TELK OMNIKA T elecommunication, Computing, Electr onics and Contr ol V ol. 24, No. 1, February 2026, pp. 228 ∼ 239 ISSN: 1693-6930, DOI: 10.12928/TELK OMNIKA.v24i1.27551 ❒ 228 New chaos function fr om the composition of DTM and Gauss iterated map f or digital image encryption Adrianus Y osia, T ok onyai T awanda J onathan Rab v emhiri, Suryadi MT Department of Mathematics, F aculty of Mathematics and Sciences, Uni v ersitas Indonesia, Depok, Indonesia Article Inf o Article history: Recei v ed Sep 15, 2025 Re vised No v 14, 2025 Accepted Dec 8, 2025 K eyw ords: Composition Dyadic transformation map Dyadic transformation-Gauss iterated map Gauss iterated map Ne w chaos function ABSTRA CT This manuscript introduces a no v el chaotic discrete function, formulated through the composition of the dyadic transformation map (DTM) and the Gauss iterated map (GIM), and designated a s DTGIM. The National Institute of Science and T echnology (NIST) randomness test suite, bifurcation diagrams, and L yapuno v e xponents are used to e xamine the chaotic characteristics of DTGIM. W ith ini- tial condition x 0 = 0 . 12345 and parameters α = − 15 and β = 0 . 3 , the func- tion sho ws chaotic beha vior in the bifurcation diagram and produces a positi v e L yapuno v e xponent. Strong randomness is further conrmed by NIST tests, which achie v e 100% for 32-bit binary sequences and 63.75% for 8-bit binary sequences. Additionally , we compare a number other chaotic discrete functions that also emplo y the composition method. These ndings sho w that DTGIM is a viable option for applications in v olving chaos-based cryptograph y . This is an open access article under the CC BY -SA license . Corresponding A uthor: Adrianus Y osia Department of Mathematics, F aculty of Mathematics and Sciences, Uni v ersitas Indonesia Depok, Indonesia Email: adrianus.yosia@ui.ac.id 1. INTR ODUCTION This age can be seen as the age of information e xchange. The utilization of information technology mak es the trade of data easier . Information can be formed as te xt, images, audio, or video, which are commonly used today . Y et, the da wn of technological information is also follo wed by security issues. As a precaution to it, application of cryptograph y is needed to ensure condentiality , data inte grity , entity authentication, or originating data authentication [1]-[3]. Cryptograph y itself is generally ackno wledged as the best method of data protection ag ainst passi v e and acti v e fraud [4]. At least there are tw o di vided camps of cryptograph y: classical cryptograph y and modern cryptograph y [1]. Classica l cryptograph y focuses on the condentiality of the algorithm that is being used, while modern cryptograph y concentrates on the secrec y of the encryption k e y [1]. Currently , the demand for ha ving f aster digital data and information encryption methods with uncompromising security is rising [5]. One of the solutions to answer the problem is a chaos function-based encryption method. This article also w ants to contrib ute to the de v elopment of chaos-function-based. There are v arious implementations of the chaos function-based encryption method [6]-[12]. Also, there are v ar ious functions that ha v e chaotic properties, such as circle maps, logistic maps, modi ed sine (MS) maps, tent maps, Gauss maps, dyadic transformation maps (DTM), Henon maps, Nahrain maps, sine–iterati v e Y u (SIY u), and others [7]-[14]. Al so, v arious methods are used to impro v e the ef fecti v eness and chaotic beha v- ior of chaotic function such as sequential method [6], modication [8], composition [15], or multi-dimensional J ournal homepage: https://telk omnika.uad.ac.id/inde x.php/TELK OMNIKA Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 229 method [16]. As an illustration (the idea is from [13]), see the Figure 1. Before continuing this article into our main purpose, as an addition, the implementation of chaos-based function encryption itself can be used in engineering [17], [18], medical eld [19]-[24], IO T [25]-[31], or satellite image encryption [32]-[34]. Figures 1(a) to (c) serv e as an illustration on the method’ s di v ersity . Our aim is to dene the ne w chaos function using composition (see Figure 1(c)). Re g arding the research g ap for our w ork, there are se v eral papers that discuss on creating a ne w chaos functions using composition method especially for DTM and Gauss iterated map (GIM). Until today , the w ork that has done is the composition of MS map and DTM [1], GIM and dyadic transformation [13]. In this paper , we will e xplore the composition from DTM and GIM. Also, as a comparison, we will als o nd the sam e rout e f o r DTM and GIM . As an addit ion, we will compare our w ork with other w ork that using composition method. (a) (b) (c) Figure 1. The method for making a ne w chaos function, we using Lena.jpe g ( 512 × 512 ); (a) impro ving chaos function through sequential method [15], (b) impro ving chaos function through multidimensional map [16], and (c) impro ving chaos function through composition [13] 2. METHOD In this section, we will discuss on ho w we deal with four things in this article: the chaos functions, the bifurcation diagram, the L yapuno v e xponent, and also the National Institute of Science and T echnology (NIST) test. - The composition of tw o chaos function The method that we will use to impro v e the chaos function is composition of tw o chaos function [1], [6], [13]. The rst function is the dyadic transformation function or Bernoulli function [35] that can be dened as: f ( x ) = 2 x mo d 1 (1) = ( 2 x, 0 ≤ x < 0 . 5 2 x − 1 , 0 . 5 ≤ x < 1 (2) Meanwhile, the Gauss iterated function [2], [13] is dened as: g ( x ) = exp( − α x 2 ) + β (3) where α , β ∈ R . Then, using the composition of tw o functions method between (2) and (3), the ne w function is, as we called it, the dyadic transformation-Bernoulli function, which can be seen (4): f ◦ g ( x ) = ( 2 exp( − α x 2 ) + 2 β mo d 1 , 0 ≤ x < 5 2 exp( − α x 2 ) + (2 β − 1) mo d 1 , 0 . 5 ≤ x < 1 . (4) Ne w c haos function fr om the composition of DTM and Gauss iter ated map for ... (Adrianus Y osia) Evaluation Warning : The document was created with Spire.PDF for Python.
230 ❒ ISSN: 1693-6930 No w , transforming the function into the discrete map function [3], where f ◦ g ( x n ) = x n +1 , the (4) can be transformed as (5): x n +1 = ( 2 exp( − α x 2 n ) + 2 β mo d 1 , 0 ≤ x n < 0 . 5 , 2 exp( − α x 2 n ) + (2 β − 1) mo d 1 , 0 . 5 ≤ x n < 1 , (5) for n ∈ Z + . W e will call the (5) DTGIM. As a brief note here, the addition mo d 1 is to mak e sure that 0 ≤ x n ≤ 1 . - Bifurcation diagram The bifurcation diagram is a graphical tool that describes stability and nonlinear beha vior from the chaos function based on the changing of parameters [36]-[38]. Then, the chaotic beha vior can be described from the bifurcation diagram [39]. W e use this Algorithm 1 belo w for nding the bifurcation diagram. - L yapuno v e xponent The L yapuno v e xponent is a v alue that can trace the chaos from the system [37]. In our article, follo wing [8], we will nd the best L yapuno v e xponent for a certain parameter . First, the L yapuno v e xponent can be found by using this calculation: L yapuno v e xponent (LE) = li m n →∞ 1 n n − 1 X i =0 ln | h ′ ( x i ) | (6) when LE < 0 , the system tends to be stable, while LE > 0 , it has chaotic beha vior [38]. As a brief note, the function h ′ ( x i ) is the deri v ation from a chaos function h ( x ) . In this paper , the chaos function h ( x ) is the function DTGIM on (5), also GIM and DTM. In thi s article, we will nd the best L yapuno v e xponent by using the algorithm from [8]. W e use Algorithms 2 and 3 for nding the best L yapuno v . Both of the algorithms will be used in one picture for the sak e of ef fecti v eness. - NIST test result W e will use the NIST testing suite [40] to e xamine the randomnes s of the DTGIM function. The NIST testing suite consists of 15 statistical tests (with 16 results) for displaying the randomness of a chaos function. W e will use the Python implementation from Ste v en Ang for our testing [41]. Also, for testing the le, we will generate 8-bit and 32-bit for testing the binary data (follo w the idea from [8], [42]). In addition, we will observ e the calculation of entrop y and autocorrelation to strengthen our result [38], [43], [44]. Algorithm 1. Bifurcation diagram Input : x 0 , α , and β Output : Plot of x n v alues 1. Input initial v alues and parameter and number of iterations ( i ) 2. F or n = 1 to i : 3. Calculate x n based on the chaos function. 4. Plot the v alue of x n 5. Ne xt n 6. Stop Algorithm 2. L yapuno v e xponent graphic: Input : x 0 , α , and β Output : Plot the v alue of h ( x ) 1. Input initial v alues and parameter and number of iterations ( j ) 2. F or n = 1 to j : 3. Calculate h ( x j ) based on the chaos function 4. Plot the v alue of h ( x ) 5. Ne xt j 6. Stop Algorithm 3. Highest L yapuno v e xponent: Input : x 0 , α , and β Output : Highest L yapuno v Exponent 1. Input initial v alues and parameter and number of iterations ( i ), parend 2. While parameter < parend: 2.1. If h ′ ( x ) < 10 − 15 , L yapuno v Exponent = −∞ end if else 2.1.1. sum = 0 2.1.2. for i = 0 to n − 1 2.1.2.1. sum = sum + h ′ ( x ) 2.1.2.2. sum = sum/n 2.2. parameter = parameter +stepsize 3. Find the Highest L yapuno v Exponent 4. Stop TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 1, February 2026: 228–239 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 231 3. RESUL TS AND DISCUSSION W e will start the discussion from the L yapuno v surf ace plot (heatmap; see Figure 2) from parameter α , β , and L yapuno v e xponent (see (6)) from the chaos map at (5) and also the GIM. Since dyadic transformation has no parameter , then there is no L yapuno v surf ace plot from it. From Figure 2(a), re g arding DT -GIM, the v alue of LE is dominantly positi v e when α < − 10 , whene v er β ∈ (0 , 1] . Then, the great candidate of α and β can be seen when α < − 10 . W e will choose α = − 15 and β = 0 . 3 as the parameter for DTGIM. Ne xt, Figure 2(b) sho ws that the v alue of LE i s dominantly positi v e when α < − 10 , whene v er β ∈ ( − 1 , 1) . Then, we will use α = 7 . 3 and β = − 0 . 6 for GIM (see also [45]). The picture w as created by us using Python. The brighter the color , the higher the probability that the system will become chaotic. (a) (b) Figure 2. L yapuno v surf ace from DT -GIM and GIM; (a) L yapuno v surf ace plot from DT -GIM for β ∈ [0 , 1] and seed x 0 = 0 . 12345 and (b) L yapuno v surf ace plot from GIM for α ∈ [ − 15 , 15] , β ∈ [0 , 1] and seed x 0 = 0 . 12345 3.1. Bifur cation diagram No w , follo wing our nding from Figure 2, we will nd the bifurcation diagram of DTGIM for α = − 15 , β = 0 . 3 and seed x 0 = 0 . 12345 (see [6]) and the results are in Figure 3. From Figure 3, we can see that the DTGIM is dense when α = − 15 (Figure 3(a)) and β = 0 . 3 (Figure 3(b)). Hence, it sho ws a great result for the parameter . F or GIM, follo wing the recommendation from Figure 2, we will sho w the Bifurcation diagram of GIM for α = 7 . 3 and β = − 0 . 6 for seed x 0 = 0 . 12345 . The result is in Figure 4. From Figure 4, Ne w c haos function fr om the composition of DTM and Gauss iter ated map for ... (Adrianus Y osia) Evaluation Warning : The document was created with Spire.PDF for Python.
232 ❒ ISSN: 1693-6930 we will use parameter α = 10 (Figure 4(a)) and β = − 0 . 6 (Figure 4(b)) because it gi v es a great result for the parameter . Since the DTM in our case does not ha v e parameters, then we will not sho w the result here. (a) (b) Figure 3. Bifurcation diagram for DT -GIM function; (a) the diagram for x ed α = 15 for β ∈ [ − 1 , 1] and (b) the diagram for x ed β = 0 . 3 for α ∈ [ − 20 , 20] . The pictures were created by us using Python (a) (b) Figure 4. Bifurcation diagram for GIM; (a) the diagram for x ed β = − 0 . 6 and (b) the diagram for x ed α = 7 . 3 3.2. L yapuno v exponent Once more, follo wing our ndi n g from Figure 2, we will e xplore the v alue of the L yapuno v e xponent for α = − 15 and β = 0 . 3 for x 0 = 0 . 12345 using Algorithms 2 and 3 [6], [26]. Then, the plot of the L yapuno v e xponent and the best v alue of it will be sho wn by Fi gure 5. From the calculation that has been sho wn in Figure 5, the L yapuno v e xponent is positi v e for α < − 1 on the function DTGIM. Also, follo wing Algorithm 3, it sho ws that α = − 15 is the highest (best ) parameter for β = 0 . 3 (Figure 5(a)). From the same picture, TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 1, February 2026: 228–239 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 233 the best L yapuno v e xponent of GIM is achie v ed for β = − 0 . 6 is when α = 7 . 3 (Figure 5(b)). Since dyadic transformation gi v es a constant in its deri v ati v e, the L yapuno v e xponent gi v es the constant result and a positi v e one ( ln 2 ) for e v ery seed. (a) (b) Figure 5. The Best L yapuno v e xponent; (a) the diagram for DTGIM with x ed β = 0 . 3 and (b) the diagram for GIM with x ed β = − 0 . 6 3.3. NIST test r esult In this section, we will sho w the NIST test results for three chaos functions, that is DT , GIM, and DTGIM. After we sho w the result. No w , for the binary le (8-bit and 32-bit), we use this step to create the binary le (see [45]): - Choose parameter v alues, in our case α = − 15 and β = 0 . 3 . - T ak e an initial v alue for our case x 0 = 0 . 12345 and record all the map’ s v alues for 127000 iterations (for 8-bit) and 33250 (for 32-bit) and remo v e the rst 2000 iterations. - T ransform the v alue into inte gers that range from 0 to 255 (for 8-bit) and 0 to 2 32 − 1 (for 32-bit). - T ransform all of the inte gers into 8-bit (or 32-bit) binary strings - Concatenate all the strings to form a 1000000-bit le and input it into the NIST test. - The test will decide the randomness. W e sho w the NIST result test for 8-bit and 32-bit binary that can be seen in T able 1 (in Appendix). As a brief description, T able 1 will sho w the randomness of the chaos function either for DTGIM, DTM, or GIM for a certain parameter . W e will add the randomness percentage to sho w the randomness. Also, as an addition, we will also sho w the entrop y per w ord and bit to sho w the utilization of randomness from the data. Also, the autocorrelation plot is sho wn to support the description of randomness of our problem. The three additions will be a support to strengthen the NIST test result. Ne w c haos function fr om the composition of DTM and Gauss iter ated map for ... (Adrianus Y osia) Evaluation Warning : The document was created with Spire.PDF for Python.
234 ❒ ISSN: 1693-6930 3.4. Comparison with other chaos function with composition Lastly , our interest goes into comparison with other functions that use the composition method. As for comparison with other data, we can see from the results of other researchers that utilize either GIM or DTM that c omposition is a method for creating a ne w chaos function. From the latest research, at leas t there are four other candidates besides our w ork [1], [6], [13], [45]. T able 2 sho ws the comparison between our w ork and theirs. T able 2. The comparison between chaos functions that use composition as a method DTGIM MS-CM [1] GIM-DT [13] GM-CM [45] MS-DT [6] L yapuno v e xponent mean 8.2334 13.7329 8.0308 8.2 3.069 NIST pass percentage 100% (32 bit) 100% 100% 25% (8-bit) 82.4% 62.5% (8 bit) Number of pix els change rate (NPCR) (%) 99.622 99.5984 99.5514 99.4214 99.6521 Unied a v erage changing intensity (U A CI) (%) 33.1144 33.5008 33.7116 27.2815 33.6363 Mean entrop y (bits) 7.968 7.9868 7.9458 7.7544 7.9874 3.5. Discussion Our rst nding, the result from L yapuno v Surf ace (Figure 2), Bifurcation diagram (Figures 3 and 4), L yapuno v e xpo ne n t (Figure 5) and NIST result (T able 1) sho ws a consistent result. In the case of DTGIM, the parameter α = − 15 and β = 0 . 3 gi v es the best result for 62 . 5% randomness (8-bit data) and 100% (32-bit data). F or the 8-bit data, the entrop y per w ord sho ws that almos t all of the possible numbers from 1-255 has been e xplored by DTGIM, yet the result is 62.5% at best. Re g arding the test itself, the NIST test proceeded smoothly . This f act is sho wn by the autocorrelation picture from the 8-bit and 32-bit. W e also try dif ferent seeds (not sho wn here), and for 32-bit data, DTGIM still gi v es 100% randomness, and the same for 8-bit. The entrop y per bit and per w ord sho ws that there is s o much room to e xplore since DTGIM has a high randomness. Therefore, DTGIM sho ws a strong candidate for a chaos function. Ne xt, T able 1 sho ws the comparison between DTGIM and tw o other functions prior to composition. F or GIM and DTM, the result also sho ws that both of the functions still lack randomness, comparing them with DTGIM. Of course, the problem with GIM is that the parameter that we ha v e chosen still does not utilize the full potential of GIM. Y et, at least for our research, we conclude that DTGIM is f ar superior with our parameter . W e also tr y dif ferent seeds, and it still gi v es the same result (not sho wn here). F or DTM, the result is the same for both 8-bit and 32-bit, since the nature of the function is “only s hifting” the number . Hence, the results sho w a v ery weak randomness from our research. The result, in line with the observ ation re g arding chaotic function, has a lo w entrop y [46]. Therefore, comparing DTGIM, GIM, and DTM, our research sho ws that the ne w chaos function surpasses its functions prior to composition. From T able 2, we conclude that in general, the image encription from the four functions gi v es a great result since the L yapuno v e xponent mean is positi v e. It means that all four functions sho w a highly chaotic beha vior . Also, high rating of NPCR and U A CI sho ws that all of the function is a great chaos functions. In terms of rating, our research is second after the composition of MS-map and circle map. Therefore, we can say that DTGIM is one of the good chaos function candidates. 4. CONCLUSION W e ha v e e xplored the possibility of a ne w chaos function that we construct from the composition of dyadic transformation and GIM. From the result, we sho w that DTGIM has a chaotic beha vior from the L yapuno v e xponent, bifurcation diagram, and also NIST test result for α = − 15 and β = 0 . 3 . W e also compare DTGIM with GIM and DTM. The result is satisfying and sho ws that DTGIM is superior to its predecessor prior to composition. Therefore, as the purpose of this ar ticle, we conclude that DTGIM is one of the good chaos functions. F or the future trajectories of this research, impro ving the randomness for the NIST test on 8-bit binary data by using another technique for generating random numbers is a viable one. Also, for the DTGIM itself, room for impro v ement can still be made since the entrop y report sho ws that we only co v er at least 40% random numbers at 32-bit data. Similar to this issue, GIM has great randomness also. W e suggest that one can use ne g ati v e α . One can also mo v e to apply the chaos function to encrypting and decrypting the image for the ne xt project. Lastly , another route can be tak en for comparing the result with other methods of chaotic function and analyzing the dif ference. TELK OMNIKA T elecommun Comput El Control, V ol. 24, No. 1, February 2026: 228–239 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 235 FUNDING INFORMA TION The authors state no funding w as in v olv ed. A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the Contrib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib u- tions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu Adrianus Y osia ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ T ok on yai T a w anda Jonathan Rab v emhiri ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Suryadi MT ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administration V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT Authors state no conict of interest. D A T A A V AILABILITY The data that support the ndings of this study are a v ailable from the corresponding author , [initial s: A Y], upon reasonable request. REFERENCES [1] I. Mursidah, S. Suryadi, S. Madenda, and S. Harmanto, “ A Ne w Chaos Function De v eloped through the Composition of the MS Map and the Circle Map, ” Resear c h T r ansformation and Digital Inno vation on Mathematics Education , 2023. [2] D. Sole v , P . Janjic, and L. K ocare v , “Introduction to chaos, ” in Studies in Computational Intellig ence , v ol. 354, 2011, pp. 1–25, doi: 10.1007/978-3-642-20542-2 1. [3] L. K ocare v , “Chaos-based cryptograph y: A brief o v ervie w , ” IEEE Cir cuits and Systems Ma gazine , v ol. 1, no. 3, pp. 6–21, 2001, doi: 10.1109/7384.963463. [4] Mishk o vski and L. K ocare v , “Chaos -Based Public-K e y Cryptograph y , ” in Studies in Computational Intellig ence , v ol. 354, 2011, pp. 27–65, doi: 10.1007/978-3-642-20542-2 2. [5] N. K P areek, “Design and Analysis of a No v el Digital Image Encryption Scheme, ” International J ournal of Network Security & Its Applications , v ol. 4, no. 2, pp. 95–108, Mar . 2012, doi: 10.5121/ijnsa.2012.4207. [6] MT . Suryadi, Y . Satria, V . Melvina, L. N. Pra w adika, and I. M. Sholihat, “ A ne w chaotic map de v elopment through the composition of the MS Map and the Dyadi c T ransformation Map, ” J ournal of Physics: Confer ence Series , v ol. 1490, no. 1, p. 012024, Jun. 2020, doi: 10.1088/1742-6596/1490/1/012024. [7] Y . Satria, MT . Suryadi, and D. J. Cah yadi, “Digital te xt and digital image encryption and ste g anograph y method based on SIY u map and least signicant bit, ” J ournal of Physics: Confer ence Series , v ol. 1821, no. 1, p. 012035, Mar . 2021, doi: 10.1088/1742- 6596/1821/1/012035. [8] MT . Suryadi, M. Y . T . Irsan, and Y . Satria, “Ne w modied map for digital image encryption and its performance, ” J ournal of Physics: Confer ence Series , v ol. 893, p. 012050, Oct. 2017, doi: 10.1088/1742-6596/893/1/012050. [9] MT . Suryadi, Y . Satria, and M. F auzi, “Implementation of digital image encryption algorithm using logistic funct ion and DN A encoding, ” J ournal of Physics: Confer ence Series , v ol. 974, p. 012028, Mar . 2018, doi: 10.1088/1742-6596/974/1/012028. [10] Y . Suryanto, Suryadi, and K. Ramli, “ A secure and rob ust image encryption based on chaotic permutation multiple circular shrinking and e xpanding, ” J ournal of Information Hiding and Multimedia Signal Pr ocessing , v ol. 7, no. 4, pp. pp. 697-713, 2016. [11] MT . Suryadi, E. Nurpeti, and D. W idya, “Performance of chaos-based encryption algorithm for digital image, ” T elk omnika (T elecom- munication Computing Electr onics and Contr ol) , v ol. 7, no. 4, pp.697-713, Sep. 2014, doi: 10.12928/TELK OMNIKA.v12i3.106. [12] Y . Dai and X. W ang, “Medical image encryption based on a composition of Logistic Maps and Chebyshe v Maps, ” in 2012 IEEE International Confer ence on Information and A utomation, ICIA 2012, IEEE , Jun. 2012, pp. 210–214, doi: 10.1109/ICInfA.2012.6246810. [13] M. Mudrika, S. Mt, and S. Made nda, “Ne w chaos function of composition function Gauss map and dyadic transformation map for digital image encryption, ” ITM W eb of Confer ences , v ol. 61, p. 01004, Jan. 2024, doi: 10.1051/itmconf/20246101004. Ne w c haos function fr om the composition of DTM and Gauss iter ated map for ... (Adrianus Y osia) Evaluation Warning : The document was created with Spire.PDF for Python.
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TELK OMNIKA T elecommun Comput El Control ❒ 237 [43] T . Stojano vski and L. K ocare v , “Chaos-based random number generators-part I: analysi s [cryptograph y], ” IEEE T r ansactions on Cir cuits and Systems I: Fundamental Theory and Applications , v ol. 48, no. 3, pp. 281–288, Mar . 2001, doi: 10.1109/81.915385. [44] T . Stojano vski, J. Pihl, and L. K ocare v , “Chaos-based random number generators - P art II: Practical realization, ” IEEE T r ansactions on Cir cuits and Systems I: Fundamental Theory and Applications , v ol. 48, no. 3, pp. 382–385, Mar . 2001, doi: 10.1109/81.915396. [45] Suryadi, Y . Satria, and L. N. Pra w adika, “ An impro v ement on the chaotic beha vior of the Gauss Map for cryptograph y purposes using the Circle Map combination, ” J ournal of Physics: Confer ence Series , v ol. 1490, no. 1, p. 012045, Mar . 2020, doi: 10.1088/1742- 6596/1490/1/012045. [46] F . Y u, L. Li, Q. T ang, S. Cai, Y . Song, and Q. Xu, “ A Surv e y on T rue Random Number Generators Based on Chaos, ” Discr ete Dynamics in Natur e and Society , v ol. 2019, pp. 1–10, Dec. 2019, doi: 10.1155/2019/2545123. APPENDIX T able 1. The comparison between the NIST result for 8-bit and 32-bit binary data for the function DT -GIM. The seed that has been used for the test is x 0 = 0 . 12345 T est name DTGIM (8-bit, α = − 15 , β = 0 . 3 ) DTGIM (32-bit, α = − 15 , β = 0 . 3 ) GIM (8-bit, α = 7 . 3 , β = − 0 . 6 ) GIM (32-bit, α = 7 . 3 , β = − 0 . 6 ) p-v alue (conclusion) p-v alue (conclusion) p-v alue (conclusion) p-v alue (conclusion) The frequenc y (monobit) test 1 . 77 × 10 − 14 (non- random) 0 . 990 (random) 83 × 10 − 5 (non- random) 0 (non-random) Frequenc y test within a block 0 . 9775 (random) 0 . 942 (random) 1 (random) 0 (non-random) The runs test 0 (non-random) 0 . 2661 (random) 0 (non-random) 0 (non-random) T est for the longest-run-of- ones in a block 0 . 16157 (random) 0 . 730 (random) 4 . 4 × 10 − 220 (non- random) 7 . 5 × 10 − 95 (non- random) The binary matrix rank test 0 . 29361 (random) 0 . 111 (random) 0 (non-random) 0 (non-random) The discrete fourier transform (spectral) test 0 . 0444 (random) 0 . 039 (random) 0 (non random) 0 (non-random) The non-o v erlapping template matching test 0 . 54874 (random) 0 . 614 (random) 0 (non-random) 1 . 884 × 10 − 214 (non- random) The o v erlapping template matching test 0 . 12514 (random) 0 . 131 (random) 0 (non-random) 6 . 94 × 10 − 168 (non- random) Maurer’ s “uni v ersal statisti- cal” test 0 . 37994 (random) 0 . 102 (random) 0 (non-random) 0 (non-random) The linear comple xity test 0 . 64399 (random) 0 . 651 (random) 0 (non-random) 0 (non-random) The serial test 0 (non-random) 0 . 250 (random) 0 (non-random) 0 (non-random) 0 (non-random) 0 . 660 (random) 0 (non-random) 0 (non-random) The approximate entrop y test 0 (non-random) 0 . 033 (random) 0 (non-random) 0 (non-random) The cumulat i v e sums (cusums) test (forw ard) 3 . 15403 × 10 − 14 (non-random) 0 . 498 (random) 0 (non-random) 0 (non-random) The cumulat i v e sums (cusums) test (backw ard) 1 . 781 × 10 − 14 (non- random) 0 . 508 (random) 0.0165 (non-random) 0 (non-random) The random e xcursions test 0 . 54972 ∗ (random) 0 . 452 ∗ (random) 1 . 4 × 10 − 6 ∗ (non- random) 0 . 982 ∗ (random) The random e xcursions v ari- ant test 0 . 6250 ∗ (random) 0 . 496 ∗ (random) 0 . 903 ∗ (random) 0 . 0601 ∗ (random) Randomness percentage 10 16 × 100% = 62 . 5% 16 16 × 100% = 100% 2 16 × 100% = 12 . 5% 2 16 × 100% = 12 . 5 % Entrop y per w ord 7.977 (out of 8) 14.931 (out of 32) 0.0816 (out of 8) 13.114 (out of 32) Entrop y per bit 0.997064 0.4666 0.0102 0.409 Autocorrelation ∗ Accumulation of se v eral v alues Ne w c haos function fr om the composition of DTM and Gauss iter ated map for ... (Adrianus Y osia) Evaluation Warning : The document was created with Spire.PDF for Python.