Tests for Permutation Functions

The Symmetric Block cipher is a considerable encryption algorithm because of its straightforwardness, rapidity and strength and this cryptographic algorithm is employed in carrying out the encryption and decryption for most current security applications. The confusion properties are attained using the substitution-Box (S-Box). Substitution and permutation functions are normally used in block ciphers to make them much firmer and more effectual ciphers. The Security of S-Box is checked using S-Box test criteria and the randomness test. The objective of this paper is to give the researchers a specific knowledge (standards) for testing the ciphers’ S-Boxes. This paper includes survey or guide for the S-box test criteria.

Reconstructing rational functions with FireFly

In this paper, we consider rational functions [Formula: see text] with some minor restrictions over the finite field [Formula: see text] where [Formula: see text] for some prime [Formula: see text] and positive integer [Formula: see text]. We establish a sufficient condition for the existence of a pair [Formula: see text] of primitive normal elements in [Formula: see text] over [Formula: see text] Moreover, for [Formula: see text] and rational functions [Formula: see text] with quadratic numerators and denominators, we explicitly find that there are at most [Formula: see text] finite fields [Formula: see text] in which such a pair [Formula: see text] of primitive normal elements may not exist.

Let \( \mathbb{F}_q \) be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over \( \mathbb{F}_q \) and \( \mathbb{F}_q \) be the rational function field over \( \mathbb{F}_q \). We seek to understand the structure of the rational invariant subfield \( \mathbb{F}_q \). In this paper, we prove that \( \mathbb{F}_q \) is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

The substitution and permutation function, which are presented in the article as elements of a number of factorial sets, are the key functions of cryptographic systems that provide diffusion and mixing of information. A new scale of notation is proposed while analyzing this problem. This is the notation scale of a number of factorial sets. This scale of notation helps to index the elements of a number of factorial sets and establish a one-to-one correspondence between the number and a specific type of substitution. This allows analyzing substitutions characteristics systematically. This paper presents the basic concepts of a number of the factorial sets. It is noted that the permutations of the factorial sets form symmetric permutation groups, and specific permutations (when raised to a power) form cyclic groups. The group axioms are fulfilled for the permutations of a number of factorial sets. Also, the definition domain, the group operation of multiplication, and identical and inverse substitutions are given for them. The number of independent cycles, decrement, inverse, parity and sign are common characteristics of the substitutions of a number of factorial sets. The criteria for choosing single substitutions with the best characteristics are proposed.

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

It is shown that permutable rational functions have not only the same Julia set but also the same periodic stable domains. In addition, an extended version of a theorem by Julia is stated.

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, * *.* ],

Let K = ¥q(T the rational function field over the finite field ¥q (T: indeterminate), and A=¥q[T]. For a non-zero ideal n of A, we can define a smooth proper geometrically connected curve X0(n) over K, called the Drinfeld modular curve of Hecke type with conductor n. In this article, we define the Eisenstein quotient J of the Jacobian variety / of Jf0(n) for n maximal and investigate its arithmetic properties. One of the main results is as follows :

This paper describes a simple procedure for synthesizing an active distributed RC network which, by using dominant poles and zeros, realizes a very accurate approximation of an arbitrary stable rational transfer function. The network uses a single uniformly distributed RC line with taps spaced along its length. A linear combination of tap voltages is added to the input signal to form the driving voltage for the RC line; the output signal is also a linear combination of the tap voltages. The network offers a number of significant advantages. Since it realizes a nearly rational transfer function, the approximation problem can be conveniently solved using readily available results on rational function approximation. Also, the network uses only one uniform RC line, the transfer function can be changed simply by changing resistor values, and the frequency can be scaled by minor connection changes. Thus one standard network with minor modification is useful for a wide variety of applications. This paper develops the design procedure and derives the various sensitivity functions of importance. Two example designs are carried out: an approximation to a second-order low-pass transfer function and an approximation to a second-order band-pass transfer function with a Q of 100. The sensitivities for the examples are very reasonable and the measurements made on laboratory models indicate excellent agreement with theoretical predictions.

Permutations in a finite field (bijective transformations) are actively studied in many applications, including in information security theory. Permutations are often used as elements for building information processing nodes. In the middle of the 20th century, K. Shannon theoretically justified the basic requirements for mapping performed on such nodes. Currently, when constructing bijective transformations, these requirements are provided by the composition of nonlinear representations given by the table in the field GF(2). The presented paper summarizes the results of work [1] on methods for increasing the dimension of stationary functional systems. Namely, in this paper, approaches to the construction of new permutations from the initial permutations in the finite field are investigated. The linear characteristic is calculated for the constructed permutations. The problem of constructing permutations given by coordinate functions is difficult. The relevance of the topic of the paper is determined by the need to search for new theoretically sound methods for constructing s permutation classes in multidimensional spaces with the required combinatorial-algebraic properties. The paper considers several methods for constructing substitutions of finite fields from initial permutations acting on vectors of smaller dimension. In the binary case, this allows us to find the nonlinearity of the substitutions under consideration, characterizing the proximity of linear combinations of coordinate functions of permutations to the entire class of affine functions. The results of the presented work somewhat expand the possibilities of constructing permutations for an arbitrary finite field.

ABSTRACTBy employing the Hirota bilinear (HB) method and starting from the solution with arbitrary functions, we independently analyze the relationship between abundant wave behavior and the superposition formulas of solutions for a (3+1)‐dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. Concurrently, we correlate findings with the behaviors of oceanic waves, yielding some valuable insights. By taking the arbitrary functions as exponential functions, polynomial functions, hyperbolic cosine functions, trigonometric functions, and mixed function, numerous solutions to the equation are obtained as follows: the rogue wave solution, the multikink wave solution, the periodic wave solution, and interaction solutions. Through asymptotic analysis, the superposition of exponential functions leads to the increase in the kink wave numbers. For the rogue wave as the rational function solution, reformulating parameter relations using vectors facilitates the precise determination of its emergence time. Additionally, the impacts of parameters and the order of the rational function on the rogue‐wave shape are independently analyzed. Finally, we examine amplitude variations in rogue and periodic waves under the gradual superposition of kink waves. All types of wave profiles are visually displayed in the form of graphics by selecting appropriate parameters.

Journal Article Heights and preperiodic points of polynomials over function fields Get access Robert L. Benedetto Robert L. Benedetto Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2005, Issue 62, 2005, Pages 3855–3866, https://doi.org/10.1155/IMRN.2005.3855 Published: 01 January 2005 Article history Published: 01 January 2005 Received: 20 October 2005 Revision received: 12 December 2005 Accepted: 25 December 2005

On derived Hall numbers for tame quivers