How Are Cryptographic Algorithms Broken???

Presented ByBhavana Tapde

June 19, 2006

Outline

Cryptographic Algorithms Cryptographic Hash Algorithms Applications of Hash Functions Properties of Hash Functions Case Study of MD5

Wang’s Method Klima’s Method Demo

Case Study of SHA-1 Conclusion

What is Cryptographic Algorithm?

Cryptography – process of scrambling information in a manner that is difficult to unscramble, and making scrambled information intelligible.

Cryptographic Algorithm – any algorithm written to achieve cryptography, and consequently confidentiality, integrity, and authentication.

Types of Cryptographic Algorithms Symmetric Key Algorithms – DES, Triple DES Asymmetric Key Algorithms – RSA Cryptographic Hash Algorithms – MD5, SHA-1

Cryptographic Hash Algorithm

Hashing Algorithm – a protocol for using the hash function, specifying how the message will be broken up and how the results from previous message blocks are chained together.

Hash Function is effectively computable. compresses information of arbitrary length to some

information of fixed length (“digital fingerprint”). possesses Avalanche (Ripple Effect) – when a input is

changed slightly, output changes significantly.

0101Hash function

How hashes are used?

Digitally Signed Documents

How hashes are used? (…cont)

Hashing Passwords

How hashes are used? (…cont)

Verifying File Integrity If you have just downloaded a large piece of software

from a website, how do you know that you've received it correctly and that it has not been tampered with?

The website publishes the hash values of its download bundles, and you can compare a published hash (e.g.MD5 sum) with checksum of downloaded file.

UtilitiesUnix OS – includes MD5 utilities in their distribution packages.

Windows – third party applications like FastSum (http://www.fastsum.com/)

When a Cryptographic Hash Function is Secured?

When it satisfies following three properties

Preimage-resistance: “Given V, find M such that h(M)= V” is infeasible.

2nd-preimage-resistance: “Given M, find M’ M such that h(M’)=h(M)” is

infeasible.

Collision-resistance: “Find M’ M such that h(M’)=h(M)” is infeasible.

Case Study of MD5

Description of MD5

MD5 (message digest algorithm) – developed at RSA Data Security, Inc.

Improved version of MD4.

Takes any message and outputs an 128-bit hash.

A message is padded so the length is multiple of 512.

Each 512-bit block is processed individually.

MD5

Description of MD5 (…cont)

The 512-bit block is divided into 16 32-bit words.

There are 4 32-bit registers A, B, C and D. These are initially loaded with IV0 and carry the hash values from one 512-bit block to the next.

It works in an iterative (chaining) process:

Hi+1 = f(Hi,Mi) IV0=H0

where Mi is a 512-bit block.

MD5

Hash Chaining

f H0=IV0

fixed

M1

H1

f

H2

… f

Hn = H

M2 Mn

Mi 512 bitsHi 128 bits

MD5

One Small Step

A,B,C,D – 4 registers. F – nonlinear function; there

are total 4 functions and one function is used in each round.

Each round has 16 steps (so, total 64 steps).

Mi – 32-bit block of the message input. (512/16=32)

Ki – 32-bit constant, different for each step.

s – Left bit rotation by s places; s varies for each operation.

– Addition modulo 232.

MD5

The Rounds and Non-Linear Functions

Mi=(w0,…,w15)

For fixed i, 4 consecutive steps will yieldai+4 =bi +((ai +Fi (bi,ci,di)+wi+ki)<<<si)

di+4=ai+((di+Fi+1 (ai,bi,ci)+wi+1+ki+1)<<<si+1)

ci+4=di+((ci+Fi+2 (di,ai,bi)+wi+2+ki+2)<<<si+2)

bi+4=ci+((bi+Fi+3 (ci,di,ai)+wi+3+ki+3)<<<si+3)

ki and si are predefined step dependant constants

Fi changes every 16 steps

Fi(X,Y,Z)=(X^Y)ν(~X^Z) 0 ≤ i ≤ 15

Fi(X,Y,Z)=(X^Z)ν(Y^~Z) 16 ≤ i ≤ 31

Fi(X,Y,Z)=X Y Z 32 ≤ i ≤ 47

Fi(X,Y,Z)=Y (X ν ~Z) 48 ≤ i ≤ 63

MD5

Finding Collisions

MD5 does 64 rounds of scrambling, so a brute force attack to find a collision requires at most 264 operations. Brute Force Attack – method of defeating cryptographic scheme by exhaustively working through all possible keys.

Xiaoyun Wang and her team – have an attack that requires 239 operations. This attack takes at most an hour and 5 minutes on a IBM P690 (supercomputer).

Vlastimil Klima and his team – have an attack that can find collisions on a Notebook PC within a minute.

MD5

Wang’s Method (August 2004)

Use of Differential Cryptanalysis: find a statistical correlation between key values and cipher transformations (typically Exclusive-OR of text pairs), then use sufficient defined plaintext to develop the key.

Find a particular M such that a particular H occurs with high probability.

In collision case, want H=0.

MD5 - Wang

Differentials

The attack uses two types of differentials

XOR differential: ΔX=X X’

Modular differential: ΔX=X-X’ mod 232

For M=(m0,…,mn-1) and M’=(m’0,…m’n-1) the full hash differential is for a message of length 512n bitsΔH0 -> ΔH1 ->…-> ΔHn= ΔHIf M and M’ are a collision pair ΔH=0

Round Differentials ΔHi -> ΔHi+1 can be split into round differentials as well ΔHi ΔR0 ΔR1 ΔR2 ΔR3 = ΔHi+1P0 P1 P2 P3

MD5 - Wang

Probability

Each of these differentials has a probabilistic relationship with the next.

Ideally, we’d like to be able to set up 2 messages where we can guarantee with probability 1 that ΔH=0.

This can be assured by modifying M so the first round differential will be what you want.

More modifications will improve the probability for the second, third and fourth round differentials.

MD5 - Wang

The Attack with Message Modification

Find M=(M0,M1 ) and M’=(M’0,M’1) ΔM0=M’0-M0=(0,0,0,0,231,0,0,0,0,0,0,215,0,0,231,0) ΔM1=M’1-M1=(0,0,0,0,231,0,0,0,0,0,0,-215,0,0,231,0)

M’0 differ in the 5th, 12th and 15th words only. Same for M1 and M’1. Message Modification Method – modify a message

word so that the first non-zero step differential (after 5th step) is anything you want with probability 1.

Modify multiple words to guarantee the round differentials with high probability.

MD5 - Wang

Results - Actual Collisions

M0 = 2dd31d1 c4eee6c5 69a3d69 5cf9af98 87b5ca2f ab7e4612 3e580440 897ffbb8 634ad55 2b3f409 8388e483 5a417125 e8255108 9fc9cdf7 f2bd1dd9 5b3c3780

M1 = d11d0b96 9c7b41dc f497d8e4 d555655a c79a7335 cfdebf0 66f12930 8fb109d1 797f2775 eb5cd530 baade822 5c15cc79 ddcb74ed 6dd3c55f d80a9bb1 e3a7cc35

M0’ = 2dd31d1 c4eee6c5 69a3d69 5cf9af98 7b5ca2f ab7e4612 3e580440 897ffbb8 634ad55 2b3f409 8388e483 5a41f125 e8255108 9fc9cdf7 72bd1dd9 5b3c3780

M1’ = d11d0b96 9c7b41dc f497d8e4 d555655a 479a7335 cfdebf0 66f12930 8fb109d1 797f2775 eb5cd530 baade822 5c154c79 ddcb74ed 6dd3c55f 580a9bb1 e3a7cc35

Hash: 9603161f a30f9dbf 9f65ffbc f41fc7ef

MD5 - Wang

Klima’s Method (March 2006)

“Tunnels in Hash Functions: MD5 Collisions Within a Minute”

Tunnel – a complex function written to find collision which takes into account individual bit of message instead of word.

Tunnels replaces multi-message modification method, and exponentially accelerate collision search.

Several tunnels are written in MD5 hash function.

Also uses ‘differential path’ – the effect of a single bit change tracked through the hash algorithm.

MD5 - Klima

Speed Comparison to Find MD5 Collisions

Software - http://cryptography.hyperlink.cz/2006/web_version_1.zip

MD5 - Klima

Machine Specification Avg. Time

Min. Time

Max. Time

Colli-sions

CPU Intel Pentium III (1 GHz),512MB RAM, Windows 2000

53.077 0.90 299.00 200

CPU Intel Pentium 4 (3 GHz),512MB RAM, Windows XP

17.542 0.20 93.30 200

Pentium M (1.7 GHz),512MB RAM, debian 2.6.14

29.104 1.03 147.54 102

AMD Athlon XP2000+(1.67 GHz),256MB RAM, Windows XP

29.733 0.30 165.70 1000

Time in seconds.

Demo of Pack3

Pack3 – software developed by one of the team members of Klima.

“Give me three files and I will give you another three with the same MD5 hash!”

The program serves as a toy example of how to get around the necessity of creating the second preimage.

Usage: pack3 file1 file2 file3 file4 file5 file6 Will create two packages – package1.exe package2.exe, having same MD5 sum. package1 extracts files 1-3. package2 extracts files 4-6. Pack3 is available at

http://cryptography.hyperlink.cz/MD5_collisions.html Verification tool used is FastSum. http://www.fastsum.com/download.php

MD5 - Klima

Screen Shots : FastSum Utility

C:\Demo\fastsum>fsum "C:\Demo\pack3\selfextract-md5_coll\FileA.txt“MD5 Checksum calculation and verification utility. [1.9.0.149] EN(C) 2003-2005 Kirill Zinov and Vitaly Rogotsevich. Web site: www.fastsum.comC:\Demo\pack3\selfextract-md5_coll\FileA.txt 12FABF28FF61D4AE9F7080F524CC3130Calculation summary: Processed 1 files in 0 folders with total size 0.04 Kb. Elapsed time: 00:00:00 Average speed: 0.00 Kb\Sec.

C:\Demo\fastsum>fsum "C:\Demo\pack3\selfextract-md5_coll\FileB.txt"MD5 Checksum calculation and verification utility. [1.9.0.149] EN(C) 2003-2005 Kirill Zinov and Vitaly Rogotsevich. Web site: www.fastsum.comC:\Demo\pack3\selfextract-md5_coll\FileB.txt 6DE787E2B6255B94B73DC39D32FC135CCalculation summary: Processed 1 files in 0 folders with total size 0.04 Kb. Elapsed time: 00:00:00 Average speed: 0.00 Kb\Sec.

Screen Shots : Pack3

C:\Demo\pack3\selfextract-md5_coll>pack3 file1.txt file2.txt file3.txt file4.txt file5.txt file6.txt

Screen Shots : Pack3 (…cont)

Verify results of Pack3 with FastSum

C:\Demo\fastsum>fsum "C:\Demo\pack3\selfextract-md5_coll\package1.exe"MD5 Checksum calculation and verification utility. [1.9.0.149] EN(C) 2003-2005 Kirill Zinov and Vitaly Rogotsevich. Web site: www.fastsum.com

C:\Demo\pack3\selfextract-md5_coll\package1.exe 0DAACC367624034BD6B4345E72241315

Calculation summary: Processed 1 files in 0 folders with total size 23.05 Kb. Elapsed time: 00:00:00 Average speed: 0.00 Kb\Sec.

C:\Demo\fastsum>fsum "C:\Demo\pack3\selfextract-md5_coll\package2.exe"MD5 Checksum calculation and verification utility. [1.9.0.149] EN(C) 2003-2005 Kirill Zinov and Vitaly Rogotsevich. Web site: www.fastsum.com

C:\Demo\pack3\selfextract-md5_coll\package2.exe 0DAACC367624034BD6B4345E72241315

Calculation summary: Processed 1 files in 0 folders with total size 23.05 Kb. Elapsed time: 00:00:00 Average speed: 23.05 Mb\Sec.

Case Study of SHA-1

Description of SHA-1

SHA-1 (Secure Hash Algorithm) – developed by NIST (National Institute of Standards and Technology).

Improved version of SHA-0.

Takes any message of length of less than 264 bits and outputs 160 bit hash.

A message is padded so the length is multiple of 512.

Each 512-bit block is processed individually.

SHA-1

Description of SHA-1 (…cont)

The 512-bit block is divided into 16 32 bit words.

There are 5 32-bit registers A, B, C, D and E. These are initially loaded with IV0 and carry the hash value from one 512-bit block to the next.

It works in an iterative process.

SHA-1

Hash Chaining

Expansion Function

512 bit blocks

Compression Function

2560 bitsInitialization vector(fixed)

160 bit hash

SHA-1

One Small Step

There are 4 rounds and each round has 20 steps (so, total 80 steps).

A,B,C,D,E – 5 registers. F – Non-linear function. Wt – 32-bit word derived

from current 512-bit input block.

t – Round number, 0 ≤ t ≤ 79.

Kt – 32-bit constant, different for each step.

s – left bit rotation by s places; s varies for each step.

– Addition modulo 232.

SHA-1

SHA-1 Functions

Expansion Function: Wi = (Wi-3 Wi-8 Wi-14 Wi-16) << 1 16 ≤i ≤79

F Functions:Ft(B,C,D)=(B^C)v(~B^D) 0 ≤ t ≤ 19

Ft(B,C,D)=B C D 20 ≤ t ≤ 39

Ft(B,C,D)=(B^C)v(B^D)v(C^D) 40 ≤ t ≤ 59

Ft(B,C,D)= B C D 60 ≤ t ≤ 79

SHA-1

Finding Collisions

SHA-1 does 80 rounds of scrambling, so a brute force attack to find a collision requires at most 280 operations.

Xiaoyun Wang and her team – have an attack that requires 269 operations (i.e. 2000 times faster than 280 brute force).

SHA-1

Wang’s Method (February 2005)

Wang found following short-comings in SHA-1

The message expansion does not offer enough avalanche effect in terms of spreading the input differences.

The structure of all the step functions is unexpectedly weak. Because of the simple step operation, the certain step properties of some Boolean functions combined with the carry effect actually facilitate, rather than prevent, differential attack.

SHA-1

Final Attack

Wang’s attack on SHA-1 consisted following techniques: Message Modification Method Differential Attack Local Collision Attack Use of Differential Path (effect of a single bit

change tracked through the hash algorithm) and Disturbance Vector (set of bit changes to the hash input designed to create a set of changes to the hash sequence).

SHA-1

Differential Attack

Differential Cryptanalysis : the study of how differences in an input can affect the resultant difference at the output.

Fundamental Observations made by the team:A change in a bit j of word Wi can be corrected by complementary changes in the following bits – bit (j+6) mod 32 of word Wi+1

bit j of word Wi+2

bit (j+30) mod 32 of word Wi+3

bit (j+30) mod 32 of word Wi+4

bit (j+30) mod 32 of word Wi+5

SHA-1

Local Collision Attack

Local Collision – a collision within a single message (or within a few steps of hash function), including intermediate hash results.

SHA-1 has a 6-step local collision that can start at any step.

SHA-1

Local Collision Attack (…cont)

Δm Δa Δb Δc Δd Δe

i 0000 0001 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000

i+1 0000 0020 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000

i+2 0000 0001 0000 0000 0000 0000 4000 0000 0000 0000 0000 0000

i+3 4000 0000 0000 0000 0000 0000 0000 0000 4000 0000 0000 0000

i+4 4000 0000 0000 0000 0000 0000 0000 0000 0000 0000 4000 0000

i+5 4000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

Collision

SHA-1

Conclusion

MD5 is breakable – 239 complexity SHA-1 is breakable – 269 complexity So, it’s time to switch from MD5 and SHA-1. What next? Longer variants published by NIST

SHA-224 SHA-256 SHA-384 SHA-512

Because “Attacks always get better; they never get worse…”

References

Xiaoyun Wang et. al. “Finding Collisions in the Full SHA-1”, http://www.infosec.sdu.edu.cn/paper/sha1-crypto-auth-new-2-yao.pdf

Xiaoyun Wang et. al. “Collisions for Hash Functions MD4, MD5, HAVAL-128 and RIPEMD”, http://eprint.iacr.org/2004/199.pdf

Vlastimil Klima “Tunnels in Hash Functions: MD5 Collisions Within a Minute” http://eprint.iacr.org/2006/105.pdf

Steve Friedl , “An Illustrated Guide to Cryptographic Hashes ”, http://unixwiz.net/techtips/iguide-crypto-hashes.html#digestonly

Hashing Function Lounge http://paginas.terra.com.br/informatica/paulobarreto/hflounge.html

http://en.wikipedia.org/wiki/SHA1 http://en.wikipedia.org/wiki/MD5

Thank You!

Questions? What is she talking about?

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