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# Public / Private keys

Posted on 2014-04-02
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Can you  have the following

1> 1 to many relationship between a private key and a public key

2> A one to many relationship between a public key and a private key
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Question by:Anthony Lucia
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Accepted Solution

Dave Howe earned 500 total points
ID: 39973624
Sort of.

Extended version? there is a mathematical relationship between the two keys such that given the original constants and one of the keys, you can compute the other.

For RSA, that is such that for the three values M (modulus), E (encryption exponent) and D (decryption exponent) then cryptotext=plaintext to the power of E modulo M, and plaintext = cryptotext to the power of D modulo M. this is reversible, in that if you swap E and D over, the math still works.

however, for any pair (E,M) there are an infinite number of possible values for D, all with a rigid mathematical relationship to each other. You don't HAVE to pick the smallest one, but it usually makes sense to do so (as otherwise you are just doing more math for no real benefit)

Summary of RSA would go like this (and I have a t-shirt with this on someplace here :)

Pick two prime numbers P and Q
Your M is the value P x Q
now, pick an E (say, 512) and calculate N as the value (P - 1) x (Q - 1)
Now, any D such that E x D = 1 Mod N is suitable.

now, given you can write "Mod N" as "i x N" for some arbitrary integer i you can rearrange that so that for any given value i, you can calculate the resulting D value.

conversely, if you have calculated some D, you can go ahead and calculate other values of E by varying the value of i.

Does that make sense?
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Expert Comment

ID: 39976472
Your thinking of the web of trust, or the method used in SMIME/PGP where one can encrypt the massage once, but send the same message to a handful of people. The way that works is, the public key of the recipients (each of them) is used to encrypt the same string (string_x). When they recieve the message they use their private-keys to decrypt string_x, and then string_x to decrypt the message itself.
MS EFS works in this way as well, where each users public-key is used to encrypt the file-encryption-key (string_x), and the FEK is used to encrypt the data.
Basically mixing asymmetric and symmetric encryption types.
-rich
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LVL 33

Expert Comment

ID: 39976534
Its possible he is thinking that. Not sure I can tell any more what he is thinking, given the storm of questions :)

Gotta love DESX though. I wonder why it wasn't used outside of EFS?
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