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so we could use RSA to verify someone's identity could we use RSA as a way of digitally signing documents the answer is No so remember RSA can be used as an identification verification scheme if Alice sets up RSA with public exponent E and public modulus n keeping private exponent D secret Bob sends Alice a message M Alice computes C congruent to M to power D mod N and sends it to Bob and then Bob verifies M is congruent to CE to power e mod N and this works to verify Alice's identity could we use this to sign documents well let's try out the obvious thing so Alice sets up RSA system with n equal to one day to nine in public encryption key e equal 37 her private decryption key is d equals 11 613 and Bob wants Alice to sign M equals 1 5 3 so 2 sine M equals 1 5 3 Alice computes 1 5 3 to her private decryption exponent and gets 5 306 and sends this message on to Bob Bob now has a pair one five three five zero six as the signed document we can verify the signatures before because if we take this encrypted value v 306 and use the public encryption key we get back the original document and again since Alice is the only one who knows the decryption exponent only Alice could have found v 306 remember technology allows us to do things we've never done before and here's something new if Alice signed a real-world document Bob could change the one five three to another number and still have a document with Alice's signature but what if Bob changed one five three to seven five eight Alice's signature is still 5 306 but when we try to verify the signature 5 306 to power 37 is not congruent to 758 and so whatever document Alice signed with v 306 was not seven five eight seven five eight is a forgery unfortunately it's easy to break the RSA signature scheme to do this suppose Bob wants Alice to sign a number M so Bob picks a random K then Bob asks Alice to sign km mod n and Bob also asks Alice to sign k inverse mod n what Alice does this Bob now has km to power e that's K to the e m to the e and also K inverse to power E but if you multiply these two things together you get m to the e which is something that only owls could have produced so bob has Alice's signature on document M so let's say that we have the same RSA setup as before but this time Bob wants to forge Alice's signature on the document M equals 758 so to carry out his nefarious scheme Bob picks K equals 11 had asks Alice to sign km 758 times 11 which is eight three three eight and Alice computes eight...