Finding Differential Patterns for the Wang Attack

1. Finding Differential Patterns for the Wang Attack CITS – Cryptology and IT-Security Faculty of Mathematics Ruhr University Bochum Magnus Daum

2. M1: M2: 02dd31d1 c4eee6c5 069a3d69 5cf9af98 87b5ca2f ab7e4612 3e580440 897ffbb8 0634ad55 02b3f409 8388e483 5a417125 e8255108 9fc9cdf7 f2bd1dd9 5b3c3780 d11d0b96 9c7b41dc f497d8e4 d555655a c79a7335 0cfdebf0 66f12930 8fb109d1 797f2775 eb5cd530 baade822 5c15cc79 ddcb74ed 6dd3c55f d80a9bb1 e3a7cc35 02dd31d1 c4eee6c5 069a3d69 5cf9af98 87b5ca2f ab7e4612 3e580440 897ffbb8 0634ad55 02b3f409 8388e483 5a417125 e8255108 9fc9cdf7 f2bd1dd9 5b3c3780 313e82d8 5b8f3456 d4ac6dae c619c936 b4e253dd fd03da87 06633902 a0cd48d2 42339fe9 e87e570f 70b654ce 1e0da880 bc2198c6 9383a8b6 2b65f996 702af76f M1‘: M2‘: 02dd31d1 c4eee6c5 069a3d69 5cf9af98 07b5ca2f ab7e4612 3e580440 897ffbb8 0634ad55 02b3f409 8388e483 5a41f125 e8255108 9fc9cdf7 72bd1dd9 5b3c3780 d11d0b96 9c7b41dc f497d8e4 d555655a 479a7335 0cfdebf0 66f12930 8fb109d1 797f2775 eb5cd530 baade822 5c154c79 ddcb74ed 6dd3c55f 580a9bb1 e3a7cc35 02dd31d1 c4eee6c5 069a3d69 5cf9af98 07b5ca2f ab7e4612 3e580440 897ffbb8 0634ad55 02b3f409 8388e483 5a41f125 e8255108 9fc9cdf7 72bd1dd9 5b3c3780 313e82d8 5b8f3456 d4ac6dae c619c936 34e253dd fd03da87 06633902 a0cd48d2 42339fe9 e87e570f 70b654ce 1e0d2880 bc2198c6 9383a8b6 ab65f996 702af76f Motivation • Crypto ’04 (Wang et al.):actual collisions for various hash functions • E.g. for MD5: Daum - Finding Differential Patterns for the Wang Attack

3. 42e7b9ca 8726b6c4 24a51ab9 c1056b84 93fb9588 9fa6e965 ff920348 793f3b2c 0634ad41 03b4adff 7a844bdf 4f01374d cb8332db a86fd419 b3c665a7 30bf16f0 2e7cff6a 9b687357 15b83319 f5e7ab64 c566cfb9 0c79fee4 367d04ee aeb077cc 307f085d 88eb60b5 404d72b3 2d667867 676484d8 809bbd7d 4ff29e98 a30e2eb8 42e7b9ca 8726b6c4 24a51ab9 c1056b84 13fb9588 9fa6e965 ff920348 793f3b2c 0634ad41 03b4adff 7a844bdf 4f01b74d cb8332db a86fd419 33c665a7 30bf16f0 2e7cff6a 9b687357 15b83319 f5e7ab64 4566cfb9 0c79fee4 367d04ee aeb077cc 307f085d 88eb60b5 404d72b3 2d65f867 676484d8 809bbd7d cff29e98 a30e2eb8 Motivation • Lenstra/Wang/de Weger:colliding (w.r.t. MD5) X.509 certificates • Differing part: Daum - Finding Differential Patterns for the Wang Attack

4. Motivation • Other actual collisions published (Klima, Lucks/D.) show the same characteristics • Reason: Attack applies a special differential pattern with fixed input differences (M0,…,M15) = (0,0,0,0,231,…,§ 215,…,231,0) • Considered bytewise these are only differences in the most significant bit • May be a problem in certain applications,e.g. when trying to find colliding ASCII texts • Possible to use other input difference patterns? Daum - Finding Differential Patterns for the Wang Attack

5. Wang‘s Attack Daum - Finding Differential Patterns for the Wang Attack

6. Wang‘s Attack • Differential attack with modular differences(i.e. differences with respect to addition modulo 232) • Starts from a given/chosen message and modifies its bits to produce a collision • Two main parts: • Choosing the differential pattern (done by hand) • Single-Step and Multi-Step Modifications ? Daum - Finding Differential Patterns for the Wang Attack

7. Choosing the Differential Pattern • Not much is known about how Wang actually found this pattern used in all the implementations • Wang: „intuitively“ and „by hand“ • Some ideas can be reconstructed by looking at what is happening during the attack Daum - Finding Differential Patterns for the Wang Attack

8. W4= 231 W14= 231 W15=-215 W18=-215 W23= 231 W25= 231 W34=-215 W35= 231 W36= 0 W37= 231 W50= 231 W60= 231 W61=-215 Attack on MD5 • Construction of the pattern starts in last rounds • design of MD5 allows differential pattern for round 3+4 which leads to a useful near-collision • Input differences are chosen such that this difference propagation happens with high probability • Look for conditions on register values which make the difference propagation in first two rounds possible Daum - Finding Differential Patterns for the Wang Attack

9. Step Operation in MD5 Daum - Finding Differential Patterns for the Wang Attack

10. MD5 • Step operation: • Message expansion by roundwise permutations of the Mi (four rounds) Daum - Finding Differential Patterns for the Wang Attack

11. MD5 • Step operation: Kt,st: constants Wt: message words f: bitwise defined Boolean function Rt: new content of register changed in step t Daum - Finding Differential Patterns for the Wang Attack

12. Step Operation • Advantage of considering modular differences: • Most operations used in the step operation have a deterministic propagation of modular differences • Analyse the other parts: • Bit rotations • Bitwise defined functions Daum - Finding Differential Patterns for the Wang Attack

13. Difference Propagation Daum - Finding Differential Patterns for the Wang Attack

14. bitwise (XOR) differences: modular differences: Various Differences • differences usually low weight: ? uniquely determined signed bitwise differences: Daum - Finding Differential Patterns for the Wang Attack

15. For fixed +x=[k]: Various Differences modular differences signed bitwise differences • Special case: • Depends on actual value of x: • Can be generalized to other differences Daum - Finding Differential Patterns for the Wang Attack

16. ? Difference Propagation:Bitwise Functions • f is applied bitwise -> modular differences are not very useful • transform to signed bitwise diff. • propagation of signed bitwise differences can be analysed easily Daum - Finding Differential Patterns for the Wang Attack

17. Difference Propagation:Bitwise Functions Daum - Finding Differential Patterns for the Wang Attack

18. ? Difference Propagation:Bitwise Functions • f is applied bitwise -> modular differences are not very useful • transform to signed bitwise diff. • propagation of signed bitwise differences can be analysed easily -> possible values for together with corresponding conditions for each of the cases • corresponding modular differencesare uniquely determined Daum - Finding Differential Patterns for the Wang Attack

19. Bit Rotationand Modular Addition Daum - Finding Differential Patterns for the Wang Attack

20. Bit Rotationand Modular Addition A random, B fixed: Daum - Finding Differential Patterns for the Wang Attack

21. Difference Propagation:Bit Rotations • Register R with a fixed difference +R =[t] • A=R, B=+R: • Applying the Theorem described earlier yields for t<n-s: for t¸n-s: Daum - Finding Differential Patterns for the Wang Attack

22. Example: Analysis ofDifference Propagation • taken from first round of MD4 Daum - Finding Differential Patterns for the Wang Attack

23. Automated Searchingof such Differential Patterns Daum - Finding Differential Patterns for the Wang Attack

24. Bits 22,25: • Bit 31: • Bit 29: Degrees of Freedom • Choices when constructing such patterns: • (Input differences Wi) • Bitwise function: 1-3 choices per nonzero bit Daum - Finding Differential Patterns for the Wang Attack

25. Degrees of Freedom • Choices when constructing such patterns: • (Input differences Wi) • Bitwise function: 1-3 choices per nonzero bit • Bit rotation: 4 choices in general (but usually one dominant case) • Assumptions on bitwise differences (“expand“ differences) Daum - Finding Differential Patterns for the Wang Attack

26. Example: Analysis ofDifference Propagation • taken from first round of MD4 Daum - Finding Differential Patterns for the Wang Attack

27. Degrees of Freedom • Choices when constructing such patterns: • (Input differences Wi) • Bitwise function: 1-3 choices per nonzero bit • Bit rotation: 4 choices in general (but usually one dominant case) • Assumptions on bitwise differences (“expand“ differences) Daum - Finding Differential Patterns for the Wang Attack

28. Searchingfor Differential Patterns • Idea: build trees of difference patterns • Each vertex represents a possible state of differences, e.g. • Possible differences resulting after following step are computable • Leads to several new vertices -> pruning necessary • For the pruning use a cost function depending on the following properties: • Probability that this difference state is actually achieved • Weights of the differences • Distance from the root of the tree Daum - Finding Differential Patterns for the Wang Attack

29. Finding Useful Patterns • Additional constraints for useful patterns,e.g. start and end with zero differences • Trivial solution: take root with zero differences and add new vertices till a vertex with zero differences is found • Build two trees, one goind foreward, one going backwardFix a layer corresponding to some step and look for common vertices • Two trees as above, but stop some steps before fixed layer, find connection by solving additional equations • Has not been fully tested up to now Daum - Finding Differential Patterns for the Wang Attack

30. Conclusion • Some analysis of background of Wang‘s attack • Theoretical basis for analysing the propagation of modular differences • Ideas for automatically finding useful difference patterns Daum - Finding Differential Patterns for the Wang Attack

31. Thank you!Questions??? Daum - Finding Differential Patterns for the Wang Attack