Chaotic systems cannot be realized using fixed or floating point arithmetic on a digital platform. Implementations using these methods always yield periodic and erroneous orbits which are not predicted by the original mathematical method. Also, using such implementations in cryptography is not appealing, because classical cryptography mostly deals with integers and bitwise operations. Therefore, new methods are needed for implementing chaotic systems on digital platforms. Integer domain chaotic systems (IDCS) constitute one such method. There is a generalized IDCS which can be realized for all dimensions. Recently, another method called LSB-extension method has been proposed. However, this method can only be used with binary shift chaotic maps (BSCM), but there is not a BSCM defined for all dimensions. In this paper, such a BSCM is defined for all dimensions by generalizing the 2D baker map and the generalized map is called higher-dimensional baker map (HDBM). It is shown that this generalization exhibits Devaney's chaos and a rigorous proof of this statement is given. Moreover, two generic building blocks are introduced for implementing the HDBM, and using these blocks, the HDBM is realized both on software and FPGA (Field Programmable Gate Array) hardware.