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I am learning about crytography and the differences between the public and private keys encryption examples and I was wondering if it is possible to have a private key encryption scheme that is completely insecure? I have not been able to find what that could look like but I'm curious? Or is this not possible? Is the encryption scheme always secure with private keys?

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Any encryption scheme can be insecure if the key space is small enough.

For example, you could perform encryption with an 8 bit RSA key. For a key of that size, it's trivial to determine the private key given the public key.

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  • $\begingroup$ would this be true even with a small message space? Or that wouldn't factor in? $\endgroup$ – droidnoob Feb 24 at 14:34
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    $\begingroup$ @droidnoob If your key is insecure nothing else matters. For example given an 8 bit RSA public key with exponent 3 modulus 187, it's trival to factor the modulus into 11 and 17, calculate lamda(187) = lcm(11-1,17-1) = 80, then calculate the private exponent d = 3^-1 mod 80 = 27. $\endgroup$ – dbush Feb 24 at 14:53
  • $\begingroup$ thanks for reply. can I ask you one more? What if the key is completely secure and protected, would there be any way to have a completely secure key but insecure scheme? $\endgroup$ – droidnoob Feb 24 at 14:57
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    $\begingroup$ @droidnoob If you mean a poorly designed encryption algorithm, then yes. $\endgroup$ – dbush Feb 24 at 15:00
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    $\begingroup$ You could derive a subkey in your scheme that is much smaller than the input key, e.g. by hashing it and taking the leftmost bits. The original, large key could be completely secure - protected by the cryptographic hash - but you have again an encryption scheme with a small key space. Presto. $\endgroup$ – Maarten - reinstate Monica Feb 24 at 16:25
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What if the key is completely secure and protected, would there be any way to have a completely secure key but insecure scheme?

Yes, certainly. For example, consider the following encryption scheme:

  • The key is a 256 bit (or, heck, 512 or 1024 bit if you want) string chosen uniformly at random by a cryptographically secure true random number generator, stored securely in a locked vault deep underground, with multiple armed guards watching the entrance 24/7.

  • The encryption method doesn't use the key for anything, and instead encrypts the data using rot13.

  • The decryption method is the same as the encryption method.

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    $\begingroup$ I have some optimizations in mind for your scheme, I think we can improve its efficiency by using the identity function instead of rot13. Furthermore, the identity function is an old, well-known and well-studied object, used often in cryptography - much more often than rot13. Hence I would suggest going for this conservative yet highly optimized option (I think there is even hardware support for identity, unlike rot13) $\endgroup$ – Geoffroy Couteau Feb 25 at 13:04

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