Over the years, new discoveries have weakened some of the mathematical primitives used in cryptography, like the broken Knapsack cryptosystem, Shor's post-quantum algorithm, etc. What we believe to be hard problems today, can become simpler tomorrow. We need to trust the judgement of mathematicians that certain algebraic problems remain hard to solve in the future, and no secret agency has found a shortcut in the meantime.
That is why I love the beauty of hash-based digital signatures (e.g. Lamport, Merkle, XMSS, Sphincs+). It is all just basic logic and bookkeeping. It is not relying on math puzzles, and only requires that the hash function is not broken. But you could even mitigate that by combining multiple different hash functions if you are truely paranoid. Even without any math background you can simply understand why hash-based signatures are secure.
Next to digital signatures, you could also create a simple hash-based symmetric cipher. For example by hashing a shared secret with a counter and XOR-ing the plain text.
I was wondering whether public-key cryptography can also be based on hashes. I was unable to find anything in literature, so my guess is that you will unfortunately need some sort of algebraic problem to construct a parameterized trap-door function (as opposed to a fixed hash function like SHA256).
Any suggestions or pointers to literature?
