The Primality Testing Problem (PTP) has now proved to be solvable in deterministic polynomial-time (P) by the AKS (Agrawal-Kayal-Saxena) algorithm whereas the Integer Factorization Problem (IFP) still remains unsolvable in (P). There is still no polynomial-time algorithm for IFP. Many practical public-key cryptosystems and protocols such as RSA (Rivest-Shamir-Adleman) rely their security on computational intractability of IFP. Primality Testing and Integer Factorization in Public Key Cryptography Second Edition provides a survey of recent progress in primality testing and integer factorization with implications to factoring based public key cryptography. Notable new features are the comparison of Rabin-Miller probabilistic test in RP Atkin-Morain elliptic curve test in ZPP and AKS deterministic test. This volume is designed for advanced level students in computer science and mathematics and as a secondary text or reference book; suitable for practitioners and researchers in industry.