Every 10 minutes, the amount of human generated data expands by more than 10 petabyes. This is equivalent to nearly one third of all literature in all languages from the beginning of recorded history. Such vast amounts of data necessitate effective techniques for data integrity, identity verification and data security. Cryptography aims to satisfy these needs. Many modern cryptographic schemes depend on generating prime numbers. Consequently, one must develop efficient tests to check whether or not a given integer is prime. Many of these tests use the properties of mathematical structures called groups. However, there are cases where non-prime numbers, called pseudoprimes, pass these tests for primality. We investigate several types of pseudoprimes based on primality tests associated to Lucas groups and Elliptic curve groups. In each case, we study the existence of and relationships among these different types of pseudoprimes
