  In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists
of $n$ edges such that there exists a cyclic ordering of the vertices where the
edges correspond to consecutive segments of $r$ vertices. We provide a first
deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton
cycles in random $r$-uniform hypergraphs with edge probability at least $C
\log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random
Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton
cycles exists already for $p=\omega(1/n)$ for $r=3$ and $p=(e + o(1))/n$ for
$r\ge 4$ using a second moment argument. Moreover our algorithm is superior to
previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures
& Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c
[arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a
randomised polynomial time algorithm working for edge probabilities $p\ge
n^{-1+\varepsilon}$, while the algorithm of Nenadov and \v{S}kori\'c is a
randomised quasipolynomial time algorithm working for edge probabilities $p\ge
C\log^8n/n$.
