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Exercises 2013-03-04

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

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Exercises 2013-03-04

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  1. Exercises2013-03-04 Information Security Course Eric Laermans – Tom Dhaene

  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? Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans

  3. Exercise 2 • Complexity • 2 efficient algorithms for factorising a number n in prime factors have a computational complexity of respectively Ln[1/2, 1] and Ln[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? Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans

  4. Exercise 3 • Complexity • Given: • to compute discrete logarithms in a subgroup of order q in the group {Zp*,} there are algorithms with a O(q1/2) complexityand algorithms with aLp[1/3, (64/9)1/3] complexity (which is the same as the complexity for solving discrete logaritmen in {Zp*,}) • Question: • if p ≈ 21024, 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 {Zp*,}? Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans

  5. Exercise 4 • Complexity • Given: • The BelgianeIDuses RSA with a 1024 bit modulus and PKCS#1 v1.5 • Usingstate-of-the-artalgorithms (GNFS) forfactorisation and state-of-the-arttechnology, it has been possiblesince December 2009 to factorise a 768 bit number • Onemayassumethatalgorithmswillnotimprove in the coming years, thatcomputation speed is the onlylimitation, and thatcomputation speed doublesevery 24 months • Question: • Howrisky is it to extend the validity of the BelgianeID to 10 years? Usecalculations to makeyour case. Information Security Vakgroep Informatietechnologie – IBCN – Eric Laermans

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