this paper, we refer to the well-known twist technique as x-twist and propose y-twist. By combining x-twist and y-twist, we can consider six elliptic curves and this paper proposes a method to obtain the orders of these six curves by counting only one order among the six curves

### Predictor Order and Error Distribution of MMAE Predictors for Lossless Image Coding

This paper investigates the relation between error distribution and predictive order of minimum mean abusolute error predictors (MMAE predictors) designed for lossless coding of grayscale images. Design of MMAE predictors reduces to the linear programming problem. Let k be the number of coefficients in a predictor (predictor order), we imagine that predictor order k may have a distribution shaping effect. Main purpose of this paper is to ensure that k has such an effect

### A Fast Implementation of Elliptic Curve Cryptosystem with Prime Order Defined over F(p8)

Public key cryptosystem has many uses, such as to sign digitally, to realize electronic commerce. Especially, RSA public key cryptosystem has been the most widely used, but its key for ensuring sufficient security reaches about 2000 bits long. On the other hand, elliptic curve cryptosystem(ECC) has the same security level with about 7-fold smaller length key. Accordingly, ECC has been received much attention and implemented on various processors even with scarce computation resources. In this paper, we deal with an elliptic curve which is defined over extension field F(p2c) and has a prime order, where p is the characteristic and c is a non negative integer. In order to realize a fast software implementation of ECC adopting such an elliptic curve, a fast implementation method of definition field F(p2c) especially F(p8) is proposed by using a technique called successive extension. First, five fast implementation methods of base field F(p2) are introduced. In each base field implementation, calculation costs of F(p2)-arithmetic operations are evaluated by counting the numbers of F(p)-arithmetic operations. Next, a successive extension method which adopts a polynomial basis and a binomial as the modular polynomial is proposed with comparing to a conventional method. Finally, we choose two prime numbers as the characteristic, and consider several implementations for definition field F(p8) by using five base fields and two successive extension methods. Then, one of these implementations is especially selected and implemented on Toshiba 32-bit micro controller TMP94C251(20MHz) by using C language. By evaluating calculation times with comparing to previous works, we conclude that proposed method can achieve a fast implementation of ECC with a prime order

### An Algorithm for Generating Irreducible Cubic Trinomials over Prime Field

This paper proposes an algorithm for generating irreducible cubic trinomials in the form x(3) + ax + b, b âˆˆ F(p), where a is a certain fixed non-zero element in the
prime field F(p). The proposed algorithm needs a certain irreducible cubic trinomial over F(p) to be previously given as a generator; however, the proposed algorithm can generate irreducible cubic polynomials one after another by changing a certain parameter in F(p). In this paper, we compare the calculation cost and the average computation time for generating an irreducible cubic polynomial, especially trinomial, among Hiramoto et al. irreducibility testing algorithm, Berlekamp-Massey minimal polynomial determining algorithm, and the proposed algorithm. From
the experimental results, it is shown that the proposed algorithm is the fastest among the three algorithms for generating irreducible cubic trinomials

### A Method for Checking the Parity of (#Jc - 1)=2 of Genus 2 and 3 Hyperelliptic Curves

This paper shows a method for checking the parity of (#Jc âˆ’ 1)/2 without calculating the order #Jc, where #Jc is the order of genus 2 or 3 hyperelliptic curve

### A Method for Generating Prime Order Elliptic Curves over F(q(2c))

This paper proposes an algorithm for generating prime order elliptic curves over extension field whose extension degree is a power of 2. The proposed algorithm is based on the fact that the order of the twisted elliptic curve is able to be a prime number when the extension degree for the twist operation is a power of 2. When the definition field is F(2(40)âˆ’87)(4) , the proposed algorithm can generate a prime order elliptic curve within 5 seconds on PentiumIII (800MHz) with C language

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