Exercises Information Security Course Eric Laermans – Tom Dhaene

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 2 Exercise 1 (1) RSA PKCS#1 v1.5 “Million Message Attack” (MMA): illustration of principle Given  C (= M e mod n), n and e  M formatted according to PKCS#1 v1.5 (M = 00||02||PS||00||D)  error message from victim if decryption of C’ fails because of erroneous formatting Question  find a strategy to recover M  hint: think of the multiplicative properties of RSA

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 3 Exercise 1 (2) RSA-formatting: MMA illustration using more limited formatting Given  formatting: M = 0010xxxx  n = 187; e = 3; C = 81 Question  find M  hint: 32 ≤ M ≤ 47

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 4 Exercise 2 ElGamal Given  in ElGamal-encryption or –signature, and also in DSA, a unique and secret random value k is used Question  what happens if an attacker knows k?  what are the consequences if the random value k is reused: –in ElGamal-encryption? –in ElGamal-signature? –in DSA?

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 5 Exercise 3 ElGamal Given:  RSA-signatures exhibit the issue of “existential forgery”, i.e. given some messages with their corresponding RSA-signatures, it is possible to generate new signed messages using RSA’s multiplicative properties, without requiring knowledge about the private key Question:  is there a similar problem with ElGamal-signatures?

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 6 Exercise 4 ElGamal Given:  in ElGamal-encryption of –signature, and in DSA, a unique and secret random value k is used Question:  how could the owner of the private key used in the digital signature add hidden information without the person receiving the signature noticing? –such a technique is called a “subliminal channel”  can you find a way to use (a small part) of this subliminal channel without needing to use the private key for this purpose? (harder)

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 7 Exercise 5 Hash functions Given  a hash function with a hash value of n bits, e.g. 128 bits  a limited storage capacity (N 1 hash values), e.g. 1 TB –you may assume N 1 ≪ 2 n/2 Question:  how many hash computations are required to find two messages with identical hash values with a given probability P (e.g. 95%)? –compute this with the given values –suppose a modern PC can compute 10 million hash values per second, how much time would be required?

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 8 Exercise 6 Hash functions Given:  a competition at XKCD to generate a hash value with as many bits as possible corresponding to the bits of a givcen hash value (Skein ) –Skein is 1 of the 5 finalists for SHA-3, used here with a 1024 bit hash value and 1024 bit internal state –winner was CMU, with only 384 wrong bits on 1024 (i.e. 640 corresponding bits)CMU Question:  compute if this result is an indication of some weakness in the weak collision resistance for the hash algorithm used –i.e. compute how many hash values should typically be generated to obtain a hash value with at most 384 bits (on 1024) differing from the bits of the original hash values, assuming that hash values are uniformly randomly distributed –does this seem a feasible number?

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 9 Exercise 6 Hash functions Hints:  N C k = N!/(k! (N–k)!) –number of combinations of k elementes from a group of N  for k sufficiently small w.r.t. N –∑(j:0..k. N C j ) ≈ (N–k–1)/(N–2*k–1)* N C k  for k more in the neighbourhood of N/2 –∑(j:0..k. N C j ) ≈ CDF_Norm(N/2,sqrt(N)/2) (k+½) »central limit theorem –CDF_Norm(mean, stdev) (x) = Φ ((x–mean)/stdev) –Φ (x) = ½ + ½ *erf(x/sqrt(2)) –erf(x) ≈ 1–(a 1 *t+ a 2 *t² + a 3 *t³)*exp(-x²) »with t=1/(1+p*x) »with p=0,47047 and a 1 =0, and a 2 =-0, and a 3 =0,  best approximation is minimum of both

Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 10 Exercise 7 Hash functions Given  11,4 million 1024 bit RSA-keys, of which the prime factors were generated randomly Question  estimate the probability that at least two keys in this set have a common prime factor  Note –according to however keys shared a prime factor with another RSA- keyhttp://eprint.iacr.org/2012/064.pdf