Digital realizations of chaos-based cryptosystems suffer from lack of a reliable method for implementation. The common choice for implementation is to use fixed or floating-point arithmetic directly. However, such an approach produces unreliable and erroneous pseudo-chaotic orbits, which are not the original orbits of the chaotic map. Also, using fixed/floating-point arithmetic is not a common approach of modern cryptography, which mostly deals with the manipulation of bits with simple logic operations. For these reasons, chaos-based cryptography and modern cryptography have a gap between them. Recently, new approaches are proposed for the digital implementation of chaotic systems. The LSB (Least Significant Bit) extension method is one such approach, which allows the production of true periodic and true chaotic orbits with simple logic operations. However, true orbits have not been utilized in any study yet. In this paper, true periodic orbits are utilized for the first time in a chaos-based cryptosystem. It turns out that true periodic orbits have many advantages over pseudo-chaotic orbits, but using them directly is not secure. Therefore, new design approaches are proposed for securely utilizing true periodic orbits. These design approaches fundamentally depend on structures called M-BSCM (Modified Binary Shift Chaotic Map). Accordingly, a chaos-based cryptosystem is modified for the utilization of true periodic orbits by replacing the chaotic maps inside it with M-BSCMs. The realizations are performed on both software and FPGA (Field-Programmable Gate Array) hardware. Such an approach will help to close the gap between chaos-based cryptography and modern cryptography.