Homework #2

J. H. Wang

Oct. 31, 2012

Homework #2

• Chap.3: – Review questions 3.4, 3.6– Problems 3.8, 3.20, 3.21

• Due: two weeks (extended to Nov. 21, 2012)

• Chap. 3– Review question 3.4: What properties must a hash

function have to be useful for message authentication?

– Review question 3.6: What are the principal ingredients of a public-key cryptosystem?

– Problem 3.8: Now consider the opposite problem: Use an encryption algorithm to construct a one-way hash function. Consider using RSA with a known key. Then process a message consisting of a sequence of blocks as follows: Encrypt the first block, XOR the result with the second block and encrypt again, and so on. Show that this scheme is not secure by solving the following problem:

Given a two-block message B1, B2, and its hash, we haveRSAH(B1, B2) = RSA(RSA(B1)B2)Given an arbitrary block C1, choose C2 so that RSAH(C1, C2)=RSAH(B1, B2), Thus, the hash function does not satisfy weak collision resistance.

– 3.20: Suppose Bob uses the RSA cryptosystem with a very large modulus n for which the factorization cannot be found in a reasonable amount of time. Suppose Alice sends a message to Bob by representing each alphabetic character as an integer between 0 and 25 (A->0, …, Z->25), and then encrypting each number separately using RSA with large e and large n. Is this method secure? If not, describe the most efficient attack against this encryption method.

– 3.21: Consider a Diffie-Hellman scheme with a common prime q=11 and a primitive root =2.a. If user A has public key YA=9, what is A’s private key XA?B. If user B has public key YB=3, what is the shared secret key K?

Homework Submission

• For hand-written exercises, please hand in your homework in class (paper version)

Thanks for Your Attention!