Efficient Reconstruction of Euler's Totient Function from Carmichael's Function in Semiprime RSA Moduli

This note presents a simple deterministic closed-form method for reconstructing Euler’s totient function φ(N) from Carmichael’s function λ(N) for semiprime RSA moduli N = pq with distinct odd primes. The reconstruction operates in constant time given λ(N), avoids explicit recovery of gcd(p−1, q−1), and enables exact recovery of the prime factors through a single quadratic equation. The proof is based on a sharp bounding relation showing that φ(N) lies within a single λ(N)-interval around an explicit estimate derived from N. Exhaustive computational verification over tens of millions of semiprime instances confirms the correctness of the reconstruction formula.