Aloha, In addition to being confused about arbitrary asymmetry in RSA cryptography and whether or not e and n were reversibly derived from d such that possession of d was the same as possession of e and n, I was making a practical assertion that many RSA implementations aren't coded in such a way as to facilitate arbitrary designation of which key is public and which private. Microsoft .NET, for example, defines a private key as inclusive of its corresponding public key. A valid XML representation of a private key in the .NET Framework includes the public key. As in: rsaDecrypt.FromXmlString( "<RSAKeyValue><Modulus>vuQkEFfmNf/XTIRL/ga4WYBsA2GMq" + "IpUpwPmCEBWIQGwXfRioppWTdIWz01u6o4h8R38alnfbh7erO/O+anmgb" + "fHdCf+8oc5G0WcCU1AYp7hV5rBHQ4gb0oaIHi+RCKkcrvzQ2PZjchLcDf" + "N15SOgsXDf88fdxFzUoZA23RXrbs=</Modulus><Exponent>AQAB</Exp" + "onent><P>4LWIuM82AHAryV3ojQ6Uzef3L5VBpn3y1wRvffg3j27w/KyB" + "ou0Zo/LnqqBc885dfLqqaBEBewxLlEpoFfaIhw==</P><Q>2XkPOpd" + "Af6sbymL41pwNvZg2CXcc49DBYbamEW+I+xAFAvBSeMP6O09fqO0jN" + "mdFeTAbACrQl7gfMteeP9JiLQ==</Q><DP>XV/yBWHNfdceytlkBiF2" + "Ai4PEE3EbwvNOj4UmlLnu4mNSGHiqLI/wlnwnH1wwrsRLABhSUcvx1L" + "voRpeMCo2xw==</DP><DQ>rhbSERYphMoGGjK2fp44BbFGeLdIgjqHw" + "+AB+u0tW8XMLTkS3CgONdJpgoIq8Q8kt0nCI5UinIHBP+MJhI+3FQ==" + "</DQ><InverseQ>e9Bf8RurDeKstBP5Awmnc78WgBiaqVTVOpxx3YF" + "fsG+Q3YHK1PgRkQKp8uMIHafAIQ0cEq7BxotXd5PYoTN2VQ==" + "</InverseQ><D>iaZFgyt/K80y2VBE5AbAhHmgace8AATQCi" + "c7hxOth9uJ7BY/0fTs6uzl2dKCeszHGPGAhMgN34CPHbFHVKz5M64" + "QvimHE1imX3LPD7bWb00KMd+G0CKJ6BUcreeYpQffcFT3FwO3fEFY" + "g44j/2UGdU2RgMiUuvOT+DTO7Os+EtE=</D></RSAKeyValue>"); The <Modulus> and <Exponent> represent the public key while the private key consists of <P>, <Q>, <DP>, <DQ>, <InverseQ>, and <D>. Based on the tests that I've done, Microsoft .NET doesn't allow you to load a private key into an instance of the RSA class and use it for encryption, you can only use it for decryption. As for encryption speed, encryption transformations with a public key (<Modulus> and <Exponent>) take far less time (approximately 1/15th as long) to complete as do decryption transformations with a private key (<P>, <Q>, <DP>, <DQ>, <InverseQ>, and <D>) under Microsoft .NET. Anyone know why? Is this a known performance differential with RSA or is Microsoft doing something strange? Sincerely, Jason Coombs jasoncat_private -----Original Message----- From: Kenneth Buchanan [mailto:K.Buchananat_private] Sent: Thursday, January 09, 2003 4:01 AM To: 'Tom Arseneault'; 'jasoncat_private'; Chris Matthews; 'Frank Knobbe' Cc: secprogat_private Subject: RE: PGP scripting... To be fair, it does depend on the cryptosystem you're using. Jason mentioned he wasn't clear on RSA, which indeed has a 'symmetry' between the keys that allows you to arbitrarily choose which is private and which is public. But his original post was correct if you are speaking of Discrete Log-based cryptosystems, as opposed to Factoring-based cryptosystems. ElGamal crypto is based on DLP. So is Elliptic Curve Cryptography, which is a variant of ElGamal. In these systems divulging your private key compromises the public key as well. ------------------------------------------------------------ Kenneth Buchanan Software Developer Kasten Chase k.buchananat_private "You do not really understand anything unless you can explain it to your grandmother." -- Albert Einstein -----Original Message----- From: Tom Arseneault [mailto:TArseneaultat_private] Sent: Wednesday, January 08, 2003 7:28 PM To: 'jasoncat_private'; Chris Matthews; 'Frank Knobbe' Cc: secprogat_private Subject: RE: PGP scripting... Not true, there is no relation between the keys in that way, you can't find one key from the other in any order. The only difference between the keys is that you keep the private key secret. Either key can be used to encrypt/decrypt messages. Here is an Algorithm for finding the public and private keys: Algorithm: Select two prime numbers p and q. Let n=p.q Let z=(p-1).(q-1) Choose a number d that does not divide z. Choose a number e such that is a multiple of z plus 1. e and n are published as the public key while d is kept secret as the private key. Example: p=3, q=11 ->n=33, z=20 Choose d=7 Choose e=3, , i.e., z+1 As you can see d and e have no relation to each other. If your private key is compromised but somehow they do not have e, since d has no relation to z (hence n) you can not determine e from d. Also although e has a relation to z (hence n) there is still no relation to either d so your still safe. Here is a quick over view of the public key encryption routines (the clearest I've yet seen) that explain the use of "n" in the above setup: Instead of sending plain text information P, transmitters compute the remainder C when Pe is divided by n. The receiver recovers the unencrypted message P by computing the remainder of Cd divided by n. ("P" stands for plain text, Pe is P modified by "e" (how exactly modified I don't recall) and Cd is C modified by "d", Pd and Ce should also be valid combinations) (The algorithm and example are taken off the web page "http://thalia.spec.gmu.edu/~pparis/classes/notes_101/node63.html") However since you normaly will freely publish your public key then it can be assumed that once someone gets ahold of your private key he/she will now have both your keys, just not for the reason you describe. As for the usage of the key in encryption and decryption, public key encryption is very compute intensive so while you could do bulk encryption with it whould be very slow.. The usual way things are done is that a symmetrical encryption will be used to encrypt a document (DES, 3DES, BLOWFISH, etc..., very fast) with a randomly generated key and that key is then encrypted with the public key of the person you sending the document to. Since only he, through the use of his private key, can decrypt the symmetrical key only he can decrypt the document. For a signature, you first take a hash of the document (MD5, SHA1, etc...) and then you encrypt it with your private key so that anyone with your public key can decrypt the signature and verify the document (since only you, thru the use of your private key, could have created the signature they can be assured that the document has not changed in transit and you were the one to send it) Tom Arseneault Security Engineer Counterpane Internet Security. "All humans are born Right-Handed...but the great ones overcome it."
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