Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security

Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security Taylan Demir1, Hasan Hüseyin Tosunoglu2 1Department of Mathematics, Ankara University, Ankara, 06100, Turkey 2Department of Management Information Systems, Istanbul University, Istanbul, 34452, Turkey Corresponding author: demir.ta[email protected] ORCID IDs: First Author: 0000-0001-7402-4468 Second Author: 0009-0004-8735-4289 DOI : 10.5281/zenodo.18033931 Abstract Classical and dynamic Caesar ciphers are mainly vulnerable to frequency attacks, since in both schemes the shift applied to each character position is memoryless. This is not the case for the proposed fractional differential shift cipher, where the running shift sequence is generated by a Caputo fractional system Dα tx ( t ) = F ( x ( t ) , k )with α∈ (0 , 1) and k denotes the secret key vector. The discrete fractional integrator associated with this Caputo fractional system yields a long-memory, non-uniform sequence of shifts, which induces a generalized Caesar mapping on any chosen character set or ASCII set. Qualitative analysis shows that relatively small variations in either α or k lead to clearly distinct shift patterns and substantially enlarge the effective key space. Numerical tests on textual data demonstrate an approximately uniform symbol distribution, a very low degree of local correlation and, conversely, a high level of entropy, while linear-time complexity and a minimal implementation footprint are preserved. Consequently, taken together, these characteristics indicate that Caputo-based fractional differential shift encryption can serve as a practical alternative to traditional substitution ciphers in resource-constrained environments. Keywords: fractional differential shift encryption, Caputo derivative, dynamic Caesar cipher, substitution cipher, fractional dynamical systems, lightweight cryptography. 1 Introduction 1.1 Background and motivation Classical substitution ciphers include the Caesar and Vigenère schemes, where the mapping from plaintext to ciphertext is done by means of a monoalphabetic rule with a fixed mapping or by a periodically repeating short key [1,2]. Typically, in the simplest versions of the classical and dynamic Caesar methods, the shift applied to each letter is determined by its position in M38-1 KOSC-2025 Proceedings an ordered alphabet and, in some variants, by a key-dependent offset determined by the index of the character in the plaintext. Each letter is transformed based solely upon the current position or local state, which makes these substitution methods effectively memoryless. Because of this property, the resulting ciphertext retains the statistical profile of the letter frequencies of the underlying language and leaves the ciphers vulnerable to frequency analysis and other simple methods for key recovery [1,3]. Although shift-based substitution ciphers have inherent weaknesses, they are still a viable option for implementations requiring a low overhead and processing cost due to their simplicity. The question remains: can the simplicity of a classical Caesar-style cipher be retained while embedding the shifts into a dynamical system with genuine memory and sensitivity, so that the resulting running shift sequence is no longer susceptible to classical frequency attacks? 1.2 Fractional calculus and dynamical encryption Fractional calculus has become a significant tool for modelling systems that exhibit time dependence due to historical data and, more generally, for processes with pronounced long-term memory effects [4,5]. Caputo’s fractional derivatives, in particular, give rise to initial value problems (IVPs) whose solutions depend on the entire past trajectory rather than only the current state, and the fractional order provides a quantitative measure of the memory strength associated with the IVP [5]. Owing to this built-in memory feature, there is a rapidly growing body of work on fractional dynamical systems in physics, engineering and control theory. In the last decade, several authors have proposed encryption schemes driven by fractional-order chaotic systems. Examples include image ciphers based on fractional-order sine maps, fractional beta chaotic maps and fractional 3D Lorenz systems, which are reported to yield large key spaces and favourable diffusion and confusion properties [6–9]. These constructions, however, typically operate on multi-dimensional state variables and target multimedia data, whereas the design of lightweight, text-oriented substitution ciphers that exploit fractional dynamics at the level of symbol shifts has received considerably less attention. 1.3 Main contributions This work seeks to address the shortfall by proposing a new fractional differential shift encryption model that expands the idea of traditional or dynamic caesar type ciphers into an entirely new domain. A summary of the key findings from this study includes: • We define a Caputo–driven shift mechanism in which the running shift sequence is obtained as the numerical solution of a one–dimensional Caputo fractional initial–value problem Dα tx ( t ) = F ( x ( t ) , k )with order α∈ (0 , 1) and secret parameters collected in the key vector k . The resulting non–uniform shift sequence is then coupled with a generalized Caesar mapping over an arbitrary alphabet or ASCII set. • As α→ 1 − it has been demonstrated that under many choices of the nonlinearity F the proposed scheme reduces to classical dynamic (or Caesar-type) ciphers. Therefore, classical and dynamic (or Caesar-type) ciphers are special cases or limiting cases of our model. M38-2 2nd Kocaeli Science Congress, November 19-21, 2025 1.4 Structure of the paper • We analyse how the long–memory property of the Caputo system enlarges the effective key space and amplifies sensitivity with respect to both the fractional order α and the key parameters k, leading to highly aperiodic and history–dependent shift patterns. • We perform a series of statistical tests on textual and gray–level data, including histogram analysis, adjacent–symbol correlation, information entropy and key–sensitivity experiments, and we compare the results with those obtained from classical Caesar and related substitution ciphers. 1.4 Structure of the paper The rest of the paper is organized as follows. Section 2collects the necessary notation, recalls the Caputo fractional derivative and introduces the discrete fractional integrator and the dynamic Caesar cipher model. Section 3presents the proposed fractional differential shift encryption scheme, including the construction of the running shift sequence, the encryption and decryption algorithms, and the relation to classical Caesar-type ciphers. Theoretical properties of the underlying fractional system and of the resulting shift sequences are analysed in Section 4. Statistical and security analyses, together with numerical experiments on representative data sets, are reported in Section 5and Section 6. Implementation issues, complexity considerations and parameter selection guidelines are discussed in Section 7. Finally, Section 8summarizes the main findings and outlines several directions for future work. 2 Preliminaries 2.1 Notation In this paper, we utilize a finite, ordered alphabet A containing ′m′ number of symbols, and where m≥ 2. The symbols of A are represented as A = {a0, a1, . . . , am−1} . A practical example of a finite collection of symbols for text-related purposes is derived from the printable characters (symbols) found within the ASCII character set; however, this construction can also be applied to any finite collection of symbols or bytes [1 , 10 , 11]. The index map ι:A −→ {0,1, . . . , m −1} corresponds each character (symbol) in Aaj to its index j within the ordered collection of symbols, and we denote the inverse relationship of the index map by ι−1. Plaintext and ciphertext are modelled as finite sequences ( pn ) N−1 n=0 and ( cn ) N−1 n=0 with values in A . When convenient we work with their index representations Pn = ι ( pn )and Cn = ι ( cn )taking values in the residue class ring Zm = Z/mZ . Addition and subtraction on indices are always understood modulo mand we write them as x⊕y= (x+y) mod m, x ⊖y= (x−y) mod m, 2nd Kocaeli Science Congress, November 19-21, 2025 M38-3 KOSC-2025 Proceedings for x, y ∈Zm. In particular, a classical Caesar shift by a fixed integer s∈Zmis the map Ss:Zm→Zm, Ss(x)=x⊕s. In the proposed scheme the shift becomes time-dependent. We denote by ( sn ) n≥0 a sequence of integer shifts in Zm , generated by an underlying dynamical system, and write the encryption and decryption rules as Cn=Pn⊕sn, Pn=Cn⊖sn, n = 0,1, . . . , N −1.(1) All subsequent constructions will be formulated at the level of the index sequences Pn, Cn, sn∈Zm ; the passage back to characters is performed via ιand ι−1. 2.2 Caputo fractional derivative The Caputo fractional derivative of order α∈ (0 , 1) , is used in this paper. Let x : [0 , T ] →R be an absolutely continuous function. The Caputo fractional derivative with respect to the lower limit 0is is represented by Dα tx(t) = 1 Γ(1 −α)Zt 0 (t−τ)−αx′(τ) dτ, t ∈(0, T],(2) and is defined using the gamma function Γ( · )[4, 5, 12]. Therefore Dα t can be expressed as an ordinary differentiation followed by a fractional integration of order 1 −α . The function ( t−τ ) −α in (2) provides a weigh-ting to the past values of x′ that is described by a slowly decreasing power-law. The result is that Dα tx ( t )depends upon the complete previous history of x on the interval [0 , t ], rather than only upon its present state (this is referred to as a long memory effect). The Caputo operator is linear and has many of the same properties that are present for the classic first derivative in that respect. In particular, all Constants have zero for the Caputo derivative, while for any xcontinuously differentiable, one has the lim α→1− Dα tx(t)=x′(t) for t∈(0, T)[4, 5]. For nonlinear initial value problems of the form Dα tx(t)=F(t, x(t)), t ∈(0, T], x(0) = x0,(3) with α∈(0,1), it is convenient to rewrite (3) in the equivalent Volterra integral form x(t)=x0+1 Γ(α)Zt 0 (t−τ)α−1F(τ, x(τ)) dτ. (4) Assuming certain standard assumptions, including a global Lipschitz condition on F for its second argument, equation (4) has a unique continuous solution over the interval [0 , T ]. Additionally, the continuous solution has a continuous dependence on initial condition x0 and function F [4, 5, 12]. All of these basic facts will be utilized in the development of the fractional differential system which describes the running shift sequence. M38-4 2nd Kocaeli Science Congress, November 19-21, 2025 2.3 Discrete fractional integrator 2.3 Discrete fractional integrator We discretize the Caputo initial value problem (3) and build the numerical representation of the running shift sequence. The fixed time step is h > 0and tn = nh for every n = 0 , 1 , . . . , N . The numerical solution of x ( tn )at point tn is denoted by xn≈x ( tn )and will be used throughout the rest of this document. For solving Caputo-type equations, we are following the work of Diethelm, Ford, and Freed [13] using a predictor-corrector method based on the Adams approach. In its most basic form, this combination uses an Adams-Bashforth predictor with fractional order followed by an AdamsMoulton corrector with fractional order on the integral equation stated in (4) . The corrector update can be written as xn+1 =x0+hα Γ(α+ 1)   n X j=0 a(n+1) jF(tj, xj)+b(n+1)F(tn+1,ˆxn+1) , n ≥0,(5) where the coefficients a(n+1) j and b(n+1) are determined by the fractional Adams weights and ˆxn+1 denotes the predictor value obtained from an explicit fractional Adams–Bashforth step [13]. This scheme has a convergence rate of O ( hmin{2,1+α} )where F is smooth enough and is zero-stable with respect to an arbitrary α∈ (0 , 1) [13]. Therefore the trajectories ( xn ) n≥0 calculated by the numerical method will be close to the true solution of equation (3) with an error that decreases as h→ 0. To provide a complete view of available discretization methods, we will also highlight the Grünwald–Letnikov (GL) discretization method for approximating a Caputo derivative using weights and backward difference [14, 15]. In this method, the operator Dα tx(tn)by Dα tx(tn)≈1 hα n X j=0 ω(α) jxn−j, will be replaced with a set of binomial coefficients, denoted as ω(α) j = ( − 1) jα j . We will demonstrate that static pressure integrators using GL methods are easy to design and implement, therefore we will use these methods for some of the numerical examples in this report. The predictor-corrector scheme given in equation (5) is the primary focus throughout our theoretical assessment. 2.4 Dynamic Caesar cipher as a discrete dynamical system A fixed shift is applied to each symbol with classical Caesar cipher; in contrast, the dynamic Caesar cipher uses either a position-based shift or an internal state that varies over time [1, 11]. The dynamic Caesar cipher is classified as a discrete-time continuous dynamical system; thus, to understand its relationship to Fractional Dynamical Systems, we investigate its relationship to a Fractional Dynamical System based on the fact that a dynamic Caesar cipher can be viewed as a discrete-time continuous dynamical system. Suppose ( xn ) n≥0 is a sequence of integers or real numbers representing time and k is the secret key that could consist of numerous parameters. The state will be updated according to a 2nd Kocaeli Science Congress, November 19-21, 2025 M38-5 KOSC-2025 Proceedings generic expression xn+1 =G(xn, k), n ≥0,(6) The running shift sequence is then obtained by applying a quantization map Q:R→Zm, sn=Q(xn), n ≥0, and encryption is performed via the shift rule (1) . The simplest example of G and Q can be taken to mean that (6) becomes a linear recurrence of order s , swept through a field Zm and forms part of the traditional running-key and/or auto-key Caesar schemes [10, 11]. For our purposes, we will replace (6) by a fractional differential system, and its discrete solution produces the series of states xn and thence the shifts sn . This concept allows the classical and dynamic functions of the Caesar cipher to follow a much larger set of functions, all of which can be viewed as limiting cases of a generalisation of the Caesar ciphers that will be developed in Section 3. 3 Fractional Differential Shift Encryption Scheme In this section we introduce the proposed fractional differential shift encryption scheme. The running shift sequence is obtained from the trajectory of a scalar Caputo fractional system, which is then quantised to the symbol ring Zmand combined with the modular shift rule (1). 3.1 Caputo-based generating system Let α∈(0,1) and consider the Caputo initial value problem Dα tx(t)=Ft, x(t); k, t ∈(0, T], x(0) = x0, where F : [0 , T ] ×R× K → R is continuously differentiable in its last two arguments and k ∈ K denotes the secret key vector. According to the hypotheses outlined in Section 2.2, this issue has one continuous answer x : [0 , T ] →R based on the parameters α , x0 and k[4, 5, 12]. The Caputo derivative has a non-local kernel, so x ( t )represents the whole past trajectory {x ( τ ) : 0 ≤τ≤t} . Hence, it essentially gives rise to long-memory effects in the driving dynamics. The nonlinearity F and the vector of parameters kare, from the perspective of cryptography, a key scheduling system: for example, different amounts of kwill generate a series of different fractional paths and give rise to multiple distinct sequences of running shifts. This approach to design is consistent with other approaches using fractional and chaotic systems as key-generators for encryption in multimedia applications [3, 6, 16, 19]. 3.2 Discretisation and construction of the shift sequence In this section we will discretise the system at uniform time steps tn = nh with n = 0 , 1 , . . . , N . With xn≈x ( tn ), we will use the predictor-corrector schemes of Diethelm et al. [13] to construct the discrete trajectory {xn}n≥0 The resulting trajectory has a qualitative behaviour similar to that of the original continuous-time system defined by Caputo. Other methods of discretisation M38-6 2nd Kocaeli Science Congress, November 19-21, 2025 3.3 Encryption algorithm such as the Grunwald–Letnikov approach [14] could be used without affecting the way the trajectory is constructed. To work at the level of the symbol indices, we introduce a quantisation map Q:R−→ Zm, Q(x) = ⌊βx⌋mod m, with a fixed scaling factor β > 0. The running shift sequence is then defined by sn=Q(xn), n = 0,1, . . . , N −1.(7) Within Section 4 an in-depth discussion on how the long-range dependence of the sequence ( sn ) n≥0 will occur as a result of its fractional dynamics as well as its near-uniform distribution will be discussed. Specific values for F , α , β and kare required in order to show that ( sn ) n≥0 will inherit both long-range dependence through fractional dynamics whilst simultaneously exhibiting characteristics of uniformity throughout Zm. 3.3 Encryption algorithm The plaintext sequence ( pn ) N−1 n=0 exists in A and corresponds to the index sequence ( Pn ) N−1 n=0 in Zm mapping via the map ι from Section 2.1. In order to encrypt, we will begin with the secret parameters Key =α, x0,k, h, β, N, and proceed as follows: 1. We will take the Key and use the predictor-corrector integration of the Caputo system from t0= 0 to tN−1= (N−1)hto obtain the numerical trajectory (xn)N−1 n=0 . 2. With the trajectory, we will compute the shift sequence (sn)as described in equation (7). 3. We will compute the ciphertext index sequence (Cn)using Cn=Pn⊕sn, n = 0,1, . . . , N −1, where ⊕is addition modulo m. 4. Finally, we will apply ι−1 to convert the ciphertext index sequence ( cn )back into ciphertext symbols (cn)in A. In summary, the encryption algorithm is a stream cipher using a fractional dynamical system to generate the keystream, as opposed to traditional linear recurrences or finite-state machines. 3.4 Decryption algorithm The person receiving the encrypted information must have access to the same parameter set Key . in order to decrypt the data. By repeating Steps 1 and 2 above, the receiver constructs the 2nd Kocaeli Science Congress, November 19-21, 2025 M38-7 KOSC-2025 Proceedings sequence of shifts ( sn ). The information contained in an index that was sent as an encrypted character is extracted with Pn=Cn⊖sn, n = 0,1, . . . , N −1, and here ⊖ indicates modulo m subtraction. Running ι−1 gives the original characters back again. The scheme can be exactly reversed, provided there are no numerical errors, because the operations for the modular group are reversible and because the fractional-integrator function will produce the same deterministic behavior every time for any set of fixed parameters. 3.5 Special cases and relation to classical schemes Traditional Caesar methods can be seen as special cases of the methods proposed by us when (1) the limiting case of the proposed system is the constant shift sequence and (2) when the function F is assumed to only depend on constants and not on other variables in the proposed system. Therefore, the trajectories ( xn )will follow an arithmetic progression. In this limiting case, the sequence ( sn )will also be constant, and the methods proposed by us will be equivalent to the traditional Caesar cipher methods, where α→ 1 − are restricted to constant shifts. If we consider the function F to depend linearly on x and on a small number of key parameters (which we will not specify), we will, in a similar manner to the dynamic or auto-key Caesar cipher methods, obtain sequences of affine recurrences of the same type as in dynamic or auto-key Caesar ciphers. Therefore, the fractional differences will allow for the development of a generalisation of the traditional Caesar and auto-key methods by incorporating the shift pathway into a nonlinear fractional dynamical framework with long-term memory [15, 16, 17, 18]. 4 Theoretical Properties In this section we collect some basic analytical properties of the Caputo system that generates the running shift sequence and discuss their implications for the resulting encryption scheme. Throughout, we consider the initial value problem Dα tx(t)=Ft, x(t); k, t ∈(0, T], x(0) = x0,(8) with 0 < α < 1and secret key vector k . Its equivalent Volterra integral form has already been given in (4). The assumptions on F stated below are standard in the analysis of Caputo fractional differential equations and are sufficient for existence, uniqueness and boundedness of solutions [4, 5, 12]. 4.1 Existence and boundedness of the generating system For every F : [0 , T ] ×R× K → R for x, y ∈R and k∈ K , F ( t, x ; k )is continuous in the variables ( t, x ) ∈ [0 , T ] × R and continuous in the variable k∈ K as described above. Further, F ( t, x ; k ) M38-8 2nd Kocaeli Science Congress, November 19-21, 2025 4.2 Long-memory and sensitivity satisfies the following conditions for all t∈[0, T ],x, y ∈Rand k∈ K: F(t, x;k)−F(t, y;k)≤L|x−y|,(9) F(t, x;k)≤a+b|x|.(10) for some fixed constants L > 0, a, b ≥ 0. Based upon these conditions, the theory behind Caputo initial value problems introduces the following results for the cases mentioned [4,5,12] above. Fact 4.1 (Existence and uniqueness). For each selected parameters ( α, x0, k )where 0 < α < 1 has one unique continuous function x : [0 , T ] →R of (8) that satisfies (8). Additionally, the solution’s dependency on (x0, k)is continuous when considering uniform heat. Fact 4.2 (A priori bound). There exists a constant B=B(T, α, a, b, L, |x0|, k)such that |x(t)|≤B, t ∈[0, T].(11) Using a Fractional Gronwall Inequality, an explicit choice for B can be made using (4) which states that (4); in particular B is always going to be finite over every bounded time interval of [5]. The above property ensures the existence of the trajectory {x ( t ) }0≤t≤T representing the shift sequence for every allowable key, and the discretisation discussed in (2 . 3) will generate numerical trajectories remaining within a bounded area of the state space as long as h is sufficiently small. More specifically, the quantised index sn = Q ( xn )remains in the symbol ring Zm for every n and every valid parameter choice. 4.2 Long-memory and sensitivity In the Volterra framework (4), the kernel ( t−τ ) α−1/ Γ( α )considers the entire past of for each present point through the implementation of a slow polynomial decay model. Thus, when 0 < α < 1, there is a gradual polynomial decrease in the effect of an early point x ( τ )on the value of x ( t ), which is classically associated with "long memory"; however, unlike the classical first-order case, the impact of all historically observed points x ( τ )never totally disappears. Let us denote by x ( t )and ˜x ( t )two different solutions of (8) that correspond to two sets of parameters ( α, x0, k ) and ( ˜α, ˜x0,˜ k )respectively, which are slightly different from each other. Then by the Lipschitz property (9) combined with a fractional Gronwall inequality, we find that we can establish an estimate |x(t)−˜x(t)|≤CΞ(α, ˜α, t)|x0−˜x0|+∥k−˜ k∥, t ∈[0, T],(12) where C is a constant which depends on F and Ξ( α, ˜α, t )increases sub-exponentially as t grows, and as the orders approach 1, there is an increase in Ξ( α, ˜α, t )as well, see, e.g. [5,20]. As demonstrated by the inequality presented, solutions are very suceptible to variations in the parameters, as well as to modifications in the initial conditions; this is particularly true over long time frames. When considering the running shift sequence defined as sn = Q ( xn ), minor adjustments to either of the triplet of values ( α, x0, k )or variations in the numerical step size 2nd Kocaeli Science Congress, November 19-21, 2025 M38-9 KOSC-2025 Proceedings desirable feature for a cryptographic keystream generator. 6.2 Image or gray-level data (optional) The same construction can be applied to gray–level data by interpreting each pixel as an element of Z256 and encrypting the image row by row using the running shift sequence ( sn ) n≥0 . In this case the criteria for a good cipher are again a nearly uniform histogram of the cipher image, low adjacent–pixel correlation and entropy values close to log2 256 per pixel [24]. Numerical experiments of this type, illustrating the behaviour on standard test images, can be included analogously but are omitted here for brevity. 6.3 Performance evaluation To give an indication of the computational cost, we implemented the scheme in C++ on a standard desktop machine and measured the average encryption time for different plaintext lengths. The predictor–corrector integrator described in section 2.3 was used without any additional optimisation. Table 2reports the average encryption times over 50 runs. Table 2: Average encryption time for different plaintext lengths (single–threaded implementation). Length N(bytes) Time (ms) 1 000 3.1 10 000 33.4 100 000 354.7 1 000 000 3 842.5 The time complexity has a nearly quadratic growth rate O ( N2 )according to an increasing integer value of N , this is primarily due to the fact that, at each iteration of the fractional predictor–corrector, it requires approximately O ( n )number of arithmetic computations for each of the N number of data elements that were used to generate the fractional prediction. Many practical applications of the method can accommodate moderate sized messages (up to a few hundred kilobytes) with computational loads well within limits. Additionally, when working with very long data streams, the theoretical upper bounds on the time complexity (asymptotic time complexity) can be substantially reduced from O ( Nlog N )to O ( nlog n )by using fast convolution methods or short–memory approximations, as outlined in Section (7.1). 7 Implementation and Practical Aspects 7.1 Complexity and memory footprint The dominant cost of the proposed cipher lies in the numerical integration of the Caputo initial value problem (3) that generates the running shift sequence. We integrate the system on a uniform grid tn = nh , n = 0 , 1 , . . . , N , by means of the fractional Adams–type predictor–corrector scheme described in section (2.3); see, e.g., Diethelm–Ford–Freed and Diethelm for a detailed analysis of this method, [5, 13]. At the ( n + 1)th step, the corrector update involves a weighted sum over all previously computed function values F ( tj, xj ; k ), j = 0 , . . . , n , so that the work per M38-16 2nd Kocaeli Science Congress, November 19-21, 2025 7.2 Parameter selection guidelines step is proportional to n . Consequently, the total number of floating point operations required to produce a trajectory of length N+ 1 is of order N X n=0 O(n) = O(N2). Since the cost for the quantisation map Q and the modular additions of the shift rule in Section 3.3 is linear in N, the full encryption run will be constrained to the O ( N2 )complexity of a fractional integrator. With regards to memory, the traditional predictor–corrector implementation stores the previous values xj, as well as the optional evaluations F (( tj, xj ; k )which occur on the grids, yielding a memory footprint of O ( N )real numbers; this is acceptable for the message lengths used in the experimental trial. If there are multiple plaintexts of equal length being encrypted using the same key, the pre-computation of the shift sequence ( sn ) n≥0 can be done once and use this pre-computed value for subsequent encryptions; in this way, the numerical computation cost for the integration is spread out over all of the encryption session costs. Many researchers have recommended an approach to implement fast versions of convolution-based techniques to solve Caputo’s equation with either FFT (Fast Fourier Transform) acceleration or by compressing the kernel using a sum-of-exponentials method. See Garrappa’s survey for examples [28]. Using these methods can allow users to solve the Caputo equation in O ( Nlog N )time as opposed to O ( N )time, with memory requirements being O ( N )or less. The methods we discuss in this paper do not utilize these techniques, but they can be applied easily to our framework without modifying the design of the cipher. In addition, these optimizations provide an efficient route to real-time execution for high-volume data, such as plaintext files with many megabytes in length. 7.2 Parameter selection guidelines In practice, the choice of the parameters Ktot = ( α, x0, k, β, h, N )is constrained by a trade–off between security, numerical accuracy and implementation cost. The analytical results of Section 4 and the experiments of Section 6 provide a set of practical guidelines, which we summarise below. The Fractional order α .The value of the fractional order α within a range of (0 , 1) determines the amount of long–memory effect. A smaller fractional order indicates that the kernel within the Volterra equation (3) has a slower decay than a larger fractional order, which approaches classical first-order behavior (e.g., see Podlubny [4]). To make the fractional order suitable for cryptography, we will instead restrict [ αmin, αmax ] ⊂ (0 , 1). In particular, selecting α closer to 0will result in increased values of numerical stiffness and an increased roundoff effect, while selecting α very close to 1results in decreased impact of fractional dynamics and creates an experience very similar to a standard dynamic, memoryless Caesar cipher. Based upon the experimental results, those values of α that are considered in the middle of the Scale between these two extremes appear to provide sufficient balance between both sensitivity and numerical robustness. Initial condition and key vector. Section 4 provides an upper bound |x ( t ) | for each k (admissible key vector) and any initial value x0 (the value at t = 0), which will contain x ( t ) 2nd Kocaeli Science Congress, November 19-21, 2025 M38-17 KOSC-2025 Proceedings within the compact interval which continuously depends on ( α, x0, k ). It is thus reasonable to provide floating point implementations with limits away from the overflow limits of the underlying arithmetic for x0 and components of k ; and provide average ranges of k , because small variations in ( α, x0, k )will also give rise to significant changes in trajectories as indicated by the results of the sensitivity analysis, a behaviour we would like to see when developing key-scheduling algorithms. Step size h and final time T = Nh .The value for total ( T )will be taken with the notation T = Nh , where: h denotes the step-size. The step size will serve two functions: First, It is associated to the level of accuracy obtained by the fractional integrator, and second, it indicates the total amount of computational effort required. As a general rule with regard to the fractional Adams-type Predictor-Corrector scheme: The Global Error will be represented by the notation Chp where the exponent p > 0would indicate the accuracy based on the regularity of the solution, as stated by Diethelm [5]. Thus, by selecting h to be a greater value will generally decrease the statistical characteristics of the generated shift sequence, and by selecting h to be a lesser value, will enhance the expense associated with producing this number sequence with an O ( N2 ) complexity (i.e. as discussed in Section 7.1). We set h in such a way that further halving of the value of h will not produce any visible changes in the values of the histograms or correlation plots shown in Section 6; this is a practical method to determine when to stop. Additionally, the effective length N of the shift sequence must be appropriate for the intended use. For short messages, it is easier to create new sequences for every encryption, while with long streams, the sequences can be produced in blocks of N size and concatenated together, assuming that the underlying numerical integration remains stable over the corresponding time interval [0,T]. Quantisation and scaling. The scale factor β > 0determines how a location xn is passed on to the Zm ring through the quantisation map Q ( x )=( ⌊βx⌋mod m ). If β is too small, consecutive values of xn may all be mapped to the same index, and this could cause statistical irregularities among the values of xn in the resulting shift sequence. Conversely, very large values of β can result in unintentional overflows or abandonment of significant digits in arithmetic with fixed–precision. In application, β is set such that the observed distribution of ( sn )over Zm is as close as is feasible to uniform as can be confirmed via the numerical experimentations and findings in Section 6. When viewed together, these factors provide evidence that there exists a large area where the numerical scheme will be stable and that it can generate a shift sequence with all the required long memory and sensitivity characteristics, while also being able to be implemented in an efficient manner on commercially available hardware. In this area, the user has the option to select specific values of ( α, x0, k, β, h, N )according to security level needs for both security and performance for the intended application. 8 Conclusions In this paper, we presented a fractional order differential shift encryption method which uses a Caputo fractional order system to generate a running shift sequence that is subsequently quantized into a symbol set using one of several possible rules. The Caputo-based initial value M38-18 2nd Kocaeli Science Congress, November 19-21, 2025 problem used to generate the shift sequences has been established as well-posed, indicated by both significant dependence on the initial condition and parameters, and the ability to exhibit both long memory effects and extreme sensitivity, creating computational complexity to support the generation of key-dependent chaotic sequences. Beyond that, properties such as the periodic nature of the generated shift sequences and an extremely large effective key space indicate that it should be able to serve well as a key stream generator in cryptography. We performed some numerical tests on a corpus of text data; our findings showed that the histograms of ciphertexts were approximately uniform in distribution and have approximately the maximum value of the measure of entropy, and only a negligible degree of correlation between adjacent ciphertext symbols. Also, tests of key sensitivity yielded error rates of nearly one-half, suggesting that the system has a high degree of resistance to statistical attacks and to methods where the attacker has access to ciphertexts generated using the key. Timing measurements indicated that goodness-of-fit measures are reasonable as the cost of implementing the existing version of a fractional Kalman predictor-corrector is still quadratic, therefore, the implementation will remain efficient for medium-sized messages, using a comparatively low amount of memory. 9 Future Work The results of the current research provide readers with numerous directions for further research. First, a primary topic for future research is to explore the implications of changing the underlying fractional operator. Researchers could use fractional derivatives based on different memory kernels (e.g., ψ -Caputo fractional derivatives) as opposed to only working with the Caputo fractional derivative. By changing the memory kernel, researchers can study the influence of the various memory kernels on a trajectory’s long-range memory and the statistical characteristics of the running shift sequence, and thereby evaluate the size of the effective key space. By performing a systematic comparison of similar types of operator with respect to these forms of fractional encryption, it will enable a clearer determination on how much of the confusion and diffusion observed within the results is specific to the Caputo kernel, or how much is common across a broader range of fractional options. A second area for future research will be the creation of block and hybrid ciphers using the new proposed key stream generator. Refinements to existing schemes such as substitution/ permutation (SPN) network may occur through an interaction with the fractal derivative shift sequences, or by integrating their use into other block cipher mode operation to take full advantage of statistical properties arising from fractional dynamics combined with structural properties of today’s symmetric enciphering primitives. Also, as future research looks into the security of hybrid designs, analyses of future attack models against chosen plain and chosen cipher text should be included. While developing new methods for minimising operational overheads associated with implementing fractional integrators should be considered for implementation design. The use of fast convolution methods, short-memory approximations, and graphics processing unit (GPU) implementations will help to reduce the asymptotic complexity from O ( N2 )to O ( Nlog N )for long data streams in order to make it feasible for high throughput applications. Coupling the fractional dynamical generator with data-driven methods for parameter estimation and performance prediction appears to be an exciting area of research. For example, techniques similar to those used in [29] using regression 2nd Kocaeli Science Congress, November 19-21, 2025 M38-19 KOSC-2025 Proceedings and deep learning will assist in identifying admissible parameter regions with maximum entropy, minimal statistical leakage, as well as automatically identify weak configurations. A fusion of fractional dynamical systems and machine learning will allow adaptive parameter tuning of encryption schemes based on the statistical behaviour of the incoming data, thus improving the overall security of the system. References [1] Stallings, W. (2011). Cryptography And Networksecurity Principles Andpractice. [2] Katz, J., & Lindell, Y. (2014). Private-key encryption. In Introduction to Modern Cryptography(pp. 77-82). CRC Press. [3] L. Brown, Classical Encryption Techniques, lecture notes on substitution ciphers and frequency analysis, available online. [4] Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). elsevier. [5] Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. [6] Lin, L., Zhuang, Y., Xu, Z., Yang, D., & Wu, D. (2023). Encryption algorithm based on fractional order chaotic system combined with adaptive predefined time synchronization. Frontiers in Physics, 11, 1202871. [7] Ibrahim, R. W., Natiq, H., Alkhayyat, A., Farhan, A. K., Al-Saidi, N., & Baleanu, D. (2022). Image Encryption Algorithm Based on New Fractional Beta Chaotic Maps. Computer Modeling in Engineering & Sciences (CMES), 132(1). [8] Feng, W., Wang, Q., Liu, H., Ren, Y., Zhang, J., Zhang, S., ... & Wen, H. (2023). Exploiting newly designed fractional-order 3D Lorenz chaotic system and 2D discrete polynomial hyper-chaotic map for high-performance multi-image encryption. Fractal and Fractional, 7(12), 887. Fractal and Fractional, 7(12), 887, 2023. [9] Vignesh, D., Fataf, N. A. A., & Banerjee, S. (2023). A novel fractional sine chaotic map and its application to image encryption and watermarking. Applied Sciences, 13(11), 6556. [10] Menezes, A. J., Van Oorschot, P. C., & Vanstone, S. A. (2018). Handbook of applied cryptography. CRC press. [11] Stinson, D. R. (2005). Cryptography: theory and practice. Chapman and Hall/CRC. [12] Kilbas, A. A. (2006). Theory and applications of fractional differential equations. NorthHolland Mathematics Studies, 204. [13] Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1), 3-22. M38-20 2nd Kocaeli Science Congress, November 19-21, 2025 REFERENCES [14] Scherer, R., Kalla, S. L., Tang, Y., & Huang, J. (2011). The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications, 62(3), 902-917. [15] Jacobs, B. A. (2015). A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme. In Abstract and Applied Analysis (Vol. 2015, No. 1, p. 952057). Hindawi Publishing Corporation. [16] Kocarev, L. (2002). Chaos-based cryptography: a brief overview. IEEE Circuits and Systems Magazine, 1(3), 6-21. [17] Jiang, S., Ding, D., Yang, Z., & Hu, Y. (2022, August). A Fractional-Order Chaotic System and its Image Encryption. In 2022 IEEE 5th International Conference on Electronic Information and Communication Technology (ICEICT) (pp. 178-182). IEEE. [18] Wei, J., Zhang, M., & Tong, X. (2021, August). Image encryption algorithm based on fractional order chaotic system. In 2021 IEEE 12th international conference on software engineering and service Science (ICSESS) (pp. 72-75). IEEE. [19] Gokyildirim, A., Çiçek, S., Calgan, H., & Akgul, A. (2024). Fractional-order Sprott K chaotic system and its application to biometric iris image encryption. Computers in Biology and Medicine, 179, 108864. [20] Sene, N. (2021). Analysis of a fractional-order chaotic system in the context of the Caputo fractional derivative via bifurcation and Lyapunov exponents. Journal of King Saud University-Science, 33(1), 101275. [21] Shannon, C. E. (1949). Communication theory of secrecy systems. The Bell system technical journal, 28(4), 656-715. [22] Stinson, D. R. (2005). textitCryptography: theory and practice. Chapman and Hall/CRC. [23] Menezes, A. J., Van Oorschot, P. C., & Vanstone, S. A. (2018). Handbook of applied cryptography. CRC press. [24] Mohammad, O. F., Rahim, M. S. M., Zeebaree, S. R. M., & Ahmed, F. Y. (2017). A survey and analysis of the image encryption methods. International Journal of Applied Engineering Research, 12(23), 13265-13280. [25] Katz, J., & Lindell, Y. (2007). Introduction to modern cryptography: principles and protocols. Chapman and hall/CRC. [26] Menezes, A. J., Van Oorschot, P. C., & Vanstone, S. A. (2018). Handbook of applied cryptography. CRC press. [27] Shannon, C. E. (1949). Communication theory of secrecy systems. The Bell system technical journal, 28(4), 656-715. [28] Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and a software tutorial. textitMathematics, 6(2), 16. 2nd Kocaeli Science Congress, November 19-21, 2025 M38-21 KOSC-2025 Proceedings [29] Hajrulla, S., Özer, Ö., & Demir, T. (2025). Polynomial Regression and Faster R-CNN Models for University Library Decision Implementation Discovery based on Deep Learning. WSEAS Transactions on Information Science and Applications, 22, 83-92. M38-22 2nd Kocaeli Science Congress, November 19-21, 2025