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Applications of SAT Solvers to Cryptanalysis of Hash Functions

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### Applications of SAT Solvers to Cryptanalysis of Hash Functions

Ilya Mironov Lintao Zhang

Microsoft ResearchSilicon Valley Campus

Overview

- Crash course on hash functions
- Collision-finding attacks (Wang et al. ’05)
- Automation via SAT solvers

Hash functions

- H: {0,1}*→{0,1}n

Cryptographic hash functions

- Several important properties
- Collision-resistance
- x, y: H(x) = H(y)

Security level

hash output

128 bits

160 bits

256 bits

512 bits

- Insecure: 264operations
- Medium-term: 280
- Long-term (~20 years): 2128
- Paranoid: 2256

Short history of hash functions

- 1990 Ron Rivest: MD4 (128-bit output)
- 1992 Ron Rivest: MD5 (128-bit output)
- 1993 NIST: SHA (Secure Hash Algorithm, 160 bits)
- 1995 NIST: Oops! SHA1
- 2003 NIST: SHA-256,384,512

0

Compression function’s building block

512 bits = 16 32-bit words

M

w

a

b

rounds

0-15

rounds

16-31

rounds

31-48

c

d

128 bits

128 bits = 4 32-bit words

Finding a collision [Wang et al’05]

- Goal: Find M, M\'such that H(M) = H(M\')
- 1. Select message difference
- M\' = M +
- 2. Select differentialpath
- bi\' = bi + bi
- 3. Find sufficient conditions
- 4. Make them happen!

Differential path

- M
- (a0,b0,c0,d0)
- b1
- b2
- …
- b48

- M\'
- (a0,b0,c0,d0)
- b1\'
- b2\'
- …
- b48\'

- differential path
- b1\' = b1 + b1
- b2\' = b2 + b2
- …
- b48\' = b48 + b48

Sufficient conditions

- (ai,bi,ci,di) (di,(ai+fi(bi,ci,di)+wi+Ki)<<<si,bi,ci,) = (ai+1,bi+1,ci+1,di+1)

fi = MAJ and si = 3 and b2,0 = 0 and c2,0 = 0,

then for b2,3 = 0 it is sufficient that lsb(b1)=0 and lsb(c1)=0

Sufficient conditions [Wang et al.]

- MD4: 122
- MD5: first block ― 294; second block ― 309
- SHA0: 260

Probabilistic method

Conditions satisfied with probability 50%*:

MD4: < 8

MD5: first block ― 37; second block ― 30

SHA0: 42

SHA1: 70

* In the original papers (better attacks are currently known)

SAT Solvers!

- Goal: Find M, M\'such that H(M) = H(M\')
- 1. Select message difference
- M\' = M +
- 2. Select differential path
- bi\' = bi + bi
- 3. Find sufficient conditions
- 4. Message modifications

MD4

- 53K variables, 221K clauses. Success!
- SatELiteGTI < 500 sec

0xe1c08802 d0001321 f3fdc66f df600178 46b5c048 06c516c5 b632403a 88e2fdd5900f8005 3f936800 4b187044 64fad83a 01d79002 68f200a8 94ab2328 2449dd7d

collides with

0xe1c08802 50001321 63fdc66f df600178 46b5c048 06c516c5 b632403a 88e2fdd5900f8005 3f936800 4b187044 64fad83a 01d69002 68f200a8 94ab2328 2449dd7d

Where to truncate?

~100 hours per full solution

Collision in MD5

- 0x80000000 98163156 d685de69 e985b795 b4320c10 cd350030 c014ca29 850b7d6d0934ad59 4871afd0 aa480edf e4fc0320 7bb68ed1 3b505ddf 5e5d5df6 b539a48d
- fcb488ff adf40003 88d9fda4 d72a8fdc a887f4ca eec4f800 b75f8b20 7f1e9b519ab427cc 45c236f1 73f20086 e000005a 3b6550cc b6cc1c59 0fe9f71a a0403064
- collides with
- 0x80000000 98163156 d685de69 e985b795 34320c10 cd350030 c014ca29 850b7d6d0934ad59 4871afd0 aa480edf e4fc0320 7bb68ed1 3b505ddf de5d5df6 b539a48d
- fcb488ff adf40003 88d9fda4 d72a8fdc a887f4ca eec4f800 b75f8b20 7f1e9b519ab427cc 45c236f1 73f20086 dfff805a 3b6550cc b6cc1c59 0fe9f71a a0403064

Open problems

- Cryptographic:
- Break SHA-1
- Automate the entire attack
- Other primitives
- SAT-solving community:
- No truncation!
- SAT solvers optimized for cryptographic applications: XOR, multiplication, table look-ups, intuition

Conclusion

- First serious SAT-solver-aided cryptanalytic effort
- Several entries into SAT Race ’06
- New applications and challenges

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