The Siddham script is derived from an older form of the North India Devanagari script, like the Tibetan script.
Many Asian scripts are derived from Indian scripts, like Javanese, Khmer, Burmese, Thai etc.

From encyclpedia4you:
The writing of mantras, and copying of Sutras using the Siddham script is still practiced in Shingon Buddhism, in Japan but has died out in other places. It was Kukai that introduced the Siddham script to Japan when he returned from studying Sanskrit, and vajrayana Buddhism in China, in 806.