Why is lcm(n) used now instead of $\varphi(n)$ in RSA?

Question

I know that lcm(n) or Carmichael's totient function $\lambda(n)$ is now used instead of $\varphi(n)$ or Euler's totient function to generate the public key in RSA encryption? Why is this so? Are there any mathematical, computer scientific reasons behind this related to key security or efficiency?

Answer 1

Defining the public and private exponents as inverses modulo φ(N), as originally done for RSA, provides a sufficient (but not necessary) condition for the decryption rule to recover the plaintext from any ciphertext. Since Carmichael’s lambda function is, by definition, the smallest number m such that $a^m \equiv 1 \pmod N$, for any integer a that is relatively prime to N. Thus, it is sufficient to define the public and private exponents as inverses modulo any multiple of λ(N).