Cryptography using key

Problem:

(a) Suppose that there are n people who want to communicate with each other securely. How many keys are needed when a symmetric key cryptosystem is used? How about public key cryptosystem? Explain.

(b) Digital signatures cannot be done using symmetric key cryptography. Explain why.

(c) Suppose we produce a “digest” of a message by simply adding up the words (e.g., treating each character as a 8-bit number). What is the problem of this approach?


Follow-up:

(a) Using symmetric keys, the number of keys required for pair-wise communication is nC2 = n (n –1)/2. Using public key cryptosystem, the number 2 of keys require is just 2n (one public key and one private for each user).

(b) One of the most important requirements in digital signatures is non-repudiation—the signer cannot deny that a signature is produced by him/her. Thus, we need a “secret” that is bound to and only to each user. Using symmetric key cryptosystem, a secret key is known to at least two users and, therefore, a digital signature produced with a secret key cannot be bound to a unique user.


(c) The problem is that a message can be easily transformed into an entirely different message with the same digest, by just re-arranging the characters or words. For example, “car” and “arc” have the same digest.