  Using a generalization due to Lerch [M. Lerch, Sur un th\'{e}or\`{e}me de
Zolotarev. Bull. Intern. de l'Acad. Fran\c{c}ois Joseph 3 (1896), 34-37] of a
classical lemma of Zolotarev, employed in Zolotarev's proof of the law of
quadratic reciprocity, we determine necessary and sufficient conditions for the
difference of two Dedekind sums to be in $8\mathbb{Z}$. These yield new
necessary conditions for equality of two Dedekind sums. In addition, we resolve
a conjecture of Girstmair [Girstmair, Congruences mod 4 for the alternating sum
of the partial quotients, arXiv: 1501.00655].
