The Itinerary of the N-Type Map and Its Application
Abstract
:1. Introduction
2. Preliminaries
3. The Itineraries and the Periodic Points of the N-Type Map
3.1. The Itinerary Computation for the N-Type Map
3.2. The Computation of the N-Type Map’s Periodic Points
4. Chaos of the N-Type Map
5. Application to Image Scrambling Encryption
Algorithm 1. (Recursive itinerary generation) |
Initialization: define the itinerary set of length 1 as the base ternary unit = (0, 1, 2). The corresponding ternary encodings are 0, 1, 2 (stored as decimal values 0, 1, 2). |
Recursive construction: given the decimal representation of , generate in four steps. |
(a) Left branch: directly inherit , equivalent to prepending 0 to each itinerary (decimal values remain unchanged): . |
(b) Middle branch: add to each element of , then reverse the entire sequence: . |
(c) Right branch: add to each element of , preserving the original order: . |
(d) Merge: combine all branches: . |
Algorithm 2. (1D itinerary-based image scrambling) |
Step 1. Read a plain image with height and width . Rearrange into a vector of length in column-priority order. |
Step 2. Let be the least integer such that is met. Apply Algorithm 1 to obtain the truncated itinerary set , which contains elements. For each truncated itinerary, reverse its order to obtain . Convert it into a decimal number . Arrange these decimal numbers orderly to form a permutation of the set . Then, reject the numbers larger than to obtain a permutation of . |
Step 3. Use to rearrange the elements of and get : |
Step 4. Rearrange into 2D matrix in column-priority order and the final scrambled image with height and width is obtained. |
Algorithm 3. (2D itinerary-based image scrambling) |
Step 1. Input a plain image . |
Step 2. Select the least integers such that . Then, use Algorithm 1 to separately generate the truncated itinerary sets and . Reverse the order of each itinerary to obtain , . Convert the elements in and into decimal numbers and arrange these decimal numbers to form permutations of and , respectively. Next, filter out the numbers in that are larger than and those in that are larger than . After filtering, we obtain the permutations of and , respectively. The permutations are denoted as and , respectively. |
Step 3. are used to rearrange the pixel positions of plain image and obtain the scrambled image : |
6. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
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Ye, R. The Itinerary of the N-Type Map and Its Application. Mathematics 2025, 13, 1159. https://doi.org/10.3390/math13071159
Ye R. The Itinerary of the N-Type Map and Its Application. Mathematics. 2025; 13(7):1159. https://doi.org/10.3390/math13071159
Chicago/Turabian StyleYe, Ruisong. 2025. "The Itinerary of the N-Type Map and Its Application" Mathematics 13, no. 7: 1159. https://doi.org/10.3390/math13071159
APA StyleYe, R. (2025). The Itinerary of the N-Type Map and Its Application. Mathematics, 13(7), 1159. https://doi.org/10.3390/math13071159