I used this explanation back in the day when I had to explain it to moderately-technically proficient people: Diffie-Hellman and a lot of the asymmetric crypto

I used this explanation back in the day when I had to explain it to moderately-technically proficient people: Diffie-Hellman and a lot of the asymmetric crypto ecosystem can be done on /any/ multiplicative cyclic groups (special sets associated by an operation that have certain properties, namely commutativity) obviously not all cyclic groups are equal, some happen to have one-way-ness backed by some fundamental cryptographic conjecture that it is hard to solve but easy to prove. the OG diffie-hellman used prime number modulo cyclic groups, but you can do that in any other cyclic group provided that it is secure. turns out when you make a cyclic group using ECC very carefully and using a crazy roundabout procedure (shown in the article), it has cryptographic security.