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REPORTS > KEYWORD > COLLISION PROBLEM:
Reports tagged with collision problem:
TR11-001 | 2nd January 2011
Scott Aaronson

#### Impossibility of Succinct Quantum Proofs for Collision-Freeness

We show that any quantum algorithm to decide whether a function $f:\left[n\right] \rightarrow\left[ n\right]$ is a permutation or far from a permutation\ must make $\Omega\left( n^{1/3}/w\right)$ queries to $f$, even if the algorithm is given a $w$-qubit quantum witness in support of $f$ being a permutation. This implies ... more >>>

TR15-041 | 25th March 2015
Mark Bun, Justin Thaler

#### Dual Polynomials for Collision and Element Distinctness

The approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\ell_\infty$ norm. In an influential result, Aaronson and Shi (J. ACM 2004) proved tight $\tilde{\Omega}(n^{1/3})$ and $\tilde{\Omega}(n^{2/3})$ lower bounds on ... more >>>

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