Dr. Watson patiently waited until Sherlock Holmes finished. “Some interesting problem to solve, Holmes?” he asked when Holmes finally logged out.

“Oh, not exactly. I merely checked my e-mail and then made a couple of network experiments instead of my usual chemical ones. I have only one client now and I have already solved his problem. If I remember correctly, you once mentioned cryptology among your other hobbies, so it may interest you.”

“Well, I am only an amateur cryptologist, Holmes. But of course I am interested in the problem. What is it about?”

“My client is Mr. Hosgrave, director of a small but progressive bank. The bank is fully computerized and of course uses network communications extensively. The bank already uses RSA to protect its data and to digitally sign documents that are communicated. Now the bank wants to introduce some changes in its procedures; in particular, it needs to digitally sign some documents by two signatories.

The first signatory prepares the document, forms its signature, and passes the document to the second signatory.

The second signatory as a first step must verify that the document was really signed by the first signatory. She then incorporates her signature into the document’s signature so that the recipient, as well as any member of the public, may verify that the document was indeed signed by both signatories. In addition, only the second signatory has to be able to verify the document’s signature after the first step; that is, the recipient (or any member of the public) should be able to verify only the complete document with signatures of both signatories, but not the document in its intermediate form where only one signatory has signed it. Moreover, the bank would like to make use of its existing modules that support RSA-style digital signatures.”

“Hmm, I understand how RSA can be used to digitally sign documents by one signatory, Holmes. I guess you have solved the problem of Mr. Hosgrave by appropriate generalization of RSA digital signatures.”

“Exactly, Watson,” nodded Sherlock Holmes. “Originally, the RSA digital signature was formed by encrypting the document by the signatory’s private decryption key ‘d’, and the signature could be verified by anyone through its decryption using publicly known encryption key ‘e’. One can verify that the signature ‘S’ was formed by the person who knows ‘d’, which is supposed to be the only signatory. Now the problem of Mr. Hosgrave can be solved in the same way by slight generalization of the process, that is …” [1]

Finish the explanation.