  We study integrals appearing in intermediate steps of one-loop open-string
amplitudes, with multiple unintegrated punctures on the $A$-cycle of a torus.
We construct a vector of such integrals which closes after taking a total
differential with respect to the $N$ unintegrated punctures and the modular
parameter $\tau$. These integrals are found to satisfy the elliptic
Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power
series in $\alpha$' -- the string length squared -- in terms of elliptic
multiple polylogarithms (eMPLs). In the $N$-puncture case, the KZB equation
reveals a representation of $B_{1,N}$, the braid group of $N$ strands on a
torus, acting on its solutions. We write the simplest of these braid group
elements -- the braiding one puncture around another -- and obtain generating
functions of analytic continuations of eMPLs. The KZB equations in the
so-called universal case is written in terms of the genus-one Drinfeld-Kohno
algebra $\mathfrak{t}_{1,N} \rtimes \mathfrak{d}$, a graded algebra. Our
construction determines matrix representations of various dimensions for
several generators of this algebra which respect its grading up to commuting
terms.
