Number Theory

Euler's Theorem



Solving which of the following problems would allow one to break RSA encryption?

Alice chooses a public key of \(n=187\) and \(e=3\). What is her private key?

Bob's public key is \(426759435605268851\) and \(e=3\). Alice uses ASCII encryption and sends the ciphertext \(c=298049520771754739\). Which of the following was Alice's original message?

Mallory discovers many public keys, including the following:

  • \(n=70441807\), \(e=3\)
  • \(n=10645627\), \(e=17\)
  • \(n=63339281\), \(e=65537\)
  • \(n=24864431\), \(e=257\)
  • \(n=89221291\), \(e=17\)

What number can Mallory discover is the prime factor of one (or more) of these keys, without needing to factor any of them, making use of a vulnerability of RSA?

Knowing which of the following would allow an attacker to efficiently break the RSA encryption?


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