Samiksha Jaiswal (Editor)

Collision problem

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The r-to-1 collision problem is an important theoretical problem in complexity theory, quantum computing, and computational mathematics. The collision problem most often refers to the 2-to-1 version: given n even and a function f : { 1 , … , n } → { 1 , … , n } , we are promised that f is either 1-to-1 or 2-to-1. We are only allowed to make queries about the value of f ( i ) for any i ∈ { 1 , … , n } . The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1.

Contents

Deterministic

Solving the 2-to-1 version deterministically requires n / 2 + 1 queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires n / r + 1 queries.

This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after n / r + 1 queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, n / r + 1 queries suffice. If we are unlucky, then the first n / r queries could return distinct answers, so n / r + 1 queries is also necessary.

Randomized

If we allow randomness, the problem is easier. By the birthday paradox, if we choose (distinct) queries at random, then with high probability we find a collision in any fixed 2-to-1 function after Θ ( n ) queries.

Quantum Solution

The BHT algorithm, which uses Grover's algorithm, solves this problem optimally by only making O ( n 1 / 3 ) queries to f.

References

Collision problem Wikipedia
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