An Evaluation: RSA Private Key Exposure Impacts All Key Vulnerabilities

As is well known, the RSA modulus n = p * q, the public key e, and the corresponding private key d satisfy modulo inverses by the extended Euler's phi function, where modular
$\phi(n)$
= (p - 1) (q - 1) [3], [11], [14]. We could generate multiple key pairs by specifying different values of e from the same modulation value
$\phi$
(n) when e and d satisfy the following conditions of the Euler Totient equation at the time of verification [12]. There are potential risks associated with Public Key Infrastructures (PKI) that generate a pair of keys using the same modulation value
$\phi$
(n) [2], [3]. The aim of this paper is to demonstrate that if the private key d has been compromised, the modulus n factoring process using the Euler Totient equation is proved [8], [10]. The risk from the previous key pair {(e, n); (d, n)} that was revealed if the PKIs produced the same modulation n [2], [3], [11], [14]. We have summarized and proved this vulnerability in a simple manner using a small number of examples to make it easier to understand how it can occur and how to avoid it.