Shamir's Secret Sharing is probably my favourite algorithm for showing the deep relationship between algebra and cryptography. The Fundamental Theorem of Algebra is a relatively straightforward result to show to a layperson, and Shamir's is basically as direct an application of it as you could want.
Then the part where using integers leaks information, and how to attack that, gets a bit hairier but is still a moderately accessible way to show cryptanalysis in practice. The fix using finite fields is then probably a step too far for most laypeople, and can be used as a lesson in how the fundamentals can be simple, but the devil is in the details.
Then the part where using integers leaks information, and how to attack that, gets a bit hairier but is still a moderately accessible way to show cryptanalysis in practice. The fix using finite fields is then probably a step too far for most laypeople, and can be used as a lesson in how the fundamentals can be simple, but the devil is in the details.