  In this paper, we introduce a novel class of embedded exponential-type
low-regularity integrators (ELRIs) for solving the KdV equation and establish
their optimal convergence results under rough initial data. The schemes are
explicit and efficient to implement. By rigorous error analysis, we first show
that the ELRI scheme provides the first order accuracy in $H^\gamma$ for
initial data in $H^{\gamma+1}$ for $\gamma>\frac12$. Moreover, by adding two
more correction terms to the first order scheme, we show a second order ELRI
that provides the second order accuracy in $H^\gamma$ for initial data in
$H^{\gamma+3}$ for $\gamma\ge0$. The proposed ELRIs further reduce the
regularity requirement of existing methods so far for optimal convergence. The
theoretical results are confirmed by numerical experiments, and comparisons
with existing methods illustrate the efficiency of the new methods.
