Assignment 2: COMP90043

Assignment 2: COMP90043
Due Date: September 23, 2016 Assignment is worth 7.5% of the total marks

1. Answer all the questions.
2. A Discussion forum thread Assignment 2 has been created on LMS. Any clarification o↵ered on this forum will be considered as a part of the specification of the Assignment.
3. The total number of points for this assignment is 30. This contributes to 7.5% of the total.
4. Answers must be submitted as a PDF file via the comp90043 Assignment 1 submission form on LMS by September 23, 2016. Late submissions will attract a penalty of 10% per day (or part thereof). Please ensure your name and login name are clearly presented.

Questions

1. (9 points) This question is concerning properties of Textbook RSA cryptosystem.

1. RSA in small parameters: Assume that Alice chooses two primes 196065871 and 102305491 to construct her RSA keys. Determine the smallest valid RSA public key and its corresponding private key for Alice. Show the detailed workings and not just the solution. You can use magma calculator from http://magma.maths.usyd.edu.au/magma/ If you use algorithms such as EEA or magma, show the workings.
2. This question is about the multiplicative property of the textbook RSA algorithm.

   We showed in the workshop that basic RSA is not secure for chosen ciphertext attack. The same idea can also be applied to create blind signatures. Assume that Alice’s public keys are [n, e] and her private key is d. Explain how Bob could create Alice’s signature on a message of choice m using the concept of blinding. Note that that Bob will not have access to private key d, but can request Alice to sign a blinded message.

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Your solution should also show the workings of the above blinding procedure using a random RSA key for Alice. Your answer here should include the following:

i. Your selection of two random primes, each of length at least 100 digits. ii. the public key e be smallest valid public key.

iii. Determine the private key d.
iv. A random message m of length at least 100 digits.

v. A blinded message mb.
vi. Signature of m through blinded process.

vii. Direct signature of m using the private key.
Note: the last two items should be identical. Any code written for the above

should be included as an appendix.

c. Assume that Alice has chosen a large RSA modulus n such that factorization is impossible with reasonable time and resources. She also then chooses a large random public exponent e < n for which the RSA problem is also not practical. However Bob decides to send a message to Alice by representing each alphabet character as an integer modulo 26 and then encrypting each number separately using Alice’s public address n,e. Is this a secure method? If not describe the most e cient attack against this method. Also, suggest a countermeasure to this attack.

2. (4 points) A variant of ElGamal cryptosystem over the prime field GF(q) given as follows. Assume the parameters as given in the ElGamal.pdf. Let yA = axA mod q, be the public address of Alice, where xA, 1 < xA < q 1, is Alice’s private key. Encryption function is defined as follows:

   E(M) = C1,C2,

   whereC1 =ak modq,wherekisarandominteger1kq 1,C2 =K M,where K = yAk mod q and is binary exclusive or function applied to binary representation of K and M.

   a. Describe the Decryption Function D(C1,C2) that Alice can use to recover the message.

   b. Show how the security of the encryption function is based on Computational Di e-Hellman (CDH) problem.

   CDH Problem: Let q be a prime number and a be a generator of the cyclic multiplica- tive group of modulo q. Given ax,ay, the CDH problem computes axy.

3. (4 points) A question on HASH functions.

   a. The textbook lists seven requirements of Hash functions. Out of these, one- way property, second image resistance and collision resistance are the three key requirements. Describe these three requirements.

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b. Now consider the following hash function. Here, messages are represented as

series of numbers from Zn, integers modulo n: M = integer t 1. The hash function is defined as follows:

{a1, a2, · · · at} for some

h(M) = ( where n is a number agreed in advance.

(ai)) mod n,

Does the above hash function satisfy any of the three key requirements mentioned in [a].? Explain your answer.

c. Now, consider a variation of the hash function for the messages represented as sequences of integers modulo n. The function is defined as follows:

Xt i=1

where n is a large number whose factorization is unknown. Does the modified hash function satisfy any of the three key requirements mentioned in [a].? Explain your answer.

4. (4 points) A question MAC and signatures.
   1. What is the main di↵erence between message authentication codes and digital signatures?
   2. Explain how Di e-Hellman(DH) key agreement protocol is vulnerable to man- in-the-middle attack. Is it possible to secure DH key agreement protocol against this attack by using each of the following primitives? If your answer is yes, sketch the method. If the answer is no, give reasons.

      i. Message Authentication Codes ii. Public Key Digital Signatures.

      iii. Hash functions.

5. (4 points) A question about standard ElGamal signature considered in lectures.
   1. (a)  What are the consequence of using same k for signing two di↵erent messages
   2. (b)  ElGamal signature algorithm given in the lectures involves computing the inverse of a number mod (q 1), where q is a prime number. Can you modify the signature algorithm avoiding the inverse computation while signing? Please explain your answer with a possible modification.
6. (5 points) This question is about Protocols. An alternative key distribution method suggested by a network vendor is illustrated in the figure below: (Fig. 14.18 of the textbook).

h(M) = (

(ai)2) mod n,

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Xt i=1

Figure 1: Fig. 14.18 of the Textbook

a. Describe the scheme.

b Compare this scheme to that of the scheme discussed in lectures (Fig 14.3 of the textbook-Given below).

c. Comment on the security of the new scheme.

4. What is the advantage of this scheme? Discuss the pros and cons.
5. Give an estimate of the memory requirements of KDC and the users with respect to storing key information.

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