Improved Generic Algorithms for 3-Collisions

An r-collision for a function is a set of r distinct inputs with identical outputs. Actually finding r-collisions for a random map over a finite set of cardinality N requires at least about N(r−1)/r units of time on a sequential machine. For r=2, memoryless and well-parallelisable algorithms are known. The current paper describes memory-efficient and parallelisable algorithms for r ≥ 3. The main results are: (1) A sequential algorithm for 3collisions, roughly using memory Nα and time N1−α for α ≤ 1/3. I.e., given N1/3 units of storage, on can find 3-collisions in time N2/3 . Note that there is a time-memory tradeoff which allows to reduce the memory consumption. (2) A parallelisation of this algorithm using N1/3 processors running in time N1/3 . Each single processor only needs a constant amount of memory. (3) An generalisation of this second approach to r-collisions for r ≥ 3: given Ns parallel processors, on can generate r-collisions roughly in time N((r−1)/r)−s ...

• Type: Conference
• Year: 2009
• Where: ASIACRYPT
• Authors: Antoine Joux, Stefan Lucks
• Keywords: Algorithm, ASIACRYPT 2009, Cryptology, Distinct Inputs, Identical Outputs