Abstract
It is known that, given an RSA modulus, n = p q , the public key e and the corresponding private key d satisfy the modulo congruence e d ≡ 1 mod ϕ n , where ϕ n = p − 1 q − 1 . Usually, the private key d can be computed efficiently using the extended Euclidean algorithm, and it is common knowledge that the private key is unique in the sense of modular ϕ n . This paper shows that there exist multiple private keys d ; they all satisfy that d < ϕ n . This paper also presents the exact relationship between an RSA public key and a corresponding private key.
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