Numerous applications that rely on asymmetric cryptography use the RSA algorithm. It can be applied to digital signatures and the encryption of sensitive data. The secure storage of the private key is essential for the algorithm’s strength. Finding ways to use factorization or other heuristics to determine the value of the private key, was the goal of several academic efforts. Both the Euler totient and the Carmichael functions are used in this study to analyze the private key properties and demonstrate the existence of many private keys for the same public key. We further demonstrate that a universal private key that complies with the FIPS standard exists. Moreover, by taking advantage of a condition imposed by FIPS recommendations, we present a new method for attacking the RSA modulus (N).