I understand the needs that lead to the development of cryptography and I am quite familiar with the uses we make of the cryptographic tools.

But, as a programmer, I am conditioned to see them as "black boxes" with specific properties. To me, SHA-X (with X being 1, 2 or 3) is some dark magic, even though I understand why I need it and use it.

That said, I am eager to find some literature to light this up.
From what I have read so far, I have seen that the common mathematical demonstrations consist of games and the evaluation of the winning advantage of an enlightened attacker over a player that would just randomly take decisions. This is exactly the kind of things I am looking for : what mathematical background lead to this construction for this hash function. Learning through an example can be worthy, but the more global this maths background is, the better.

To say it in other words, how does a cryptographer prove the essential properties of his design ? In the case of cryptographic hash functions, these properties are one-wayness, collision resistance and preimage resistances.

What are the logical steps, starting from the required properties, that lead to the specification of a hash function ? How are these properties translated into mathematical definitions ?

Note : I think I needed to ask that last question because for a cryptographic hash function, I feel there is the need to define some thresholds somewhere.


Question: Clearest water beach close to Athens (Greece)
Answer: The island of Euboea (Greek: Εύβοια, Evvia) driving-distance from Athens. Indeed, its main town Chalcis (or Chalkida) (Greek: Χαλκίδα, Halkida) is approximately 1 hour away from Athens, according to google maps. The island is scattered with beaches, some of which are quite amazing. For more inspiration, here is a top 5 list from the Greek Reporter, and here is the Visit Greece page on Chalcis. Below is a sample (marketing) image of a beach in Evia, courtesy of the Greek Reporter: