Riemann–Hurwitz formula

In cryptography, MD5CRK was a distributed effort (similar to distributed.net) launched by Jean-Luc Cooke and his company, CertainKey Cryptosystems, to demonstrate that the MD5 message digest algorithm is insecure by finding a collision — two messages that produce the same MD5 hash. The project went live on March 1, 2004. The project ended on August 24, 2004 after researchers independently demonstrated a technique for generating collisions in MD5 using analytical methods by Xiaoyun Wang, Feng, Xuejia Lai, and Yu[1]. CertainKey awarded a 10,000 Canadian Dollar prize to Wang, Feng, Lai and Yu for their discovery.

Pollard's Rho collision search for a single path

A technique called Floyd's cycle-finding algorithm was used to try to find a collision for MD5. The algorithm can be described by analogy with a random walk. Using the principle that any function with a finite number of possible outputs placed in a feedback loop will cycle, one can use a relatively small amount of memory to store outputs with particular structures and use them as "markers" to better detect when a marker has been "passed" before. These markers are called distinguished points, the point where two inputs produce the same output is called a collision point. MD5CRK considered any point whose first 32 bits were zeroes to be a distinguished point.

Complexity

The expected time to find a collision is not equal to $2^{N}$ where $N$ is the number of bits in the digest output. It is in fact $2^{N}! \over {(2^{N}-K)!\times {2^{N}}^{K}}$, where $K$ is the number of function outputs collected.

For this project, the probability of success after $K$ MD5 computations can be approximated by: $1 \over {1-e^{K\times (1-K) \over 2^{N+1}}}$.

The expected number of computations required to produce a collision in the 128-bit MD5 message digest function is thus: ${1.17741\times 2^{N/2}}={1.17741\times 2^{64}}$

To give some perspective to this, using Virginia Tech's System X with a max performance of 12.25 Teraflops, it would take approximately ${2.17\times 10^{19}/12.25\times 10^{12}\approx 1,770,000}$ seconds or about 3 weeks. Or for commodity processors at 2 gigaflops it would take 6,000 machines approximately the same amount of time.

See also

• List of distributed computing projects
• Brute force attack

References

• Paul C. van Oorschot, Michael J. Wiener: Parallel Collision Search with Application to Hash Functions and Discrete Logarithms. ACM Conference on Computer and Communications Security 1994: pp210–218 Online version.