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Randomization of the Fourier transform.

Zhengjun Liu1, Shutian Liu

  • 1Department of Physics, Harbin Institute of Technology, Harbin, 150001 China.

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Researchers explored the fractional Fourier transform, discovering eigenvalue ambiguity suggests randomness. A novel method to randomize the Fourier transform was developed, applicable to image encryption and decryption.

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Area of Science:

  • Mathematics
  • Signal Processing
  • Information Security

Background:

  • The fractional Fourier transform (FrFT) is a generalization of the traditional Fourier transform, offering unique properties in signal processing.
  • Investigating the spectral properties of the FrFT, particularly its eigenvalues, is crucial for understanding its behavior and potential applications.
  • Ambiguity in the eigenvalues of mathematical transforms can sometimes hint at underlying random processes.

Purpose of the Study:

  • To analyze the multiplicity and complexity of eigenvalues associated with the fractional Fourier transform.
  • To explore the potential link between eigenvalue ambiguity and randomness in the FrFT.
  • To propose and validate a method for randomizing the Fourier transform based on these findings.

Main Methods:

  • Mathematical analysis of the fractional Fourier transform and its eigenvalue spectrum.
  • Development of a novel algorithm to introduce controlled randomness into the Fourier transform.
  • Simulation and testing of the proposed random Fourier transform for image encryption.

Main Results:

  • The study identified significant multiplicity and complexity in the eigenvalues of the fractional Fourier transform.
  • A strong correlation was observed between eigenvalue ambiguity and inherent randomness within the FrFT.
  • A practical method for generating a random Fourier transform was successfully developed.

Conclusions:

  • The eigenvalue characteristics of the FrFT provide insights into its potential for randomization.
  • The proposed random Fourier transform offers a new approach for secure image encryption and decryption.
  • This work bridges theoretical mathematical analysis with practical applications in cryptography.