Hard and easy components of collision search in the Zémor- Tillich hash function: New attacks and reduced variants with equivalent security Christophe Petit UCL Crypto Group 04/22/09 | CRYP-201 Collisions for hash functions C. Petit, J.J. Quisquater, J.P. Tillich, G. Zémor

2 Cryptographic hash functions

3 Graph-based hash functions Most hash functions can be seen as While Zémor-Tillich is more like

4 Outline The Zémor-Tillich hash functionIntroductionNew attacksReduced variantsConclusion

The Zémor- Tillich hash function

6 Introduced at CRYPTO’94 [TZ94] Let irreducible over with and let Let For a message Output set has size

7 The Zémor-Tillich hash function Graph and group interpretations of main properties Representation problem : given a group and a set, find a product Balance problem : find

8 The Zémor-Tillich hash function Previous cryptanalysis: –Malleability –Invertibility for short messages [SGGB00] –Trapdoor attacks on [CP94,AK98,SGGB00] –Projection to finite fields [G96] –Subgroup attacks for composite [SGGB00] This paper: –Generic collision and preimage subgroup attacks in time (instead of and for birthday and exhaustive)

New attacks

10 Generic collision attack Sketch: 1.Find lower triangular matrices with meet-in-the-middle random search 2.Combine lower triangular matrices to have a lower diagonal matrix with ones in the diagonal by solving discrete logarithms 3.The resulting matrix has order 2 In each step, we use

11 Generic collision attack, 1st step If for some Then for some To solve the equation: –Compute and on various random messages –For each obtained, store the projective point ( ) –After messages, likely to be done

12 Generic collision attack, 2nd step Combine triangular matrices to get a matrix with ones in the diagonal Use Representation problem in finite fields: Given find Equivalent to Discrete Logarithm [BM97]… that is easy here !

13 Generic collision attack, 3d step For any,

14 Improvements Preimage attack: –A bit more technical, but same ideas –Same complexity Memory-free versions –Transform the birthday search in the first step into a cycle detection problem –Use standard techniques (distinguished points,…)

15 Hard and easy components Finding a message hashing to a triangular matrix is “nearly’’ as hard as Finding a message hashing to the identity Similarly: –Finding a message hashing to a diagonal matrix –Given some vector, finding a message hashing to a matrix with left / right eigenvector are nearly as hard as finding a message hashing to the identity

16 Hard and easy components The output of ZT is bits while its security is bits: how to extract the secure bits ?

Reduced variants

18 Vectorial Zémor-Tillich The output of ZT is bits while its security is bits: how to extract the secure bits ? Vectorial version –Outputs bits –For a given initial vector, returns If the initial vector is chosen randomly, just as secure as the original matrix version

19 Equivalence between vectorial and matrix versions Suppose there is an algorithm finding collision for the vectorial version… –Run it on a random We get where and are the ZT hash values of the colliding messages –Run it on We get –Repeat times

20 Equivalence between vectorial and matrix versions Key observations: – –« Homomorphism » To find a collision: –Let –Find such that

21 Equivalence between vectorial and matrix versions Colliding messages: – – where if The two messages collide to the value

22 Projective version The output of ZT is bits while its security is bits: how to extract the secure bits ? Projective version –Outputs bits –Returns if the vectorial version returns If the initial vector is chosen randomly, « nearly » as secure as the initial matrix version

23 « Quasi » equivalence between projective and vectorial versions Suppose there is an algorithm finding collision for the projective version… –Run it on to get and –After steps, find such that Complexity of last step –Hard asymptotically ( discrete logarithms problems + one subset sum problem) –Feasible for

Conclusion

25 Conclusion New generic attacks –Collision attack in time (instead of ) –Preimage attack in time (instead of ) New variants –Vectorial variant as secure –Projective variant « nearly » as secure –Best attack against projective variant is birthday search Zémor-Tillich is not broken – is too small –Still a very interesting design

Questions ?