Graph Structure of Chebyshev Permutation Polynomials over Binary and Ternary Adic Rings
This study investigates the functional graph structure of Chebyshev permutation polynomials over the composite ring ℤ_{2^{k₁}3^{k₂}}, where elements serve as vertices and the polynomial mapping defines directed edges, thereby characterizing the topology of cycles and tails. By integrating tools from algebraic number theory and graph theory, and leveraging newly established properties of Chebyshev polynomials modulo powers of 2 and 3, the work reveals a striking regularity in the functional graphs when binary and ternary components coexist: the number of cycles of any fixed length remains constant, while branching patterns grow predictably with k₁ and k₂. Extending prior results on prime-power rings, this research demonstrates that despite their seemingly complex behavior, such nonlinear mappings exhibit a highly structured nature, offering a theoretical foundation for analyzing the security of cryptographic systems based on these mappings.