Applications of SAT Solvers to Cryptanalysis of Hash Functions Ilya Mironov Lintao Zhang Microsoft Research Silicon Valley Campus.

Applications of SAT Solvers to Cryptanalysis of Hash Functions

Ilya Mironov Lintao Zhang

Microsoft ResearchSilicon Valley Campus

Overview1. Crash course on hash functions

2. Collision-finding attacks (Wang et al. ’05)

3. 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)

Birthday paradoxFinding collision: ~|S| = 2n/2

outputH

S

Security levelInsecure: 264

operations

Medium-term: 280

Long-term (~20 years): 2128

Paranoid: 2256

hash output128 bits

160 bits

256 bits

512 bits

Short history of hash functions1990 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

1990 MD4

1991

1992 MD5

1993 SHA0

1994

1995 SHA1

1996

1997

1998

1999

2000

2001

2002

2003 SHA-256,384,512

2004

2005

2006

MD4 is broken

theoretical attack on SHA0

MD5, SHA0 broken, theoretical attack on SHA1

SHA1MD5

MD4

SHA1

MD4 and MD5’s structure- Basic building block:

512 bits

128 bits 128 bits48 rounds

compression function

Compression function’s building block

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

rounds 0-15

abcd

0 1 2 3 4 5 6 7 8 9 101112131415

M

0 4 8 12 1 5 9 13 2 6 1014 3 7 1115

0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

rounds 16-31

rounds 31-48

512 bits = 16 32-bit words

128 bits = 4 32-bit words 128 bits

w

One round

ai

bi

ci

di

<<<si ai+1

bi+1

ci+1

di+1

+ + +

fi

wi Ki

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 differential path

bi' = bi + bi

3. Find sufficient conditions

4. Make them happen!

Disturbance vector

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

rounds 0-15

abcd

0 1 2 3 4 5 6 7 8 9 101112131415

M

0 4 8 12 1 5 9 13 2 6 1014 3 7 1115

0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

rounds 16-31

rounds 31-48

Differential pathM

(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

ai

bi

ci

di

<<<si ai+1

bi+1

ci+1

di+1

+ + +

fi

wi Ki

Sufficient conditions [Wang et al.]MD4: 122

MD5: first block ― 294; second block ― 309

SHA0: 260

Message modification technique

rounds 0-15

abcd

0 1 2 3 4 5 6 7 8 9 101112131415

0 4 8 12 1 5 9 13 2 6 1014 3 7 1115

0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

rounds 16-31

rounds 31-48

0 4 8 12 1 5 9 13 2 6 1014 3 7 1115

0 1 2 3 4 5 6 7 8 9 1011121314150 1 2 3 4 5 6 7 8 9 101112131415

Probabilistic method

0 1 2 3 4 5 6 7 8 9 101112131415 0 4 8 12 1 5 9 13 2 610

14 3 7 1115 0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

Conditions satisfied with probability 50%*:MD4: < 8MD5: first block ― 37; second block ― 30SHA0: 42SHA1: 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

MD453K variables, 221K clauses. Success!

SatELiteGTI < 500 sec

0xe1c08802 d0001321 f3fdc66f df600178 46b5c048 06c516c5 b632403a 88e2fdd5 900f8005 3f936800 4b187044 64fad83a 01d79002 68f200a8 94ab2328 2449dd7d

collides with

0xe1c08802 50001321 63fdc66f df600178 46b5c048 06c516c5 b632403a 88e2fdd5 900f8005 3f936800 4b187044 64fad83a 01d69002 68f200a8 94ab2328 2449dd7d

MD5Hmm… Truncated MD5?

truncated MD5

CNF formula

SAT solver

filter

solution

Probabilistic method

all messages

reduced-round solutions

full solutions

0

20000

40000

60000

80000

100000

120000

140000

160000

16 18 20 22 24 26 28 30 32 34 36 38 40 42

rounds

yield

Where to truncate?~100 hours per full solution

Collision in MD50x80000000 98163156 d685de69 e985b795 b4320c10 cd350030 c014ca29 850b7d6d 0934ad59 4871afd0 aa480edf e4fc0320 7bb68ed1 3b505ddf 5e5d5df6 b539a48d

fcb488ff adf40003 88d9fda4 d72a8fdc a887f4ca eec4f800 b75f8b20 7f1e9b51 9ab427cc 45c236f1 73f20086 e000005a 3b6550cc b6cc1c59 0fe9f71a a0403064

collides with0x80000000 98163156 d685de69 e985b795 34320c10 cd350030 c014ca29 850b7d6d 0934ad59 4871afd0 aa480edf e4fc0320 7bb68ed1 3b505ddf de5d5df6 b539a48d

fcb488ff adf40003 88d9fda4 d72a8fdc a887f4ca eec4f800 b75f8b20 7f1e9b51 9ab427cc 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