Chaotification of 1D maps by Multiple Remainder Operator Additions: Application to B-Spline Curve Encryption

Lazaros Moysis* (Corresponding Author), Marcin Lawnik* (Corresponding Author), Ioannis P. Antoniades, Ioannis Kafetzis, Murilo Baptista, Christos Volos

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
2 Downloads (Pure)


In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By appropriate parameter tuning, the resulting map can achieve a higher Lyapunov exponent value, a result that was first proven theoretically and then showcased through numerical simulations for a collection of chaotic maps. As a proposed application of the transformed maps, the encryption of B-spline curves and patches was considered. The symmetric encryption consisted of two steps: a shuffling of the control point coordinates and an additive modulation. A transformed chaotic map was utilised to perform both steps. The resulting ciphertext curves and patches were visually unrecognisable compared to the plaintext ones and performed well on several statistical tests. The proposed work gives an insight into the potential of the remainder operator for chaotification, as well as the chaos-based encryption of curves and computer graphics.
Original languageEnglish
Article number726
Number of pages27
Issue number3
Early online date14 Mar 2023
Publication statusPublished - 14 Mar 2023

Bibliographical note

The authors are thankful to Chongyang Deng for providing the control points for the three curves used for encryption. The authors are thankful to the anonymous Reviewers for their comments.


  • chaos
  • chaotification technique
  • Lyapunov exponent
  • B-splines
  • Encryption


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