Home > UP (complexity)

A common reformulation of NP states that a language is in NP if and only if a given answer can be verified by a deterministic machine in polynomial time. Similarly, a language is in UP if a given answer can be verified in polynomial time, and the verifier machine only accepts at most one answer for each problem instance. More formally, a language L belongs to UP if there exists a two input polynomial time algorithm A and a constant c such that

L = {x in {0,1}* | ∃! certificate, y with |y| = O(|x|^c) such that A(x,y) = 1}

Algorithm A verifies L in polynomial time.

Papadimitriou discusses UP in the context of cryptography, where it is shown that UP=P if and only if no one-way functions exist. ¹ Since UP lies between P and NP, this implies that finding a one-way function would suffice to show P≠NP.

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