The REX cryptosystem presented herein is a variant of the NTRU cryptosystem. In the REX cryptosystem, a primary ring RXOR and two secondary rings RXOR,q and RXOR,p are used to reduce the number of operations required to compute the keys, to perform the encryption process, and to perform the decryption process. The REX cryptosystem may also be implemented using Walsh-Hadamard transformations to significantly increase speed.