1. Exercises 2013-03-04 Information Security Course Eric Laermans – Tom Dhaene

2. Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 2 Exercise 1 Strong primes Given  p–1 has a sufficiently large prime factor r  p+1 has a sufficiently large prime factor s  r+1 has a sufficiently large prime factor t Question  How would you determine such a number?

3. Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 3 Exercise 2 Complexity 2 efficient algorithms for factorising a number n in prime factors have a computational complexity of respectively L n [1/2, 1] and L n [1/3, (64/9) 1/3 ] Questions:  which of both algorithms is asymptotically most efficient  which algorithm would you use to factorize a number of 128 binary digits (and for a number of 256, 512 of 1024 bits)?  from which size of number (decimal digits) would you switch to the other algorithm?

4. Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 4 Exercise 3 Complexity Given:  to compute discrete logarithms in a subgroup of order q in the group {Z p *,  } there are algorithms with a O(q 1/2 ) complexity and algorithms with a L p [1/3, (64/9) 1/3 ] complexity (which is the same as the complexity for solving discrete logaritmen in {Z p *,  }) Question:  if p ≈ 2 1024, which value of q must be chosen so that solving discrete logarithms in the subgroup is not significantly easier than the original discrete logarithm problem in {Z p *,  }?

5. Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans p. 5 Exercise 4 Complexity Given:  The Belgian eID uses RSA with a 1024 bit modulus and PKCS#1 v1.5  Using state-of-the-art algorithms (GNFS) for factorisation and state-of-the-art technology, it has been possible since December 2009 to factorise a 768 bit number  One may assume that algorithms will not improve in the coming years, that computation speed is the only limitation, and that computation speed doubles every 24 months Question:  How risky is it to extend the validity of the Belgian eID to 10 years? Use calculations to make your case.