1. Assume that users A and B have implemented the RSA public-key cryptography in
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Question
1. Assume that users A and B have implemented the RSA public-key cryptography in order to establish secure communications. Adversary C has discovered one of the prime numbers for determining the public-key pairs of users A and B.
2. Assume user A has implemented RSA public-key cryptography. User A leaks its private key and instead of generating a new modulus, it decides to generate a new public and a new private key. Is this a safe approach? Why?
3. Assume user A has implemented RSA public-key cryptography. A small number of repeated encodings give back the plaintext. What might be the cause?
Explanation / Answer
1. Power of RSA is valid until the modulus n is not factorized. Since the adversary have already discovered one of the prime numbers which are used for generating the modulus. He can derive the public and private key easily using which he will break the system.
Users A and B have to regenerate their public and private keys with larger and different prime numbers. Then they can secure their communication against adversary C.
2. Yes. Regenerating modulus alone won't be sufficient when the private key is leaked. Let us assume that user B is encrypting the message with new modulus generated by A and e (part of A's public key). Adversary C knows the private key d which is not changed, and modulus value n which is already available as part of A's public key.
He can decrypt the message easily as message = Cipher-text ^ d mod n
Hence the user should generate new public and private key.
3. If the prime numbers p,q used for key generation are not large enough or the encryption key e derived from (p-1)(q-1) is not a random integer but selected in relationship with n, then RSA will become vulnerable to cyclic attack. Cyclic attack will give back the plain text / cipher text with repeated encoding using public key.
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