  We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice $Z^d$.
The method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high $d$, and in fact agree with the
first four terms of the $1/d$ expansion for the connective constant. The bounds
are the best to date for dimensions $d \geq 3$, but do not produce good results
in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within
2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.
