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The concealment of information passed over a non-secure communication link lies in the complex field of cryptography. Furthermore, when absolutely no secure channel exists for the exchange of a secret key with which data is encrypted and decrypted, the remedy lies in a branch of cryptography known as public-key cryptosystem (PKS). This thesis provides an in-depth study of the public-key cryptosystem. Essential background knowledge is covered leading up to a VLSI implementation of a fast modulo exponentiator based on the sum of residues (SOR) method. Fast modulo exponentiation is vital in the most popular PKS schemes. Furthermore, since all cryptosystems make use of some form of mapping functions, a neural network - an excellent non-linear mapping technique - provides a viable medium upon which a possible cryptosystem can be based. In examining this possibility, this thesis presents an adaptation of the back- propagation neural network to a 'pseudo' public-key arrangement. Following examinations of the network, a key management system is then devised. Finally, a complete top-down block diagram of an entire cryptosystem based on the neural network of this study is proposed. Cryptography, Public-Key, Secret-Key, Discrete Logarithm, Fast Exponentiation, Diffie-Hellman, RSA, Inverse, GCD, Neural Networks, Back-Propagation, Factorization, Sum of Residues, Modulo Reduction.