Flat Power Series over a Finite Field

where |y1P #1. These were first discussed by Schur [6]1 with |7pl <1 in connection with functions bounded in the unit circle. In ?2 it is shown that to an arbitrary power series there correspond continued fractions (1.1), and, conversely, to every such continued fraction there corresponds a power series which agrees with the power series for the 2pth and (2p+1)th approximants of the continued fraction up to and including the terms involving zP and zP-1, respectively. An important new feature which is given in ?3 is the simple algorithm for the expansion of a power series into a continued fraction (1.1), and, conversely, for the computation of the corresponding power series from the continued fraction. Heretofore this computation has been exceedingly cumbersome. In ?4 conditions under which these continued fractions converge to the corresponding power series are considered, in particular, if the generating function is meromorphic. In ?5 convergence regions for the expansions (1.1) are found, and in ?6 convergence regions for the continued fraction whose approximants are the even and odd approximants of (1.1). In ?7 certain transformations of (1.1) are discussed.

Hankel continued fraction and its applications

A note on extending Euler's connection between continued fractions and power series

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f ( x ) f(x) and g ( x ) g(x) are integral polynomials of degree n ≥ 2 n \geq 2 and p is a prime exceeding ( n 2 − 3 n + 4 ) 2 {({n^2} - 3n + 4)^2} for which f and g are both permutation polynomials of the finite field F p {F_p} , then their difference h = f − g h = f - g cannot be such that h ( x ) = c x h(x) = cx for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in F p {F_p} and t is the degree of h, then t ≥ 3 n / 5 t \geq 3n/5 and, provided t ≤ n − 3 t \leq n - 3 , t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

In this paper, we will discuss the period length of the continued fraction of the product of a polynomial with a quadratic power series over a finite field. Furthermore, we will give the first example of bounded continued fraction in characteristic 3 with not flat partial quotients.

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field $${\mathbb {F}}_{q}$$.

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

Let \(\mathbb{F}_{q}\) be a finite field, where \(q\) is an odd prime such that \(q&gt;3\). Let \(f\left(t\right) =t^{3}-t\) \(\in \mathbb{F}_{q}\left[ t\right]\) be a polynomial of degree 3. For \(\lambda \neq 0\) in \(\mathbb{F}_{q}\), consider families of elliptic curves \(\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast}}\) and \(\left\{ H_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast }}\) defined respectively by\[v^{2}=\lambda f(u)\text{ and }f\left( v\right) =\lambda f(u).\]In this paper, I investigate the relation between the rational points over finite field on \(\left\{E_{\lambda }\left( \mathbb{F}_{q}\right) \right\}_{\lambda \in \mathbb{F}_{q}^{\ast }}\) and \(\left\{ H_{\lambda }\left(\mathbb{F}_{q}\right) \right\} _{_{\lambda \in \mathbb{F}_{q}^{\ast }}}\), and determine the number of rational points on both of these family of curves.

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].

Rational functions over finite fields having continued fraction expansions with linear partial quotients

In this paper, we give the value of the approximation exponent of the hyperquadratic power series satisfying the equation $$Cx^{r}-Ax^{r-1}-1=0$$ where $r> 2$ is a power of a prime number $p$, $A$ and $C$ are nonzero polynomials over a finite field $\mathbf{K}$ of characteristic $p$ and $\deg A> \deg C$. Further, we exhibit explicitly its continued fraction expansion when $C$ divides $A$.

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-

1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational functions F(x) has a unique infinite continued fraction expansion f = a. + 1/(al + 1/(a2 + * * *)), or more briefly f = [a0; a,, a2, * *.* ],