Question

3) Out of the following, name which kind of attack you carried out in part 1 and part2: a. ciphertext only, b. known plaintext, c. chosen plaintext, d. chosen ciphertext. Explain your answer Problem 3

10 points] A 4-bit long message was encrypted using one-time pad to yield a cipher-text “1010” Assuming the message space consists of all 4-bit long messages, what is the probability that the corresponding plaintext was “1001”? Explain your answer.

Problem 4

Assume we perform a known-plaintext attack on DES with one pair of plaintext and ciphertext. How many keys do we have to test in a worst-case scenario if we apply an exhaustive key search in a straightforward way? How many on average? Explain your answer.

Problem 5 [10*3=30pts] In this case, you are asked to attack an RSA encrypted message. Imagine being the attacker: You obtain the ciphertext C=1141 by eavesdropping on a certain connection. The public key is kpub=(n,e)=(2623,2111). a. Consider the encryption formula, C=Me mod n. All variables except the plaintext M are known. Why can’t you simply solve the equation for M? b. In order to determine the private key d, you have to calculate d=e-1 mod Ø(n).

There is an efficient expression for calculating Ø(n). Can we use this formula here? c. Calculate the plaintext M by computing the private key d through factoring n=p.q. Does this approach remain suitable for number with a length of 1024 bit or more?

Problem 6

1) Use the Euclidian algorithm to compute (Show all of the steps involved)

a. The GCD of 7469 and 2464.

b. The GCD of 198 and 243 (use the Extended form).

2) We also discussed the use of the Extended Euclidian algorithm to calculate modular inverses. Use this algorithm to compute the following values. Show all of the steps involved. a. 9570-1(mod 12935) b. 550-1 (mod 1769) 3) Let n be a product of two large primes p and q (i.e., n=pq). Assume that x, y, and g are relatively prime to n. 1) If x ≡ y (mod n), is gx ≡ gy (mod n)? Show why or why not. 2) If x ≡ y (mod Φ(n)) instead, is gx ≡ gy (mod n)? Show why or why not.

Answer 1

Problem 6:

1.a) GCD of 7469 and 2464

step1 : Divide the larger number by the smaller one: 7469/2464=3+77;

step2 : Divide the smaller number by the above operations remainder: 2464/77=32+0;

As the remainder is 0,we stop the process: 77 is the greatest common factor for the above numbers.

b)GCD of 198 and 243  

step1 : Divide the larger number by the smaller one: 243/198=1+45;

step2 : Divide the smaller number by the above operations remainder: 198/45=4+18;

step3 : Divide the remainder from the step1 by the remainder from step2: 45/18=2+9;

step4 : Divide the remainder from step2 by the remainder from step3: 18/9=2+0;

As the remainder is 0,we stop the process. 9 is the GCD of two numbers.