Establishing code equivalences between index coding and network coding provides important insights for code design. Previous works showed an equivalence relation between any index-coding instance and a network-coding instance, for which a code for one instance can be translated to a code for the other instance with the same decoding-error performance. The equivalence also showed a surprising result that any network-coding instance can be mapped to an index-coding instance with a properly designed code translation. In this article, we extend the existing equivalence (instance map and code translation) to one between secure index coding and secure network coding, where eavesdroppers are present in the network. In the secure setting, any code construction needs to guarantee security constraints in addition to decoding-error performance. A rate equivalence between these two problems is also established.