  Elliptic curve cryptography has received great attention in recent years due
to its high resistance against modern cryptanalysis. The aim of this article is
to present efficient generators to generate substitution boxes (S-boxes) and
pseudo random numbers which are essential for many well-known cryptosystems.
These generators are based on a special class of ordered Mordell elliptic
curves. Rigorous analyses are performed to test the security strength of the
proposed generators. For a given prime, the experimental results reveal that
the proposed generators are capable of generating a large number of distinct,
mutually uncorrelated, cryptographically strong S-boxes and sequences of random
numbers in low time and space complexity. Furthermore, it is evident from the
comparison that the proposed schemes can efficiently generate secure S-boxes
and random numbers as compared to some of the well-known existing schemes over
different mathematical structures.
