What cryptography is suitable to motivate High School students to learn math?
15-18 is an age where kids are looking for independence and question authority of adults.  Cryptography is often horribly implemented, and can serve as a teaching tool of where professionals have gone wrong.

So pick some real world blunders, and do some analysis on those blunders.   There's enough cryptographic blunders that are easy enough for HS students to understand to fill an entire course with.  

A good recent example is the Adobe password leak.  A good writeup is available here  You could very easily find the leaked data, create a small, anonymized subset of it, and repeat the same analysis done in the article on that subset.  

It's a great example because it uses something kids use every day that's highly mathematical (passwords on websites), but they're unaware of.  It also shows how adults screwed up, which every kid loves, but also how to do things right.  The mathematics is simple enough that you don't need advanced degrees in mathematics, and you don't really need to go into depth on how the crypto works.