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The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
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Properties of Fourier series II01:21

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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
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The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
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Convergence of Fourier Series01:21

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The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
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The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
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A New Composite Fractal Function and Its Application in Image Encryption.

Shafali Agarwal1

  • 1Independent Researcher, 9600 Coit Road, Plano, TX 75025, USA.

Journal of Imaging
|August 30, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a novel composite fractal function (CFF) for enhanced image encryption. The fractal-based algorithm offers high security and robustness against various cyberattacks.

Keywords:
composite fractal functiondiffusionhenon mappermutationrandom fractal matrixz-scan

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Area of Science:

  • Cryptography
  • Applied Mathematics
  • Computer Science

Background:

  • Fractals exhibit complex, spatially nonuniform phenomena and chaotic behavior.
  • These properties are valuable for developing advanced cryptographic applications.
  • Existing fractal-based methods can be further enhanced for improved security.

Purpose of the Study:

  • To propose a new composite fractal function (CFF) for image encryption.
  • To develop a secure and robust image encryption algorithm utilizing fractal properties.
  • To analyze the security and performance of the proposed fractal-based encryption method.

Main Methods:

  • A novel composite fractal function (CFF) is designed by combining two Mandelbrot set (MS) functions.
  • An image encryption algorithm is developed using fractal-based pixel permutation (Henon map, Z-scanned fractal matrix) and substitution (CFF-generated complex sequences).
  • The algorithm employs row-wise and column-wise diffusion using modified fractal sequences.

Main Results:

  • The CFF exhibits high initial value sensitivity, complex structure, and wider chaotic regions.
  • The proposed image encryption algorithm demonstrates high security and robustness.
  • The algorithm effectively resists brute-force, known/chosen-plaintext, differential, and occlusion attacks.

Conclusions:

  • The proposed composite fractal function (CFF) provides complex dynamical behavior suitable for cryptography.
  • The developed fractal-based image encryption algorithm offers a reliable and high-security solution.
  • The algorithm's robustness against diverse attacks validates its effectiveness in practical applications.