I am trying to solve a problem where private key exponent d, ciphertext c, and public key exponent e (65537) are known. How can I calculate φ(n) or n itself?

An extended version of the problem would be : If we can get many d and ciphers for the same plaintext, where e is always 65537, is there a way to decrypt the cipher?

  • $\begingroup$ Couple of hints - what is the definition of $e$ and $d$? How do they relate to $\phi(N)$? $\endgroup$
    – Kolja
    Commented Mar 16, 2023 at 18:31
  • $\begingroup$ Hints given are more than enough to solve this without number theory. $\endgroup$
    – kodlu
    Commented Mar 16, 2023 at 19:10
  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Cryptography Meta, or in Cryptography Chat. Comments continuing discussion may be removed. $\endgroup$
    – fgrieu
    Commented Mar 17, 2023 at 11:54


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