  Given a set $X$ of $n$ binary words of equal length $w$, the 3XOR problem
asks for three elements $a, b, c \in X$ such that $a \oplus b=c$, where $
\oplus$ denotes the bitwise XOR operation. The problem can be easily solved on
a word RAM with word length $w$ in time $O(n^2 \log{n})$. Using Han's fast
integer sorting algorithm (2002/2004) this can be reduced to $O(n^2
\log{\log{n}})$. With randomization or a sophisticated deterministic dictionary
construction, creating a hash table for $X$ with constant lookup time leads to
an algorithm with (expected) running time $O(n^2)$. At present, seemingly no
faster algorithms are known. We present a surprisingly simple deterministic,
quadratic time algorithm for 3XOR. Its core is a version of the Patricia trie
for $X$, which makes it possible to traverse the set $a \oplus X$ in ascending
order for arbitrary $a\in \{0, 1\}^{w}$ in linear time.
  Furthermore, we describe a randomized algorithm for 3XOR with expected
running time $O(n^2\cdot\min\{\log^3{w}/w, (\log\log{n})^2/\log^2 n\})$. The
algorithm transfers techniques to our setting that were used by Baran, Demaine,
and P\u{a}tra\c{s}cu (2005/2008) for solving the related int3SUM problem (the
same problem with integer addition in place of binary XOR) in expected time
$o(n^2)$. As suggested by Jafargholi and Viola (2016), linear hash functions
are employed. The latter authors also showed that assuming 3XOR needs expected
running time $n^{2-o(1)}$ one can prove conditional lower bounds for triangle
enumeration just as with 3SUM. We demonstrate that 3XOR can be reduced to other
problems as well, treating the examples offline SetDisjointness and offline
SetIntersection, which were studied for 3SUM by Kopelowitz, Pettie, and Porat
(2016).
