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

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 D _t ^α x(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.