A binary stream cipher cryptosystem can be used to encrypt/decrypt many types of digital files, especially those can be considered huge data files like images. To guarantee that the encryption or decryption processes need a reasonable time to encrypt/decrypt the images, so we have to make the stream cipher key generator that acts quickly without effect in the complexity or randomness of the output key binary sequences. In this paper, we increase the size of the output sequence from binary to digital sequence in the field  to obtain byte sequence, then we test this new sequence not only as binary but also -sequence. So we have to test the new output sequence in the new mathematical field. This is done by changing the base of the randomness tests and extending Golomb’s postulates from binary to . Some theorems and lemmas are proved to find the new testing laws that are suitable to the new sequences field. The results of using the extended randomness tests are compared to the results of binary randomness tests to guarantee that the decision of pass or fail is identical, the results also prove the precision of identicalness.