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Synthetic Disvision of Polynomials01:28

Synthetic Disvision of Polynomials

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Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
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Partial Fractions01:28

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A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
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Real Zeros of Polynomials01:27

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Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0​, then every rational zero is...
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Quadratic Equations01:29

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A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
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Quadratic Equations in the Complex Number System01:29

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A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
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Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

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The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
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A Novel Construction of Efficient Substitution-Boxes Using Cubic Fractional Transformation.

Amjad Hussain Zahid1,2, Muhammad Junaid Arshad2, Musheer Ahmad3

  • 1Department of Computer Science, University of Management and Technology, Lahore 54000, Pakistan.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

This study introduces a novel method for creating secure substitution boxes (S-Boxes) using cubic fractional transformation (CFT). The proposed S-Boxes demonstrate strong cryptographic properties, making them suitable for modern block ciphers.

Keywords:
block cipherscubic fractional transformationsecuritysubstitution box

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Area of Science:

  • Cryptography
  • Information Security
  • Applied Mathematics

Background:

  • Symmetric block ciphers are crucial for information security.
  • Substitution boxes (S-Boxes) are key components in modern block ciphers, requiring specific design criteria for strength.
  • Existing S-Box construction methods are continuously being improved to meet evolving security demands.

Purpose of the Study:

  • To present an innovative technique for constructing substitution boxes (S-Boxes) using a cubic fractional transformation (CFT).
  • To critically evaluate the cryptographic strength of the proposed S-Boxes against established performance criteria.
  • To demonstrate the efficacy of the CFT-based S-Box construction method for block cipher applications.

Main Methods:

  • Developed a novel S-Box construction technique based on cubic fractional transformation (CFT).
  • Evaluated the S-Boxes using key cryptographic criteria: bijection, nonlinearity, bit independence criterion, strict avalanche effect, and linear and differential approximation probabilities.
  • Performed comparative analysis against recently investigated S-Boxes through simulations.

Main Results:

  • The proposed CFT-based S-Boxes satisfy essential cryptographic criteria, including bijection and high nonlinearity.
  • The S-Boxes exhibit strong resistance against linear and differential cryptanalysis.
  • Comparative analysis confirms the cryptographic strength and efficiency of the proposed S-Boxes.

Conclusions:

  • The cubic fractional transformation (CFT) offers an effective method for generating strong S-Boxes.
  • The proposed S-Boxes meet the stringent requirements for use in secure symmetric block ciphers.
  • This technique provides a valuable contribution to the field of cryptographic primitive design.