hashing

Wrap the number line into a circle and addition and multiplication survive intact while division mostly does not. This lesson builds congruences, shows why you can reduce early to avoid overflow, works through Euclid's algorithm and modular inverses, and explains how a million-digit exponent becomes twenty multiplications.

Combinatorics answers how many arrangements exist without producing any of them, which is what makes password strength, hash collisions and search-space size computable at all. This lesson builds the product rule, permutations, combinations, inclusion-exclusion and the pigeonhole principle, then applies them to problems where intuition is reliably wrong.

A Bloom filter answers set membership using a bit array and a handful of hash functions, with no items stored anywhere. This lesson builds it, derives the sizing formula that trades memory against false positives, explains exactly why deletion is impossible, and covers the variants that buy it back.

A hash table promises constant-time lookup, and the promise holds only because of how it handles collisions. This lesson builds one from the array up: hashing, chaining versus open addressing, why load factor is the tuning dial, and how modern tables scan sixteen slots at once.

Hash functions are not all equal and MACs are not just hashes with a secret. Learn the three resistance properties, why length-extension attacks break naive HMAC constructions, how KDFs differ from hashes, and why bcrypt/scrypt/Argon2 exist for passwords.